Recent zbMATH articles in MSC 35https://zbmath.org/atom/cc/352024-09-27T17:47:02.548271ZUnknown authorWerkzeugApplied mathematics and computational intelligence. ICAMCI-2020, Tripura, India, December 23--24, 2020. Selected papers based on the presentations at the 1st international conferencehttps://zbmath.org/1541.000672024-09-27T17:47:02.548271ZPublisher's description: This book contains select papers presented at the International Conference on Applied Mathematics and Computational Intelligence (ICAMCI-2020), held at the National Institute of Technology Agartala, Tripura, India, from 19--20 March 2020. It discusses the most recent breakthroughs in intelligent techniques such as fuzzy logic, neural networks, optimization algorithms, and their application in the development of intelligent information systems by using applied mathematics. The book also explains how these systems will be used in domains such as intelligent control and robotics, pattern recognition, medical diagnosis, time series prediction, and complicated problems in optimization.
The book publishes new developments and advances in various areas of type-3 fuzzy, intuitionistic fuzzy, computational mathematics, block chain, creak analysis, supply chain, soft computing, fuzzy systems, hybrid intelligent systems, thermos-elasticity, etc. The book is targeted to researchers, scientists, professors, and students of mathematics, computer science, applied science and engineering, interested in the theory and applications of intelligent systems in real-world applications. It provides young researchers and students with new directions for their future study by exchanging fresh thoughts and finding new problems.
The articles of mathematical interest will be reviewed individually.Computational and mathematical approach for recent problems in mathematical scienceshttps://zbmath.org/1541.000772024-09-27T17:47:02.548271ZFrom the text: This special issue on \textit{Computational and mathematical approach for recent problems in mathematical sciences} is the compilation of the short listed papers of 4th International Conference on Mathematical Techniques in Engineering Applications (ICMTEA2020) held during 4--5 December 2020 in Graphic Era (Deemed to be University), Dehradun, Uttarakhand, India.The CD\(_p\) curvature condition on a graphhttps://zbmath.org/1541.051162024-09-27T17:47:02.548271Z"Xu, Xi"https://zbmath.org/authors/?q=ai:xu.xi"Shen, Wang"https://zbmath.org/authors/?q=ai:shen.wang"Wang, Linfeng"https://zbmath.org/authors/?q=ai:wang.linfeng.2Summary: In this paper we firstly prove that the CD\(_p\) curvature condition always satisfies for \(p \geq 2\) on any connected locally finite graph. We show this property does not hold for \(1<p<2\). We also derive a lower bound for the first nonzero eigenvalue of the \(p\)-Laplace operator on a connected finite graph with the \(\mathrm{CD}_p (m, K)\) condition for the case that \(1<p\leq 2\), \(m> \frac{2(p-1)^2}{p}\) and \(K>0\).Discrete restriction for \((x,x^3)\) and related topicshttps://zbmath.org/1541.110842024-09-27T17:47:02.548271Z"Hughes, Kevin"https://zbmath.org/authors/?q=ai:hughes.kevin-j|hughes.kevin"Wooley, Trevor D."https://zbmath.org/authors/?q=ai:wooley.trevor-dConjecture. For each real \(p \geq 1\) there exists a positive constant \(C_p\) such that, for all \(N \in \mathbb{N}\) and all sequences \(a \in \ell_2\) the following bound holds
\[
{\vert\vert E a \vert \vert}_{L^p(\mathbb{T^2})} \leq C_p \cdot (1+N^{\frac{1}{2}-\frac{4} {p}}) \cdot {\vert\vert a \vert\vert}_{\ell_2(\mathbb{Z})}
\]
The object of the paper consists of proving (using the circle method) the conjecture for \(p=10\) and extending the result to other appropriate curves. The operator \(E\) is linked to the curve \((x,x^3)\) by
\[
Ea(u,v) = \sum_{\vert n \vert \leq N} a(n) \exp(2 \pi i \cdot (u n^3+ v n)),
\]
and \(\mathbb{T} = \mathbb{R}/ \mathbb{Z}\). The paper is in honor to Jean Bourgain
Reviewer: Luis Gallardo (Brest)Projective rigidity and Alexander polynomials of certain nodal hypersurfaceshttps://zbmath.org/1541.140622024-09-27T17:47:02.548271Z"Escudero, Juan García"https://zbmath.org/authors/?q=ai:garcia-escudero.juanSummary: We present nodal algebraic hypersurfaces in the complex projective space which are projectively rigid. Defects and Alexander polynomials associated with the hypersurfaces are obtained. There are families of nodal hypersurfaces with nontrivial Alexander polynomials and nodal threefolds with projective rigidity which are potentially infinite.Non-local gradients in bounded domains motivated by continuum mechanics: fundamental theorem of calculus and embeddingshttps://zbmath.org/1541.260172024-09-27T17:47:02.548271Z"Bellido, José Carlos"https://zbmath.org/authors/?q=ai:bellido.jose-carlos"Cueto, Javier"https://zbmath.org/authors/?q=ai:cueto.javier"Mora-Corral, Carlos"https://zbmath.org/authors/?q=ai:mora-corral.carlosSummary: In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz \(s\)-fractional gradient and peridynamics, in the sense that its integration domain depends on a ball of radius \(\delta > 0\) (horizon of interaction among particles, in the terminology of peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for non-local models in calculus of variations and partial differential equations. Our motivation is to develop the proper functional analysis framework to tackle non-local models in \textit{continuum mechanics}, which requires working with bounded domains, while retaining the good mathematical properties of Riesz \(s\)-fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the \(L^p\) norms of the function and its non-local gradient. Among the results showed in this investigation, we highlight a non-local version of the fundamental theorem of calculus (namely, a representation formula where a function can be recovered from its non-local gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger, and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this non-local situation are also established, and those can be viewed as a new class of non-local partial differential equations in bounded domains.A new fractional derivative operator and its application to diffusion equationhttps://zbmath.org/1541.260292024-09-27T17:47:02.548271Z"Sharma, Ruchi"https://zbmath.org/authors/?q=ai:sharma.ruchi"Goswami, Pranay"https://zbmath.org/authors/?q=ai:goswami.pranay"Dubey, Ravi Shanker"https://zbmath.org/authors/?q=ai:dubey.ravi-shanker"Belgacem, Fethi Bin Muhammad"https://zbmath.org/authors/?q=ai:belgacem.fethi-bin-muhammad(no abstract)Differential equations. A first course on ODE and a brief introduction to PDEhttps://zbmath.org/1541.340012024-09-27T17:47:02.548271Z"Ambrosetti, Antonio"https://zbmath.org/authors/?q=ai:ambrosetti.antonio"Ahmad, Shair"https://zbmath.org/authors/?q=ai:ahmad.shairThis book is the second edition of a first course on Ordinary Differential Equations (ODE) and a very brief introduction to Partial Differential Equations (PDE) [Zbl 1433.34001]. The structure of the book consists of 16 chapters, followed by a final Solutions chapter and the list of References and the Index of terms.
The chapters are the following:
1. A brief survey of some topics in calculus
2. First order linear differential equations
3. Analytical study of first order differential equations
4. Solving and analyzing some nonlinear first order equations
5. Exact differential equations
6. Second order linear differential equations
7. Higher order linear equations
8. Systems of first order equations
9. Phase plane analysis
10. Introduction to stability
11. Series solutions for linear differential equations
12. Laplace transform
13. A primer on equations of Sturm-Liouville type
14. A primer on linear PDE in 2D. I: first order equations
15. A primer on linear PDE in 2D. II: second order equations
16. The Euler-Lagrange equations in the Calculus of Variations: an introduction
Each chapter contains a theoretical part, describing the most important approaches to the topic of that chapter, and a practical part, containg examples, applications and illustrative exercices for better understanding of the subject. Some chapters also contain an Appendix section, usually for the proofs of the main results. Each chapter ends with a list of proposed exercises and problems, whose solutions are briefly given in the last chapter.
This book is a very nice and well-written contribution to the main topics of ODE and an useful introduction to first order and second order PDE, with applications to Calculus of Variations. It is a valuable and very useful book for undergraduate students in Mathematics, Computer Science and Engineering, as well as for researchers in the field of applied mathematics to various sciences (Biology, Chemistry, Physics).
Reviewer: Adrian Petruşel (Cluj-Napoca)Spreading speeds for time heterogeneous prey-predator systems with nonlocal diffusion on a latticehttps://zbmath.org/1541.340272024-09-27T17:47:02.548271Z"Ducrot, Arnaud"https://zbmath.org/authors/?q=ai:ducrot.arnaud"Jin, Zhucheng"https://zbmath.org/authors/?q=ai:jin.zhuchengSummary: We investigate the spreading behaviour for the solutions of a non-autonomous prey-predator system on a discrete lattice. These time variations are assumed to enjoy an averaging property. This includes periodicity, almost periodicity and unique ergodicity as special cases. The spatial motion of individuals from one site to another is modelled by a discrete convolution operator. In order to take into account external fluctuations such as seasonality, daily variations and so on, the convolution kernels and reaction terms may vary with time. Our analysis of the spreading speeds of invasion of the species is based on the careful and detailed study of the hair-trigger effect and spreading speed for a non-autonomous scalar Fisher-KPP equation on a lattice. Then, we are able to compare the solutions of the prey-predator system with those of a suitable scalar Fisher-KPP equation and derive the invasion speeds of the prey and of the predator.Beyond the Bristol book: advances and perspectives in non-smooth dynamics and applicationshttps://zbmath.org/1541.340312024-09-27T17:47:02.548271ZSummary: Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic ``Bristol book'' [\textit{M. di Bernardo} et al., Piecewise-smooth dynamical systems. Theory and applications. New York, NY: Springer (2008; Zbl 1146.37003)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.The particle paths of hyperbolic conservation lawshttps://zbmath.org/1541.340332024-09-27T17:47:02.548271Z"Fjordholm, Ulrik S."https://zbmath.org/authors/?q=ai:fjordholm.ulrik-skre"Mæhlen, Ola H."https://zbmath.org/authors/?q=ai:maehlen.ola-h"Ørke, Magnus C."https://zbmath.org/authors/?q=ai:orke.magnus-cSummary: Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.Unified asymptotic analysis and numerical simulations of singularly perturbed linear differential equations under various nonlocal boundary effectshttps://zbmath.org/1541.340362024-09-27T17:47:02.548271Z"Chen, Xianjin"https://zbmath.org/authors/?q=ai:chen.xianjin"Lee, Chiun-Chang"https://zbmath.org/authors/?q=ai:lee.chiun-chang"Mizuno, Masashi"https://zbmath.org/authors/?q=ai:mizuno.masashiSummary: While being concerned with a singularly perturbed linear differential equation subject to integral boundary conditions, the exact solutions, in general, cannot be specified, and the validity of the maximum principle is unassurable. Hence, a problem arises: \textit{how to identify the boundary asymptotics more precisely?} We develop a rigorous asymptotic method involving recovered boundary data to tackle the problem. A key ingredient of the approach is to transform the ``nonlocal'' boundary conditions into ``local'' boundary conditions. Then, we perform an ``\(\varepsilon \log \varepsilon\)-estimate'' to obtain the refined boundary asymptotics of its solutions with respect to the singular perturbation parameter \(\varepsilon\). Furthermore, for the inhomogeneous case, diversified asymptotic behaviors including uniform boundedness and asymptotic blow-up are obtained. Numerical simulations and validations are also presented to further support the corresponding theoretical results.Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcationshttps://zbmath.org/1541.340732024-09-27T17:47:02.548271Z"Goh, Ryan"https://zbmath.org/authors/?q=ai:goh.ryan-n"Kaper, Tasso J."https://zbmath.org/authors/?q=ai:kaper.tasso-j"Scheel, Arnd"https://zbmath.org/authors/?q=ai:scheel.arnd"Vo, Theodore"https://zbmath.org/authors/?q=ai:vo.theodoreSummary: This work studies front formation in the Allen-Cahn equation with a parameter heterogeneity which slowly varies in space. In particular, we consider a heterogeneity which mediates the local stability of the zero state and subsequent pitchfork bifurcation to a nontrivial state. For slowly varying ramps which are either rigidly propagating in time or stationary, we rigorously establish existence and stability of positive, monotone fronts and give leading order expansions for their interface location. For nonzero ramp speeds, and sufficiently small ramp slopes, the front location is determined by the local transition between convective and absolute instability of the base state and leads to an \(\mathcal{O}(1)\) delay beyond the instantaneous pitchfork location before the system jumps to a nontrivial state. The slow ramp induces a further delay of the interface controlled by a slow passage through a fold of strong- and weak-stable eigenspaces of the associated linearization. We introduce projective coordinates to desingularize the dynamics near the trivial state and track relevant invariant manifolds all the way to the fold point. We then use geometric singular perturbation theory and blow-up techniques to locate the desired intersection of invariant manifolds. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation with inner expansion given by the unique Hastings-McLeod connecting solution of Painlevé's second equation. We once again use geometric singular perturbation theory and blow-up techniques to track invariant manifolds into a neighborhood of the nonhyperbolic point where the ramp passes through zero and to locate intersections.An excursion through partial differential equationshttps://zbmath.org/1541.350012024-09-27T17:47:02.548271Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgievThis book is basically a collection of problems in partial differential equations with the solutions to these problems. It does not have any new scientific results.
It does not have any references to earlier published books, in particular, to collections of problems in PDE. There are many of these, for example, by Tikhonov and Samarskii, by Steeb, by Smirnov, and by many other authors. It may be useful for the students to have some references to the books on the theory of partial differential equations and to the collections of problems in PDE (partial differential equations).
This book does not have any derivations of the PDE from the physical laws. Probably, the author believes that the students learned these derivations from some courses in physics.
There are no notions from functional analysis in this book. The norm of an element of a functional space is not defined and is not used. In particular, on p. 2 the Hadamard's definition of well-posedness is given, but the meaning of ``small change in the solution'' is not discussed in the Hadamard's example on p. 142.
There is no definition of distributions, but the word delta-function is used on p. 150, where it is mentioned that it is the Dirac measure. The fundamental solution for the Laplace operator is not defined, but formula is given for this fundamental solution.
There are no mentioning of the Maxwell's equations (see, e.g., [\textit{A. Zangwill}, Modern electrodynamics. Cambridge: Cambridge University Press (2013; Zbl 1351.78001)]), or of the Navier-Stokes equations (see, e.g., [\textit{L. Landau} and \textit{E. Lifschitz}, Fluid mechanics. Oxford: Pergamon Press (1984); \textit{A. G. Ramm}, Analysis of the Navier-Stokes problem. Solution of a millennium problem. 2nd expanded edition. Cham: Springer (2023; Zbl 1532.35353)]), or of the scattering problems (see, e.g., [\textit{A. G. Ramm}, Scattering by obstacles and potentials. Hackensack, NJ: World Scientific (2018; Zbl 1394.35306)]), or of the inverse problems (see, e.g. [\textit{A. G. Ramm}, Inverse Problems. Mathematical and analytical techniques with applications to engineering. New York, NY: Springer (2005; Zbl 1083.35002)]). All these topics are quite important. They can be explained to the students which have the same background that is assumed in the book under review.
On p. 10 the second equation in the shallow water equations is given with an error: \(v_t+vv_x=gh_x=0\). On p. 244, at the bottom of the page, the name of d'Alembert is misspelled.
On p. 8 the Helmholtz equation the Poisson equation and the Schrödinger equation are given in ``two dimensional form''. It is not clear why these equations are not given in the three-dimensional space.
In this book there is no notion of the radiation condition.
In the preface the author writes that the book presents an introduction to the PDE theory and is suitable for any courses in PDE, that the material ``is presented in highly readable, mathematically solid format''.
The book contains 9 chapters:
\begin{itemize}
\item[1.] General introduction,
\item[2.] First order partial differential equations,
\item[3.] Classification of second order partial differential equations,
\item[4.] Classification and canonical forms for second order partial differential equations,
\item[5.] The Laplace equation,
\item[6.] The heat equation,
\item[7.] The wave equation.
\item[8.] Solutions, hints, and answers to the exercises,
\item[9.] Solutions, hints, and answers to the problems.
\end{itemize}
Index. pp. 421--422.
Reviewer: Alexander G. Ramm (Manhattan, KS)Introduction to traveling waveshttps://zbmath.org/1541.350022024-09-27T17:47:02.548271Z"Ghazaryan, Anna R."https://zbmath.org/authors/?q=ai:ghazaryan.anna-r"Lafortune, Stéphane"https://zbmath.org/authors/?q=ai:lafortune.stephane.1"Manukian, Vahagn"https://zbmath.org/authors/?q=ai:manukian.vahagnPublisher's description: Introduction to Traveling Waves is an invitation to research focused on traveling waves for undergraduate and masters level students. Traveling waves are not typically covered in the undergraduate curriculum, and topics related to traveling waves are usually only covered in research papers, except for a few texts designed for students. This book includes techniques that are not covered in those texts.
Through their experience involving undergraduate and graduate students in a research topic related to traveling waves, the authors found that the main difficulty is to provide reading materials that contain the background information sufficient to start a research project without an expectation of an extensive list of prerequisites beyond regular undergraduate coursework. This book meets that need and serves as an entry point into research topics about the existence and stability of traveling waves.
Features
\begin{itemize}
\item Self-contained, step-by-step introduction to nonlinear waves written assuming minimal prerequisites, such as an undergraduate course on linear algebra and differential equations.
\item Suitable as a textbook for a special topics course, or as supplementary reading for courses on modeling.
\item Contains numerous examples to support the theoretical material.
\item Supplementary MATLAB codes available via GitHub.
\end{itemize}A coupled system of differential-algebraic equation and hyperbolic partial differential equation. Analysis and optimal controlhttps://zbmath.org/1541.350032024-09-27T17:47:02.548271Z"Groh, Dennis"https://zbmath.org/authors/?q=ai:groh.dennisPublisher's description: Coupled systems of differential-algebraic equations (DAEs) and partial differential equations (PDEs) appear in various fields of applications such as electrical engineering, bio-mathematics, or multi-physics. They are of particular interest for the modeling and simulation of flow networks, for instance energy transport networks. In this thesis, we discuss a system in which an abstract DAE and a second order hyperbolic PDE are coupled through nonlinear coupling functions.
The analysis presented is split into two parts: In the first part, we introduce the concept of matrix-induced linear operators which arise naturally in the context of abstract DAEs but have surprisingly not been discussed in literature on abstract DAEs so far. We also present a novel index-1-like criterion that allows to separate dynamical and non-dynamical parts of the abstract DAE while allowing for a considerable reduction of required assumptions, compared to existing theoretical results for abstract DAEs.
In the second part, we build upon the developed techniques. We show how to combine the theoretical frameworks for abstract DAEs and second order hyperbolic PDEs in a way such that both parts of the solution are of similar regularity. We then use a fixed-point approach to prove existence and uniqueness of local as well as global solutions to the coupled system.
In the last part of this thesis, we throw a glance at a related optimal control problem and prove existence of a global minimizer.A short survey on overdetermined elliptic problems in unbounded domainshttps://zbmath.org/1541.350042024-09-27T17:47:02.548271Z"Sicbaldi, Pieralberto"https://zbmath.org/authors/?q=ai:sicbaldi.pieralbertoSummary: We present some recent results about overdetermined elliptic problems in unbounded domains.
For the entire collection see [Zbl 1497.42002].Modern problems in PDEs and applications. Extended abstracts of the 2023 GAP Center summer school, Ghent, Belgium, August 23 -- September 2, 2023https://zbmath.org/1541.350052024-09-27T17:47:02.548271Z"Restrepo, Joel"https://zbmath.org/authors/?q=ai:restrepo.joel-estebanPublisher's description: The principal aim of the volume is gathering all the contributions given by the speakers (mini courses) and some of the participants (short talks) of the summer school ``Modern Problems in PDEs and Applications'' held at the Ghent Analysis and PDE Center from 23 August to 2 September 2023. The school was devoted to the study of new techniques and approaches for solving partial differential equations, which can either be considered or arise from the physical point of view or the mathematical perspective. Both sides are extremely important since theories and methods can be developed independently, aiming to gather each other in a common objective. The aim of the summer school was to progress and advance in the problems considered. Note that real-world problems and their applications are classical study trends in physical or mathematical modelling. The summer school was organised in a friendly atmosphere and synergy, and it was an excellent opportunity to promote and encourage the development of the subject in the community.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Barbatis, Gerassimos}, The Hardy constant: a review, 3-11 [Zbl 07919524]
\textit{Casarino, Valentina; Ciatti, Paolo; Sjögren, Peter}, Some harmonic analysis in a general Gaussian setting, 13-18 [Zbl 07919525]
\textit{Cerejeiras, Paula}, Introduction to hypercomplex analysis, 19-27 [Zbl 07919526]
\textit{Ciatti, Paolo}, Some norm bounds for the spectral projections of the Heisenberg sublaplacian, 29-36 [Zbl 07919527]
\textit{Dindoš, Martin; Pipher, Jill}, Boundary value problems for elliptic operators satisfying Carleson condition, 37-45 [Zbl 07919528]
\textit{Grieger, Elisabeth; Scott, Simon G.}, An elementary computation of heat trace invariants, 47-71 [Zbl 07919529]
\textit{Kähler, Uwe}, Can we divide vectors? -- Geometric calculus in science and engineering, 73-80 [Zbl 07919530]
\textit{Monniaux, Sylvie}, Maximal regularity as a tool for partial differential equations, 81-93 [Zbl 07919531]
\textit{Pucci, Patrizia}, Recent existence results for some critical subelliptic problems, 95-103 [Zbl 07919532]
\textit{Schrohe, Elmar}, Introduction to the analysis on manifolds with conical singularities, 105-118 [Zbl 07919533]
\textit{Smyrnelis, Panayotis}, Elliptic systems of phase transition type, 119-128 [Zbl 07919534]
\textit{Yannakakis, Nikos}, Cordes condition, Campanato nearness and beyond, 129-137 [Zbl 07919535]
\textit{Cardona, Duván; Kowacs, André}, Global hypoellipticity on homogeneous vector bundles: necessary and sufficient conditions, 141-151 [Zbl 07919537]
\textit{Maes, Frederick; van Bockstal, Karel}, On inverse source problems for space-dependent sources in thermoelasticity, 153-161 [Zbl 07919538]
\textit{Nurakhmetov, Daulet; Jumabayev, Serik; Aniyarov, Almir}, Symmetric properties of eigenvalues and eigenfunctions of uniform beams with axial loads, 163-166 [Zbl 07919539]
\textit{Shaimerdenov, Yerkin; Yessirkegenov, Nurgissa}, Cylindrical and horizontal extensions of critical Sobolev type inequalities and identities, 167-174 [Zbl 07919540]
\textit{Yeskermessuly, Alibek}, Very weak solution of the wave equation for Sturm-Liouville operator, 175-183 [Zbl 07919541]
\textit{Yimer, Markos Fisseha; Persson, Lars-Erik; Ayele, Tsegaye Gedif}, On Cochran-Lee and Hardy-type inequalities in some classical and homogeneous group cases, 185-190 [Zbl 07919542]Mathematical physics and its interactions. In honor of the 60th birthday of Tohru Ozawa, Tokyo, Japan, August 25--27, 2021. Conference proceedingshttps://zbmath.org/1541.350062024-09-27T17:47:02.548271ZPublisher's description: This publication comprises research papers contributed by the speakers, primarily based on their planned talks at the meeting titled `Mathematical Physics and Its Interactions,' initially scheduled for the summer of 2021 in Tokyo, Japan. It celebrates Tohru Ozawa's 60th birthday and his extensive contributions in many fields.
The works gathered in this volume explore interactions between mathematical physics, various types of partial differential equations (PDEs), harmonic analysis, and applied mathematics. They are authored by research leaders in these fields, and this selection honors the spirit of the workshop by showcasing cutting-edge results and providing a forward-looking perspective through discussions of problems, with the goal of shaping future research directions.
Originally planned as an in-person gathering, this conference had to change its format due to limitations imposed by COVID, more precisely to avoid inducing people into unnecessary vaccinations.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Hirata, Kentaro}, Positive solutions of superlinear elliptic equations with respect to the Schrödinger operator, 1-34 [Zbl 07925336]
\textit{Ishiwata, Tetsuya; Yazaki, Shigetoshi}, Convexity phenomena arising in an area-preserving crystalline curvature flow, 35-62 [Zbl 07925337]
\textit{Koike, Shigeaki; Kosugi, Takahiro}, Rate of convergence for approximate solutions in obstacle problems for nonlinear operators, 63-93 [Zbl 07925338]
\textit{Kozono, Hideo; Shimizu, Senjo}, On a compatibility condition for the Navier-Stokes solutions in maximal \(L^p\)-regularity class, 95-117 [Zbl 07925339]
\textit{Masaki, Satoshi; Segata, Jun-Ichi; Uriya, Kota}, Asymptotic behavior in time of solution to system of cubic nonlinear Schrödinger equations in one space dimension, 119-180 [Zbl 07925340]
\textit{Tsutaya, Kimitoshi; Wakasugi, Yuta}, Remarks on blow up of solutions of nonlinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime, 181-197 [Zbl 07925341]
\textit{Kumar, Sandeep; Ponce-Vanegas, Felipe; Roncal, Luz; Vega, Luis}, The Frisch-Parisi formalism for fluctuations of the Schrödinger equation, 199-223 [Zbl 07925342]
\textit{Cossetti, Lucrezia; Fanelli, Luca; Schiavone, Nico M.}, Recent developments in spectral theory for non-self-adjoint Hamiltonians, 225-253 [Zbl 07925343]
\textit{Saut, Jean-Claude; Xu, Li}, Boussinesq, Schrödinger and Euler-Korteweg, 255-282 [Zbl 07925344]
\textit{Hiroshima, Fumio}, Representations of Pauli-Fierz type models by path measures, 283-410 [Zbl 07925345]On asymptotic properties of solutions to \(\sigma \)-evolution equations with general double dampinghttps://zbmath.org/1541.350072024-09-27T17:47:02.548271Z"Dao, Tuan Anh"https://zbmath.org/authors/?q=ai:dao.tuan-anh"Duong, Dinh Van"https://zbmath.org/authors/?q=ai:van-duong.dinh"Nguyen, Duc Anh"https://zbmath.org/authors/?q=ai:nguyen.duc-anhIn the present paper, the authors turn to the following \(\sigma\)-evolution models:
\[
u_{tt}+ (-\Delta)^\sigma u + \mu_1(-\Delta)^{\sigma_1} u_t + \mu_2(-\Delta)^{\sigma_2} u_t=0,\,u(0,x)=u_0(x),\,u_t(0,x)=u_1(x),\tag{1}
\]
\[
u_{tt}+ (-\Delta)^\sigma u + \mu_1(-\Delta)^{\sigma_1} u_t + \mu_2(-\Delta)^{\sigma_2} u_t=|\partial_t^j u|^p,\,u(0,x)=u_0(x),\,u_t(0,x)=u_1(x),\tag{2}
\]
where \(j=0,1\). The main goal of the paper is to understand the influence of interaction of the ``parabolic like damping term'' (\(\sigma_1 \in [0,\sigma/2)\)) and the ``\(\sigma\)-evolution damping term'' (\(\sigma_2 \in (\sigma/2,\sigma]\)) on qualitative properties of solutions.
For this reason the authors propose for Sobolev solutions to (1) energy decay estimates and asymptotic profile. They assume additional integrability of the data.
For Sobolev solutions to (2) the authors study the global (in time) existence of Sobolev solutions. For the proof the authors use modern tools from Harmonic Analysis. Here the ``parabolic like damping term'' influences decay estimates. The ``\(\sigma\)-evolution damping term'' determines regularity assumptions for the data and solution spaces as well. Moreover, the asymptotic profile is studied. Here the main interesting point is how the nonlinear source term influences the profile in comparison with the profile of Sobolev solutions to (1).
The question for blow-up remains open. After recent papers of Aslan/Reissig a challenging problem is to study instead of (1) the model
\[
u_{tt}+ (-\Delta)^\sigma u + \mu_1(t)(-\Delta)^{\sigma_1} u_t + \mu_2(t)(-\Delta)^{\sigma_2} u_t=0,\,u(0,x)=u_0(x),\,u_t(0,x)=u_1(x),\tag{3}
\]
with time-dependent coefficients \(\mu_1=\mu_1(t)\) and \(\mu_2=\mu_2(t)\).
Reviewer: Michael Reissig (Freiberg)Well-posedness and numerical results to 3D periodic Burgers' equation in Lebesgue-Gevrey classhttps://zbmath.org/1541.350082024-09-27T17:47:02.548271Z"Selmi, Ridha"https://zbmath.org/authors/?q=ai:selmi.ridha"Chaabani, Abdelkerim"https://zbmath.org/authors/?q=ai:chaabani.abdelkerimSummary: We prove that a unique global in time solution to the three dimensional periodic Burger's equation exists, in the Lebesgue-Gevrey class. Also, we establish that the long time to this solution is determined by a finite number of Fourier modes; this is useful as a numerical result. Energy methods, compactness methods, maximum principle and Fourier analysis are the main tools.
For the entire collection see [Zbl 1497.42002].On uniqueness of steady 1-D shock solutions in a finite nozzle via asymptotic analysis for physical parametershttps://zbmath.org/1541.350092024-09-27T17:47:02.548271Z"Fang, Beixiang"https://zbmath.org/authors/?q=ai:fang.beixiang"Jiang, Su"https://zbmath.org/authors/?q=ai:jiang.su"Sun, Piye"https://zbmath.org/authors/?q=ai:sun.piyeThis paper uses asymptotic analysis to study 1-D steady transonic shocks for inviscid compressible flows in a finite nozzle. Convergence of heat conductive shock solutions to a shock solution for the Euler system is discussed for barotropic gases vis-à-vis polytropic gases. Conditions, under which different viscous shock solutions exist, are specified. It is shown that the position of the shock front depends strongly on the viscosity function of the temperature.
Reviewer: Vishnu Dutt Sharma (Ghandinagar)Boundedness in a chemotaxis-May-Nowak model with exposed statehttps://zbmath.org/1541.350102024-09-27T17:47:02.548271Z"Zhang, Qingshan"https://zbmath.org/authors/?q=ai:zhang.qingshan"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.15|li.yan.5|li.yan.66|li.yan.21|li.yan.72|li.yan.71|li.yan.16|li.yan.57|li.yan.58|li.yan.43|li.yan.7|li.yan.14|li.yan.90|li.yan.2|li.yan.12|li.yan.56|li.yan.41|li.yan.19|li.yan.54|li.yan.55|li.yan.11|li.yan.67|li.yan.24|li.yan.28|li.yan.9|li.yan.25Summary: This paper is concerned with the Neumann initial-boundary value problem of the chemotaxis-May-Nowak model with exposed state for virus dynamics. It is proved that the problem admits a unique global classical solution which is uniformly bounded for all sufficiently smooth initial data in smoothly bounded domains \(\Omega \subset \mathbb{R}^n\), \(n \leq 3\).Surface-subsurface filtration transport with seawater intrusion: multidomain mixed variational evolution problemshttps://zbmath.org/1541.350112024-09-27T17:47:02.548271Z"Alduncin, Gonzalo"https://zbmath.org/authors/?q=ai:alduncin.gonzaloSummary: Surface-subsurface filtration transport with seawater intrusion phenomena are formulated and variationally analyzed, as coupled multimedia mixed pairs of free boundary interface problems. Physically, multidomain subsurface mixed velocity-pressure fractional Darcian flow models coupled with surface evolution Stokesian mixed flows are considered. Specifically, two-phase air-fresh water above the sea level and fresh water-seawater characterizations are considered. Internal boundary synchronizing transmission conditions of multidomain nonoverlapping decompositions are modeled in terms of variational Lagrangian dual subpotential maximal monotone inclusions. Similarly, filtration transport coupling interface transmission constraints are implemented by mass flux-velocity-pressure Lagrange dual multipliers as solutions of subpotential subdifferential equations.Normalized solutions for the fractional Choquard equations with Hardy-Littlewood-Sobolev upper critical exponenthttps://zbmath.org/1541.350122024-09-27T17:47:02.548271Z"Meng, Yuxi"https://zbmath.org/authors/?q=ai:meng.yuxi"He, Xiaoming"https://zbmath.org/authors/?q=ai:he.xiaomingSummary: In the present paper, we study the existence of normalized ground states for the following nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent:
\[
(-\Delta)^su = \lambda u + \mu (I_\alpha\ast|u|^p)|u|^{p-2}u+(I_\alpha\ast|u|^{2^\ast_{\alpha ,s}}) |u|^{2^\ast_{\alpha, s} - 2}u,\quad x\in\mathbb{R}^N,
\]
having prescribed mass
\[
\int_{\mathbb{R}^N}|u|^2dx=a>0,
\]
where \(N > 2s\), \(s\in(0, 1)\), \(\alpha\in(0, N)\), \(\underline{p} := \frac{N+\alpha}{N} < p < \overline{p} := \frac{N+2s+\alpha}{N} < 2^\ast_{\alpha, s}\), \(2^\ast_{\alpha, s} := \frac{N+\alpha}{N-2s}\) is the upper Hardy-Littlewood-Sobolev critical exponent, \(\mu > 0\) and \(\lambda\in\mathbb{R}\). Furthermore, the qualitative behavior of the ground states as \(\mu\rightarrow0^+\) is also studied.On the radius of spatial analyticity for Ostrovsky equation with positive dispersionhttps://zbmath.org/1541.350132024-09-27T17:47:02.548271Z"Yang, Pan"https://zbmath.org/authors/?q=ai:yang.pan"Zhao, Yajuan"https://zbmath.org/authors/?q=ai:zhao.yajuan(no abstract)Considering the wave processes in oceanography, acoustics and hydrodynamics by means of an extended coupled (2+1)-dimensional Burgers systemhttps://zbmath.org/1541.350142024-09-27T17:47:02.548271Z"Gao, Xin-Yi"https://zbmath.org/authors/?q=ai:gao.xinyiSummary: Recent activities in oceanography, acoustics and hydrodynamics are impressive. In respect to oceanography, acoustics and hydrodynamics, the Burgers-type systems/equations attract people's attention, including an extended coupled (2+1)-dimensional Burgers system describing the wave processes in oceanography, acoustics and hydrodynamics, which we investigate hereby. We construct two families of the hetero-Bäcklund transformations, each from that system to a (2+1)-dimensional Broer-Kaup-Kupershmidt system, as well as a family of the similarity reductions, from that system to a known ordinary differential equation. We take notice of all our results dependent on the coefficients in that system.Korn-Maxwell-Sobolev inequalities for general incompatibilitieshttps://zbmath.org/1541.350152024-09-27T17:47:02.548271Z"Gmeineder, Franz"https://zbmath.org/authors/?q=ai:gmeineder.franz"Lewintan, Peter"https://zbmath.org/authors/?q=ai:lewintan.peter"Neff, Patrizio"https://zbmath.org/authors/?q=ai:neff.patrizioSummary: We establish a family of coercive Korn-type inequalities for generalized incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [Calc. Var. Partial Differ. Equ. 62, No. 6, Paper No. 182, 33 p. (2023; Zbl 1518.35024)], where we focused on the case \(p=1\) and incompatibilities governed by the matrix curl, the case \(p>1\) considered in this paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalizations of recently proved inequalities by \textit{P. Lewintan} et al. [ibid. 60, No. 4, Paper No. 150, 46 p. (2021; Zbl 1471.35009)] in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher-order scenarios as well, our approach equally gives the first and sharp inequalities involving Kröner's incompability tensor \textbf{inc}.Endpoint Sobolev inequalities for vector fields and cancelling operatorshttps://zbmath.org/1541.350162024-09-27T17:47:02.548271Z"van Schaftingen, Jean"https://zbmath.org/authors/?q=ai:van-schaftingen.jeanSummary: The injectively elliptic vector differential operators \(A (\mathrm{D})\) from \(V\) to \(E\) on \(\mathbb{R}^n\) such that the estimate
\[
\Vert \mathrm{D}^\ell u\Vert_{L^{n/(n - (k - \ell))} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)}
\]
holds can be characterized as the operators satisfying a cancellation condition
\[
\displaystyle \bigcap_{\xi \in \mathbb{R}^n \setminus \{0\}} A (\xi)[V] = \{0\}.
\]
These estimates unify existing endpoint Sobolev inequalities for the gradient of scalar functions (Gagliardo and Nirenberg), the deformation operator (Korn-Sobolev inequality by M.J. Strauss) and the Hodge complex (Bourgain and Brezis). Their proof is based on the fact that \(A (\mathrm{D}) u\) lies in the kernel of a cocancelling differential operator.
For the entire collection see [Zbl 1537.35003].Instability analysis and regularization approximation to the forward/backward problems for fractional damped wave equations with random noisehttps://zbmath.org/1541.350172024-09-27T17:47:02.548271Z"Song, Zefang"https://zbmath.org/authors/?q=ai:song.zefang"Di, Huafei"https://zbmath.org/authors/?q=ai:di.huafeiSummary: Considered herein is the forward/backward problems for fractional damped wave equations \(u_{t t} + \alpha (- \Delta)^{s_1} u_t + \beta (- \Delta)^{s_2} u = F(u, x, t)\) subject to the random Gaussian white noise initial and final data. First of all, we construct the mild solutions to forward/backward problems under all cases for parameters \(\alpha\), \(\beta\), \(s_1\) and \(s_2\), and then investigate their stability and instability properties in the sense of Hadamard. Secondly, we propose the regularized solutions by using the Fourier truncation method and establish well-posedness for regularized solutions in above unstable cases. Thirdly, we derive some error estimates between the exact solutions and their regularized solutions in \(\mathbb{E} \| \cdot \|_2^2\)-norm, and theoretically characterize the Fourier truncation approximation effect from regularized solutions to exact solutions. Finally, we give a series of numerical examples used to illuminate the regularization approximation effect of above method.Partial differential equationshttps://zbmath.org/1541.350182024-09-27T17:47:02.548271Z"Srikanth, P. N."https://zbmath.org/authors/?q=ai:srikanth.p-nFor the entire collection see [Zbl 1242.00057].Lie group analysis for obtaining the abundant group invariant solutions and dynamics of solitons for the Lonngren-wave equationhttps://zbmath.org/1541.350192024-09-27T17:47:02.548271Z"Hussain, A."https://zbmath.org/authors/?q=ai:hussain.akhtar"Usman, M."https://zbmath.org/authors/?q=ai:usman.mahamood|usman.muhammad|usman.mohammad|usman.muhammad.1|usman.mustofa|usman.murat|usman.muhammad-rashid"Zaman, F. D."https://zbmath.org/authors/?q=ai:zaman.fiazud-din"Almalki, Yahya"https://zbmath.org/authors/?q=ai:almalki.yahyaSummary: The Lonngren-wave equation (LW Equation), one of the many nonlinear evolution equations (N-EEs) that arise in the field of mathematical physics, is the subject of this study, which uses an extremely strong analytical technique known as ``Lie group analysis'' to create novel solutions. We obtain a five-dimensional optimal system based on four-dimensional Lie algebra. We compute the group invariant solutions via subalgebras. Our obtained solutions are based on the trigonometric, hyperbolic, and polynomial functions. By varying the parameters, the solutions exhibit wavelike properties that include bright, dark, singular, dark-singular-combined solitons, periodic singular and dark-bright-combined. The physical dynamics of the obtained solutions are explored by the 3D and 2D Mathematica simulations which are explaining the new properties of the model considered in this paper.Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equationhttps://zbmath.org/1541.350202024-09-27T17:47:02.548271Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Li, Yong"https://zbmath.org/authors/?q=ai:li.yong.7|li.yong.27|li.yong.24|li.yong.22|li.yong.11|li.yong.1|li.yong.12|li.yong.10|li.yong.5|li.yong.17|li.yong.3|li.yong.13|li.yong|li.yong.15"Xu, Fei"https://zbmath.org/authors/?q=ai:xu.fei.7Summary: In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the \(\mathfrak{p}\)-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying \textbf{the higher dimensional discrete convolution operation for several functions}:
\[
\underbrace{c\times\cdots\times c}_{\mathfrak{p}}\,(\text{total distance}):=\sum_{\begin{matrix}\clubsuit_1,\cdots, \clubsuit_{\mathfrak{p}}\in\mathbb{Z}^\nu\\ \clubsuit_1+\cdots+\clubsuit_{\mathfrak{p}}=\text{total distance}\end{matrix}}\prod\limits_{j=1}^{\mathfrak{p}} c(\clubsuit_j).
\]
In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for \(\mathfrak{p}=2\) [\textit{D. Damanik} and \textit{M. Goldstein}, J. Am. Math. Soc. 29, No. 3, 825--856 (2016; Zbl 1342.35300)], and the combinatorics become harder for larger values of \(\mathfrak{p}\). For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, \(\mathfrak{p}=3\). Next, we prove existence and uniqueness in the general case \(\mathfrak{p}\geq 2\), which then covers the remaining cases \(\mathfrak{p}\geq 4\). As a byproduct, we recover the local result from [loc. cit.]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index \(\sigma\) (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general \(\mathfrak{p}\).New concentrated solutions for the nonlinear Schrödinger-Newton systemhttps://zbmath.org/1541.350212024-09-27T17:47:02.548271Z"Chen, Haixia"https://zbmath.org/authors/?q=ai:chen.haixia"Yang, Pingping"https://zbmath.org/authors/?q=ai:yang.pingping(no abstract)Chemical diffusion limit of a chemotaxis-Navier-Stokes systemhttps://zbmath.org/1541.350222024-09-27T17:47:02.548271Z"Hou, Qianqian"https://zbmath.org/authors/?q=ai:hou.qianqian(no abstract)Approximate boundary conditions for a Mindlin-Timoshenko plate surrounded by a thin layerhttps://zbmath.org/1541.350232024-09-27T17:47:02.548271Z"Madjour, Farida"https://zbmath.org/authors/?q=ai:madjour.farida"Rahmani, Leila"https://zbmath.org/authors/?q=ai:rahmani.leilaSummary: We consider the model of Mindlin-Timoshenko for a multi-structure composed of an elastic plate surrounded by a thin layer of uniform thickness. From the viewpoint of numerical simulation, the treatment of the behavior of this structure is difficult because of the presence of the thin coating. In order to overcome this difficulty, we use the asymptotic expansion method to identify an approximate model that does not involve the thin layer geometrically but which accounts for its effect through new approximate boundary conditions. These conditions are set on the junction interface between the two sub-structures and depend on the thickness and the physical characteristics of the thin layer. Moreover, we give optimal error estimates between the exact and the approximate solutions of the considered transmission problem, which validate this approximation.Controlling pulse stability in singularly perturbed reaction-diffusion systemshttps://zbmath.org/1541.350242024-09-27T17:47:02.548271Z"Veerman, F."https://zbmath.org/authors/?q=ai:veerman.frits"Schneider, I."https://zbmath.org/authors/?q=ai:schneider.ingo|schneider.isabelle|schneider.ian-c|schneider.ivo-h(no abstract)Asymptotic analysis for non-local problems in composites with different imperfect contact conditionshttps://zbmath.org/1541.350252024-09-27T17:47:02.548271Z"Amar, M."https://zbmath.org/authors/?q=ai:amar.micol"Andreucci, D."https://zbmath.org/authors/?q=ai:andreucci.daniele"Timofte, C."https://zbmath.org/authors/?q=ai:timofte.claudia(no abstract)Evolutionary equations are \(G\)-compacthttps://zbmath.org/1541.350262024-09-27T17:47:02.548271Z"Burazin, Krešimir"https://zbmath.org/authors/?q=ai:burazin.kresimir"Erceg, Marko"https://zbmath.org/authors/?q=ai:erceg.marko"Waurick, Marcus"https://zbmath.org/authors/?q=ai:waurick.marcusSummary: We prove a compactness result related to \(G\)-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for \(G\)-convergence for time-dependent equations solely in terms of static equations.Aperiodical isoperimetric planar homogenization with critical diameter: universal non-local strange term for a dynamical unilateral boundary conditionhttps://zbmath.org/1541.350272024-09-27T17:47:02.548271Z"Díaz, J. I."https://zbmath.org/authors/?q=ai:diaz.jesus-ildelfonso|diaz-diaz.jesus-ildefonso|diaz.jesus-idelfonso"Shaposhnikova, T. A."https://zbmath.org/authors/?q=ai:shaposhnikova.tatiana-ardolionovna|shaposhnikova.tatiana-a"Podolskiy, A. V."https://zbmath.org/authors/?q=ai:podolskiy.alexander-vadimovich|podolskii.alexander-vSummary: We study the asymptotic behavior of the solution to the diffusion equation in a planar domain, perforated by tiny sets of different shapes with a constant perimeter and a uniformly bounded diameter, when the diameter of a basic cell, \( \varepsilon \), goes to 0. This makes the structure of the heterogeneous domain aperiodical. On the boundary of the removed sets (or the exterior to a set of particles, as it arises in chemical engineering), we consider the dynamic unilateral Signorini boundary condition containing a large-growth parameter \(\beta (\varepsilon )\). We derive and justify the homogenized model when the problem's parameters take the ``critical values''. In that case, the homogenized problem is universal (in the sense that it does not depend on the shape of the perforations or particles) and contains a ``strange term'' given by a non-linear, non-local in time, monotone operator \textbf{H} that is defined as the solution to an obstacle problem for an ODE operator. The solution of the limit problem can take negative values even if, for any \(\varepsilon \), in the original problem, the solution is non-negative on the boundary of the perforations or particles.The double scale convergence for problem of dissipative evolution in periodically perforated environmenthttps://zbmath.org/1541.350282024-09-27T17:47:02.548271Z"Dumitrache, Mihaela"https://zbmath.org/authors/?q=ai:dumitrache.mihaela"Gheldiu, Camelia"https://zbmath.org/authors/?q=ai:gheldiu.cameliaSummary: This article discusses the homogenization of a dissipative system governed by nonlinear wave equation in a periodically perforated domain. The difficulty and the originality of this paper is the homogenization problem with semilinear condition on the boundary of holes, applying double scale convergence method.
For the entire collection see [Zbl 1388.00025].Uniform convergence for linear elastostatic systems with periodic high contrast inclusionshttps://zbmath.org/1541.350292024-09-27T17:47:02.548271Z"Fu, Xin"https://zbmath.org/authors/?q=ai:fu.xin|fu.xin.1|fu.xin.2"Jing, Wenjia"https://zbmath.org/authors/?q=ai:jing.wenjiaSummary: We consider the Lamé system of linear elasticity with periodically distributed inclusions whose elastic parameters have high contrast compared to the background media. We develop a unified method based on layer potential techniques to quantify three convergence results when some parameters of the elastic inclusions are sent to extreme values. More precisely, we study the \textit{incompressible inclusions} limit where the bulk modulus of the inclusions tends to infinity, the \textit{soft inclusions} limit where both the bulk modulus and the shear modulus tend to zero, and the \textit{hard inclusions} limit where the shear modulus tends to infinity. Our method yields convergence rates that are independent of the periodicity of the inclusions array, and are sharper than some earlier results of this type. A key ingredient of the proof is the establishment of uniform spectra gaps for the elastic Neumann-Poincaré operator associated to the collection of periodic inclusions that are independent of the periodicity.Convergence rates and fluctuations for the Stokes-Brinkman equations as homogenization limit in perforated domainshttps://zbmath.org/1541.350302024-09-27T17:47:02.548271Z"Höfer, Richard M."https://zbmath.org/authors/?q=ai:hofer.richard-m"Jansen, Jonas"https://zbmath.org/authors/?q=ai:jansen.jonasSummary: We study the homogenization of the Dirichlet problem for the Stokes equations in \(\mathbb{R}^3\) perforated by \(m\) spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order \(m^{-1}\), the homogenization limit \(u\) is given as the solution to the Brinkman equations. We provide optimal rates for the convergence \(u_m \to u\) in \(L^2\), namely \(m^{-\beta}\) for all \(\beta < 1/2\). Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in \(L^2 (\mathbb{R}^3)\), with an explicit covariance. Our analysis is based on explicit approximations for the solutions \(u_m\) in terms of \(u\) as well as the particle positions and their velocities. These are shown to be accurate in \(\dot{H}^1 (\mathbb{R}^3)\) to order \(m^{-\beta}\) for all \(\beta < 1\). Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.Homogenization for Poisson equations in domains with concentrated holeshttps://zbmath.org/1541.350312024-09-27T17:47:02.548271Z"Ishida, Hiroto"https://zbmath.org/authors/?q=ai:ishida.hirotoSummary: We consider solutions \(u^{\varepsilon}\) of Poisson problems with the Dirichlet condition on domains \(\Omega_{\varepsilon}\) with holes concentrated at subsets of a domain \(\Omega\) nonperiodically. We show \(u^{\varepsilon}\) converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by \textit{D. Cioranescu} and \textit{F. Murat} [Prog. Nonlinear Differ. Equ. Appl. 31, 45--93 (1997; Zbl 0912.35020)]. They showed a result for case that holes are distributed at \(\Omega\) periodically.A homogenization approach to the effect of surfactant concentration and interfacial slip on the flow past viscous dropshttps://zbmath.org/1541.350322024-09-27T17:47:02.548271Z"Mahato, H. S."https://zbmath.org/authors/?q=ai:mahato.hari-shankar|mahato.hari-shankar.2"Raja Sekhar, G. P."https://zbmath.org/authors/?q=ai:rajasekhar.g-p|sekhar.g-p-raja(no abstract)Wave-breaking phenomena and persistence properties for a nonlinear dissipative Camassa-Holm equationhttps://zbmath.org/1541.350332024-09-27T17:47:02.548271Z"Fu, Shanshan"https://zbmath.org/authors/?q=ai:fu.shanshan"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.2|wang.ying.23|wang.ying.35|wang.ying.19|wang.ying.16|wang.ying.36|wang.ying.8|wang.ying.31|wang.ying|wang.ying.42|wang.ying.12|wang.ying.9|wang.ying.53|wang.ying.38(no abstract)The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEshttps://zbmath.org/1541.350342024-09-27T17:47:02.548271Z"Harbrecht, Helmut"https://zbmath.org/authors/?q=ai:harbrecht.helmut"Schmidlin, Marc"https://zbmath.org/authors/?q=ai:schmidlin.marc"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophSummary: This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under \(s\)-Gevrey assumptions on the residual equation, we establish \(s\)-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.Weakly nonlinear analysis of a two-species non-local advection-diffusion systemhttps://zbmath.org/1541.350352024-09-27T17:47:02.548271Z"Giunta, Valeria"https://zbmath.org/authors/?q=ai:giunta.valeria"Hillen, Thomas"https://zbmath.org/authors/?q=ai:hillen.thomas"Lewis, Mark A."https://zbmath.org/authors/?q=ai:lewis.mark-a"Potts, Jonathan R."https://zbmath.org/authors/?q=ai:potts.jonathan-rSummary: Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection-diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg-Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg-Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.Bifurcations and dynamical behaviors for a generalized delayed-diffusive Maginu modelhttps://zbmath.org/1541.350362024-09-27T17:47:02.548271Z"Ju, Xiaowei"https://zbmath.org/authors/?q=ai:ju.xiaoweiSummary: This paper is committed to study the dynamical behaviors of a generalized Maginu model with discrete time delay. We investigate the stability of the positive equilibrium and the existence of periodic solutions bifurcating from the positive equilibrium. Further, by using the center manifold theorem and the normal form theory, we derive the precise condition to judge the bifurcation direction and the stability of the bifurcating periodic solutions. Also, we deduce the exact condition to determine the Turing instability of the Hopf bifurcating periodic solutions for diffusive system. Numerical simulations are used to support our theoretical analysis.Bifurcation and stability of a reaction-diffusion-advection model with nonlocal delay effect and nonlinear boundary conditionhttps://zbmath.org/1541.350372024-09-27T17:47:02.548271Z"Li, Chaochao"https://zbmath.org/authors/?q=ai:li.chaochao"Guo, Shangjiang"https://zbmath.org/authors/?q=ai:guo.shangjiangSummary: In this paper, a reaction-diffusion-advection model with nonlocal delay effect and nonlinear boundary condition is investigated. By employing the Lyapunov-Schmidt reduction method, we not only establish the existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions, but also obtain some sufficient conditions ensuring the occurrence of a Hopf bifurcation at the steady-state solutions. It is observed that time delay determines the existence of Hopf bifurcation when the interior reaction term is stronger than the boundary reaction term. Finally, the general theoretical results are applied to a diffusive Logistic model with advection term under monostable nonlinear boundary condition and the effect of advection on Hopf bifurcation values is also considered. The results show that Hopf bifurcation is more likely to occur in the case of small advection.Cross-diffusion induced Turing instability of Hopf bifurcating periodic solutions in the reaction-diffusion enzyme reaction modelhttps://zbmath.org/1541.350382024-09-27T17:47:02.548271Z"Liu, Haicheng"https://zbmath.org/authors/?q=ai:liu.haicheng"Yuan, Wenshuo"https://zbmath.org/authors/?q=ai:yuan.wenshuo"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.bin"Shen, Jihong"https://zbmath.org/authors/?q=ai:shen.jihongSummary: Aiming at the spatial pattern phenomenon in biochemical reactions, an enzyme-reaction Sporns-Seelig model with cross-diffusion is chosen as study object. Applying the central manifold theory, normal form method, local Hopf bifurcation theorem and perturbation theory, we study Turing instability of the spatially homogeneous Hopf bifurcation periodic solutions. At last, the theoretical results are verified by numerical simulations.Turing-Hopf bifurcation analysis of the Sel'kov-Schnakenberg systemhttps://zbmath.org/1541.350392024-09-27T17:47:02.548271Z"Liu, Yuying"https://zbmath.org/authors/?q=ai:liu.yuying"Wei, Xin"https://zbmath.org/authors/?q=ai:wei.xin.2(no abstract)Classification of linear bifurcations of double-diffusive micropolar convection in two dimensionhttps://zbmath.org/1541.350402024-09-27T17:47:02.548271Z"Raesi, Behruz"https://zbmath.org/authors/?q=ai:raesi.behruzSummary: This paper is devoted to classifying linear bifurcations and the stability of the double-diffusive convective micropolar fluid in a two-dimensional horizontal domain at a rest state. We prove that the linear part of the operator is sectorial; and in the plane of thermal and saline Rayleigh numbers, for any fixed values of other parameters, the boundary of the region of stability comprises infinitely many segments of lines followed by finitely many segments of hyperbolas, patched together. Indeed, for any length scale, there is an infinite sequence of critical wave vectors, in which saddle-node bifurcations occur, and there is a finite sequence of critical wave vectors, in which Hopf bifurcations occur. The region of the stability and sequences of critical wave vectors is sensitive to length scale. It is shown that considering microinertia increases the marginal instability range in both fingering and double-diffusive convections.Global dynamics of a Leslie-Gower predator-prey model in open advective environmentshttps://zbmath.org/1541.350412024-09-27T17:47:02.548271Z"Zhang, Baifeng"https://zbmath.org/authors/?q=ai:zhang.baifeng"Zhang, Guohong"https://zbmath.org/authors/?q=ai:zhang.guohong"Wang, Xiaoli"https://zbmath.org/authors/?q=ai:wang.xiaoli.1In this article, the authors report on the global dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey model in open advective environments. The dynamics of the equation in an open advection environment are complex and interested. Some of their results reveal some significant differences with the classical specialist and generalist predator-prey systems. Especially, the authors find that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers. Overall, these conclusions are helpful to figure out the complex interation profiles among the predators and prey in advection environment.
Reviewer: Mengxin Chen (Xinxiang)Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas methodhttps://zbmath.org/1541.350422024-09-27T17:47:02.548271Z"Chatziafratis, Andreas"https://zbmath.org/authors/?q=ai:chatziafratis.andreas"Grafakos, Loukas"https://zbmath.org/authors/?q=ai:grafakos.loukas"Kamvissis, Spyridon"https://zbmath.org/authors/?q=ai:kamvissis.spyridonSummary: We study the inhomogeneous Airy partial differential equation (also called Stokes or linearized Korteweg-de Vries equation with a negative sign) on the half-line with generic initial and boundary data in a classical smooth setting, via the formula provided by the Fokas unified transform method for linear evolution equations. We first present a suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in it, thus securing convergence in a strict sense. Writing the solution in an Ehrenpreis-Palamodov form, our analysis allows for rigorous \textit{a posteriori} verification of the full initial-boundary-value problem and a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to prescribed values and the solution admits a smooth extension up to the boundary only under certain data compatibility conditions (with implications for well-posedness, control theory and efficient numerical computations). Importantly, based on this analysis, we perform an effective asymptotic study of far-field dynamics. This yields new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the `corner' of the quadrant. In particular, the asymptotic behavior of the solution is sensitive to perturbations of the data at the origin. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property at soon as time becomes positive. Hereby, we report on the discovery of a novel type of a long-range instability phenomenon for linear dispersive differential equations. Our ideas are extendable to other Airy-like and more general problems for dispersive evolution equations.A comparison of solutions of two convolution-type unidirectional wave equationshttps://zbmath.org/1541.350432024-09-27T17:47:02.548271Z"Erbay, H. A."https://zbmath.org/authors/?q=ai:erbay.husnuata-a"Erbay, S."https://zbmath.org/authors/?q=ai:erbay.saadet"Erkip, A."https://zbmath.org/authors/?q=ai:erkip.albert-kohen(no abstract)Linear stability of elastic \(2\)-line solitons for the KP-II equationhttps://zbmath.org/1541.350442024-09-27T17:47:02.548271Z"Mizumachi, Tetsu"https://zbmath.org/authors/?q=ai:mizumachi.tetsuSummary: The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of \(2\)-line solitons whose line solitons interact elastically each other. Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.Linear stability for a periodic tumor angiogenesis model with free boundary in the presence of inhibitorshttps://zbmath.org/1541.350452024-09-27T17:47:02.548271Z"Peng, Huiyan"https://zbmath.org/authors/?q=ai:peng.huiyan"Feng, Zhaoyong"https://zbmath.org/authors/?q=ai:feng.zhaoyong"Wei, Xuemei"https://zbmath.org/authors/?q=ai:wei.xuemeiSummary: In this paper, we research the issue of the free boundary of vascularized tumor growth using a T-periodic supply \(\psi (t)\) of outside nutrients and inhibitors. The model consists of two reaction diffusion equations, an elliptic equation and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor concentrations. The internal pressure distribution is described by the elliptic equation. The ODE describes the boundary value condition of the tumor model. After some meticulous mathematical analysis of the model system, we prove the existence and uniqueness of the radially symmetric T-periodic positive solution with \(\tilde{u} \leq \min\limits_{0 \leq t \leq T} \psi (t)\), where \(\tilde{u}\) is a parameter, denoting a threshold concentration for proliferation. Next, we further demonstrate the existence of a \(\mu_{\ast} >0\) such that \((u_{\ast}(r, t), v_{\ast}(r, t), p_{\ast}(r, t), R_{\ast}(t))\) is linearly stable for \(\mu < \mu_{\ast}\) and linearly unstable for \(\mu > \mu_{\ast}\) under perturbations that are not radially symmetric, where \(\mu\) is a constant, representing the ``intensity'' of mitosis-induced cell growth.Symmetry groupoids for pattern-selective feedback stabilization of the Chafee-Infante equationhttps://zbmath.org/1541.350462024-09-27T17:47:02.548271Z"Schneider, I."https://zbmath.org/authors/?q=ai:schneider.ingo|schneider.isabelle|schneider.ian-c|schneider.ivo-h"Dai, J.-Y."https://zbmath.org/authors/?q=ai:dai.jinyu|dai.jiayin|dai.jianyu|dai.jiangyan|dai.jia-yuan|dai.jingyi|dai.jianyun|dai.james-y|dai.jiayi|dai.jiyang|dai.jiayang|dai.jiongyu|dai.jingyu(no abstract)Early warning of tipping in a chemical model with cross-diffusion via spatiotemporal pattern formation and transitionhttps://zbmath.org/1541.350472024-09-27T17:47:02.548271Z"Lu, Yunxiang"https://zbmath.org/authors/?q=ai:lu.yunxiang"Xiao, Min"https://zbmath.org/authors/?q=ai:xiao.min"Huang, Chengdai"https://zbmath.org/authors/?q=ai:huang.chengdai"Cheng, Zunshui"https://zbmath.org/authors/?q=ai:cheng.zunshui"Wang, Zhengxin"https://zbmath.org/authors/?q=ai:wang.zhengxin"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde(no abstract)Asymptotics for a wave equation with critical exponential nonlinearityhttps://zbmath.org/1541.350482024-09-27T17:47:02.548271Z"Boudjeriou, Tahir"https://zbmath.org/authors/?q=ai:boudjeriou.tahir"Thin, Nguyen Van"https://zbmath.org/authors/?q=ai:thin.nguyen-vanSummary: In this paper, we discuss some qualitative analysis of solutions to the following Cauchy problem of wave equations involving the \(1 / 2\)-Laplace operator with critical exponential nonlinearity
\[
\begin{cases}
u_{t t} + (- \Delta)^{\frac{1}{2}} u + \delta u_t + u = \lambda | u |^{q - 2} u e^{\alpha_0 u^2} & \text{ in } \mathbb{R} \times (0, + \infty), \\
u (x, 0) = u_0 (x), u_t (x, 0) = u_1 (x) & \text{ in } \mathbb{R},
\end{cases}
\]
where \(\lambda > 0\), \(\delta \geq 0\), \(q > 2\), and \(\alpha_0 > 0\). By using the contraction mapping principle, we show that the above Cauchy problem has a unique local solution. With the help of the potential well argument, we characterize the stable sets by the asymptotic behavior of solutions as \(t\) goes to infinity, as well as the unstable sets by the blow-up of solutions in finite time.Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitionshttps://zbmath.org/1541.350492024-09-27T17:47:02.548271Z"Chekroun, Mickaël D."https://zbmath.org/authors/?q=ai:chekroun.mickael-d"Liu, Honghu"https://zbmath.org/authors/?q=ai:liu.honghu"McWilliams, James C."https://zbmath.org/authors/?q=ai:mcwilliams.james-c(no abstract)Asymptotics and scattering for massive Maxwell-Klein-Gordon equationshttps://zbmath.org/1541.350502024-09-27T17:47:02.548271Z"Chen, Xuantao"https://zbmath.org/authors/?q=ai:chen.xuantaoSummary: We study the asymptotic behavior and the scattering from infinity problem for the massive Maxwell-Klein-Gordon system in the Lorenz gauge, which were previously only studied for the massless system. For a general class of initial data, in particular of nonzero charge, we derive the precise asymptotic behaviors of the solution, where we get a logarithmic phase correction for the complex Klein-Gordon field, and a combination of interior homogeneous field, radiation fields, and an exterior charge part for the gauge potentials. Moreover, we also derive a formula for the charge at infinite time, which shows that the charge is concentrated at timelike infinity, a phenomenon drastically different from the massless case. After deriving the forward asymptotics, we formulate the scattering from infinity problem by defining the correct notion of scattering data, and then solve this problem. We show that one can determine the correct charge contribution using the information at timelike infinity, which is a crucial step for us to obtain backward solutions not only for the reduced equations in the Lorenz gauge but also for the original physical system.Exponential decay of solutions of damped wave equations in one dimensional space in the lp framework for various boundary conditionshttps://zbmath.org/1541.350512024-09-27T17:47:02.548271Z"Chitour, Yacine"https://zbmath.org/authors/?q=ai:chitour.yacine"Nguyen, Hoai-Minh"https://zbmath.org/authors/?q=ai:nguyen.hoai-minhSummary: We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.Asymptotic stability and the hair-trigger effect in Cauchy problem of the parabolic-parabolic Keller-Segel system with logistic sourcehttps://zbmath.org/1541.350522024-09-27T17:47:02.548271Z"De-Ji-Xiang-Mao"https://zbmath.org/authors/?q=ai:de-ji-xiang-mao."Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.15"Yin, Jingxue"https://zbmath.org/authors/?q=ai:yin.jingxueSummary: In this paper, we study the asymptotic stability and the hair-trigger effect for Cauchy problem of the following parabolic-parabolic Keller-Segel system with logistic term
\[
\begin{cases}
u_t = \Delta u-\chi \nabla\cdot\left ( u\nabla v \right )+u\left(a-b u\right)\quad & x\in \mathbb{R}^N,\, t>0,\\
\tau v_t = \Delta v+\lambda u-\mu v\quad & x\in \mathbb{R}^N,\, t>0,
\end{cases} \tag{1}
\]
where \(\chi, a, b, \lambda, \mu\) and \(\tau\) are positive constants and \(N\) is a positive integer. To this end, for small \(\chi \), we firstly obtain the global boundedness of solution by loop-argument based on \(L^p-L^q\) estimates of heat semigroup, with which we can further obtain the asymptotic stability of the positive constant equilibria in \(L^\infty(\mathbb{R}^N)\) for any initial data with positive lower bound. Moreover, for the special case \(\tau = 1 \), if \(\int_{B(x,\delta)}\ln u_0(s)ds\in L^\infty(\mathbb{R}^N)\) for some \(\delta>0 \), by constructing localized Lyapunov type functional, the solutions are shown to converge to the positive constant equilibria uniformly on any compact subset of \(\mathbb{R}^N \), which is known as the hair-trigger effect. Our contribution lies in the generalization of the results on asymptotic stability from the special case \(\tau = 1\) [\textit{W. Shen} and \textit{S. Xue}, Discrete Contin. Dyn. Syst. 42, No. 6, 2893--2925 (2022; Zbl 1495.35040)] to any \(\tau>0 \), and the generalization of classical results on hair-trigger effect for Fisher-KPP equation to Keller-Segel system.Linear decay property for the hyperbolic-parabolic coupled systems of thermoviscoelasticityhttps://zbmath.org/1541.350532024-09-27T17:47:02.548271Z"Dharmawardane, Priyanjana M. N."https://zbmath.org/authors/?q=ai:dharmawardane.priyanjana-m-n"Kawashima, Shuichi"https://zbmath.org/authors/?q=ai:kawashima.shuichi"Ogawa, Takayoshi"https://zbmath.org/authors/?q=ai:ogawa.takayoshi"Segata, Jun-ichi"https://zbmath.org/authors/?q=ai:segata.jun-ichiSummary: We consider the initial value problem of a linearized hyperbolic-parabolic coupled systems of thermoviscoelasticity around the constant state with arbitrary spatial dimensions \((n\geq1)\) and study the decay property of the systems. In the first place, we construct the fundamental solutions in Fourier space. Then we prove the decay property for the linearized systems provided that the initial data are in \(L^2\cap L^1\).Dynamics of a one-dimensional nonlinear poroelastic system weakly dampedhttps://zbmath.org/1541.350542024-09-27T17:47:02.548271Z"Dos Santos, Manoel"https://zbmath.org/authors/?q=ai:dos-santos.manoel-jeremias"Freitas, Mirelson"https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Ramos, Anderson"https://zbmath.org/authors/?q=ai:ramos.anderson-j-aSummary: In this paper, we study the long-time behavior of a nonlinear porous elasticity system. The system is subject to a viscoporous damping and a nonlinear source term which is locally Lipschitz and depends only on the volume fraction. The dynamical system associated with the solutions of the model is gradient, and under the hypothesis of equal speeds of propagation for the waves, we prove that it is also quasi-stable, which allows us to show the existence of a global attractor for the system, which is the main result of the paper.Stabilization of laminated beam with structural damping and a heat conduction of Gurtin-Pipkin's lawhttps://zbmath.org/1541.350552024-09-27T17:47:02.548271Z"Fayssal, Djellali"https://zbmath.org/authors/?q=ai:fayssal.djellali(no abstract)Precise asymptotic spreading behavior for an epidemic model with nonlocal dispersalhttps://zbmath.org/1541.350562024-09-27T17:47:02.548271Z"Guo, Jong-Shenq"https://zbmath.org/authors/?q=ai:guo.jong-shenq"Poh, Amy Ai Ling"https://zbmath.org/authors/?q=ai:poh.amy-ai-ling"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahikoSummary: This paper is to derive the precise asymptotic spreading behavior for an epidemic model with nonlocal dispersal. The proof is based on a Liouville type theorem on the positive bounded entire solutions. This Liouville theorem holds for a general class of reaction-diffusion systems with nonlocal dispersal which can be useful for reaction-diffusion systems arising in ecology and epidemiology.The wave equation on the Schwarzschild spacetime with small non-decaying first-order termshttps://zbmath.org/1541.350572024-09-27T17:47:02.548271Z"Holzegel, Gustav"https://zbmath.org/authors/?q=ai:holzegel.gustav"Kauffman, Christopher"https://zbmath.org/authors/?q=ai:kauffman.christopher-jSummary: In this paper, we present an elementary physical space argument to establish local integrated decay estimates for the perturbed wave equation \(\square_g\phi=\epsilon \beta^a \partial_a\phi\) on the exterior of the Schwarzschild geometry \((\mathcal{M},g)\). Here \(\beta\) is a regular vector field on \(\mathcal{M}\) decaying suitably in space but not necessarily in time. The proof is formulated to cover also perturbations of the Regge-Wheeler equation.Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorptionhttps://zbmath.org/1541.350582024-09-27T17:47:02.548271Z"Iagar, Razvan Gabriel"https://zbmath.org/authors/?q=ai:iagar.razvan-gabriel"Laurençot, Philippe"https://zbmath.org/authors/?q=ai:laurencot.philippe"Sánchez, Ariel"https://zbmath.org/authors/?q=ai:sanchez.arielSummary: We study the dynamics of the following porous medium equation with strong absorption
\[
\partial_tu=\Delta u^m-|x|^\sigma u^q,
\]
posed for \((t,x)\in(0,\infty)\times\mathbb{R}^N\), with \(m>1\), \(q\in(0,1)\) and \(\sigma>2(1-q)/(m-1)\). Considering the Cauchy problem with non-negative initial condition \(u_0\in L^\infty(\mathbb{R}^N)\), \textit{instantaneous shrinking} and \textit{localization of supports} for the solution \(u(t)\) at any \(t>0\) are established. With the help of this property, \textit{existence and uniqueness} of a non-negative compactly supported and radially symmetric \textit{forward self-similar solution} with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the \textit{pattern for large time behavior} of these general solutions.Limiting behavior of quasilinear wave equations with fractional-type dissipationhttps://zbmath.org/1541.350592024-09-27T17:47:02.548271Z"Kaltenbacher, Barbara"https://zbmath.org/authors/?q=ai:kaltenbacher.barbara"Meliani, Mostafa"https://zbmath.org/authors/?q=ai:meliani.mostafa"Nikolić, Vanja"https://zbmath.org/authors/?q=ai:nikolic.vanjaSummary: In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin-Pipkin type. Aiming at minimal assumptions on the involved memory kernels -- which we allow to be weakly singular -- we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.Well-posedness and exponential stability of Timoshenko system of second sound with time-varying delay and forcing termshttps://zbmath.org/1541.350602024-09-27T17:47:02.548271Z"Khaldi, Oussama"https://zbmath.org/authors/?q=ai:khaldi.oussama"Rahmoune, Abdelaziz"https://zbmath.org/authors/?q=ai:rahmoune.abdelaziz"Ouchenane, Djamel"https://zbmath.org/authors/?q=ai:ouchenane.djamel"Yazid, Fares"https://zbmath.org/authors/?q=ai:yazid.faresSummary: As a continuity to the study by \textit{D. Ouchenane} [Georgian Math. J. 21, No. 4, 475--489 (2014; Zbl 1304.35103)]. We consider a nonlinear thermoelastic system of Timoshenko type with a time-varing delay and forcing term. We show the well-posedness of the system by using the semigroup theory, and we prove an exponential stability result under the usual assumption on the wave speed by the energy method.Unbounded Sturm attractors for quasilinear parabolic equationshttps://zbmath.org/1541.350612024-09-27T17:47:02.548271Z"Lappicy, Phillipo"https://zbmath.org/authors/?q=ai:lappicy.phillipo"Fernandes, Juliana"https://zbmath.org/authors/?q=ai:fernandes.julianaSummary: We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.Dynamics of locally monotone stochastic evolution equationshttps://zbmath.org/1541.350622024-09-27T17:47:02.548271Z"Liu, Chunjie"https://zbmath.org/authors/?q=ai:liu.chunjie"Gu, Anhui"https://zbmath.org/authors/?q=ai:gu.anhuiSummary: This paper is concerned with the dynamics of the locally monotone stochastic evolution equations in Bochner spaces. We first prove the well-posedness of the abstract stochastic evolution equations with locally Lipschitz nonlinear diffusion terms. Then we establish a mean random dynamical system and prove the existence and uniqueness of weak pullback mean random attractors generated by the mean random dynamical system. Furthermore, we prove the existence of invariant measures for the locally monotone model. As applications, the related dynamics for several stochastic models such as stochastic semilinear equations, stochastic Burgers equation and stochastic 2D Navier-Stokes equations are established.Existence and uniqueness of random nonlocal differential equations with colored noisehttps://zbmath.org/1541.350632024-09-27T17:47:02.548271Z"Liu, Ruonan"https://zbmath.org/authors/?q=ai:liu.ruonanSummary: The existence and uniqueness of one kind of random differential equations with a Caputo fractional time derivative driven by colored noise are investigated by using approximating sequences. To this end, a generalized Gronwall inequality containing singular kernel which is first proved as an auxiliary tool to handle random time fractional differential equations, which is based on the ideas mentioned in [\textit{S. S. Dragomir}, Some Gronwall type inequalities and applications. Hauppauge, NY: Nova Science Publishers (2003; Zbl 1094.34001)].On the stability and asymptotic behavior for a quasi-linear parabolic flowhttps://zbmath.org/1541.350642024-09-27T17:47:02.548271Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li"Shi, Zizhen"https://zbmath.org/authors/?q=ai:shi.zizhenSummary: In this paper, we study the stability property and asymptotic behavior for a quasi-linear parabolic flow in the whole line. We first show the existence and uniqueness of global solutions of the problem. Then we study the stability of the solution to the straight line. We prove the asymptotic behavior or the convergence of the global solution. Similar to the behavior of solutions to heat equation, we prove that the stationary line attracts the graphical curves which surround it.On the doubly non-local Hele-Shaw-Cahn-Hilliard system: derivation and \(2D\) well-posednesshttps://zbmath.org/1541.350652024-09-27T17:47:02.548271Z"Peter, Malte A."https://zbmath.org/authors/?q=ai:peter.malte-andreas"Woukeng, Jean Louis"https://zbmath.org/authors/?q=ai:woukeng.jean-louisThis paper is concerning the flow of two incompressible immiscible fluids in a thin highly heterogeneous Hele-Shaw cell. The basic model is the nonlocal Cahn-Hilliard-Navier-Stokes system (at the microscale). The sigma-convergence theory is used, previously obtained as a generalization of two-scale convergence for thin periodic structures -- see [\textit{W. Jäger} and \textit{J. L. Woukeng}, ``Sigma-convergence for thin heterogeneous domains and application to the upscaling of Darcy-Lapwood-Brinkman flow'', Preprint, \url{arXiv:2309.09004}; \textit{M. Neuss-Radu} and \textit{W. Jäger}, SIAM J. Math. Anal. 39, No. 3, 687--720 (2007; Zbl 1145.35017)]. The new element of this paper is the obtained upscaled model. Existence results for the considered system and uniform estimates are given in Section 2, by using the Wiener amalgam and some local Sobolev spaces. Fundamentals of algebras with mean value are used to give very interesting and important elements concerning sigma-convergence for thin heterogeneous domains, in Section 3. Useful results for the homogenization process and the homogenized system are given in Section 4. The obtained ``homogenized system'' (4.35) is a Hele-Shaw equation with memory, which can be also used for transient flow modelling the tumours growth. To the best of the authors knowledge ``this is the first time that such a system is obtained in the literature''. The analysis of the homogenized system (4.35) is given in Section 5, with important assumptions concerning the regularity of the coefficients appearing in (1.2). Some results of \textit{F. Della Porta} and \textit{M. Grasselli} [Commun. Pure Appl. Anal. 15, No. 2, 299--317 (2016; Zbl 1334.35226)] are also used. Important tools are the Gagliardo-Nirenberg inequality, the Laplace transform in some particular spaces and the Moser-type iteration. Very interesting examples are given in the last section. Periodic, almost periodic and perturbed periodic microstructures are analyzed.
Reviewer: Gelu Paşa (Bucureşti)Blow-up and decay for a pseudo-parabolic equation with nonstandard growth conditionshttps://zbmath.org/1541.350662024-09-27T17:47:02.548271Z"Quach Van Chuong"https://zbmath.org/authors/?q=ai:quach-van-chuong."Le Cong Nhan"https://zbmath.org/authors/?q=ai:le-cong-nhan."Le Xuan Truong"https://zbmath.org/authors/?q=ai:le-xuan-truong.Summary: This paper deals with a pseudo-parabolic equation involving variable exponents under homogeneous Dirichlet boundary value condition. The authors first develop the potential well method to prove a threshold result on the existence or nonexistence of global solutions to the equation when the initial energy is less than the mountain pass level \(d\). In the case of high energy initial data, a new characterization for the nonexistence of global solution is also given. These results extend and improve some recent results obtained by \textit{H. Di} et al. [Appl. Math. Lett. 64, 67--73 (2017; Zbl 1353.35072)].On the stability of the swelling porous elastic soils with fluid saturation and Gurtin-Pipkin thermal lawhttps://zbmath.org/1541.350672024-09-27T17:47:02.548271Z"Ramos, A. J. A."https://zbmath.org/authors/?q=ai:ramos.anderson-j-a"Nonato, C. A."https://zbmath.org/authors/?q=ai:nonato.carlos-a-s|nonato.carlos-alberto"Raposo, C. A."https://zbmath.org/authors/?q=ai:raposo.carlos-alberto"Freitas, M. M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Coayla-Teran, E. A."https://zbmath.org/authors/?q=ai:coayla-teran.edson-albertoSummary: The present paper is devoted to studying the well-posedness and exponential stability of the one-dimensional system in the linear isothermal theory of swelling porous elastic soils with fluid saturation and Gurtin-Pipkin thermal law. For the well-posedness, we apply the well-known Hille-Yosida theorem of semigroup theory. To prove exponential stability without assuming that the wave speeds are the same, we use the energy method which consists of constructing a Lyapunov functional equivalent to the system's total energy.Long-time behavior for fourth order nonlinear wave equations with dissipative and dispersive termshttps://zbmath.org/1541.350682024-09-27T17:47:02.548271Z"Wang, Xingchang"https://zbmath.org/authors/?q=ai:wang.xingchang"Xu, Runzhang"https://zbmath.org/authors/?q=ai:xu.runzhang"Yang, Yanbing"https://zbmath.org/authors/?q=ai:yang.yanbingSummary: The initial boundary value problem for a class of fourth-order nonlinear wave equations with dissipative terms and dispersive terms is considered in this paper. We first establish the global existence and the exponential decay of the solution. Next, based on Condition (C) and semigroup theory arguments, the existence of the global attractor is derived.Global dynamics of a competition-diffusion-advection system with general boundary conditionshttps://zbmath.org/1541.350692024-09-27T17:47:02.548271Z"Wei, Jinyu"https://zbmath.org/authors/?q=ai:wei.jinyu"Liu, Bin"https://zbmath.org/authors/?q=ai:liu.bin.6Summary: We study a general Lotka-Volterra competition-diffusion-advection system with general boundary conditions from river ecology. A complete classification on all possible long-time dynamical behaviors is established. Moreover, we investigate the joint effects of diffusion rates, advection rates, the inter-specific competition intensities and boundary conditions on global dynamics of the system. Finally, several numerical simulations are performed to verify the theoretical results. These results improve previously known ones by removing one condition and considering an interesting boundary condition where the species can be exposed to a net loss of individuals.Boundedness and large time behavior for flux limitation in a two-species chemotaxis systemhttps://zbmath.org/1541.350702024-09-27T17:47:02.548271Z"Wu, Chun"https://zbmath.org/authors/?q=ai:wu.chun"Huang, Xiaojie"https://zbmath.org/authors/?q=ai:huang.xiaojieThis paper examines a two-species chemotaxis cross-diffusion system in a bounded domain \(\Omega \subset \mathbb{R}^n\) where \( n \geq 2\) with smooth boundary \(\partial \omega.\) Under the condition that \(\alpha > \displaystyle \frac{2n-nm-2}{2(n-1)}\) and \(m \geq 1,\) it is shown that the problem possesses a global bounded classical solution. Moreover, this paper also investigates the large time behavior of the solution, and obtains the corresponding solution exponentially converges to a constant stationary solution when the initial data is sufficiently small.
Reviewer: Lingeshwaran Shangerganesh (Ponda)Quantitative rapid and finite time stabilization of the heat equationhttps://zbmath.org/1541.350712024-09-27T17:47:02.548271Z"Xiang, Shengquan"https://zbmath.org/authors/?q=ai:xiang.shengquanSummary: The finite time stabilizability of the one dimensional heat equation is proved by \textit{J.-M. Coron} and \textit{H.-M. Nguyen} [Arch. Ration. Mech. Anal. 225, No. 3, 993--1023 (2017; Zbl 1417.93067)], while the same question for multidimensional spaces remained open. Inspired by \textit{J.-M. Coron} and \textit{E. Trélat} [SIAM J. Control Optim. 43, No. 2, 549--569 (2004; Zbl 1101.93011)] we introduce a new method to stabilize multidimensional heat equations quantitatively in finite time and call it Frequency Lyapunov method. This method naturally combines spectral inequality [\textit{G. Lebeau} and \textit{L. Robbiano}, Commun. Partial Differ. Equations 20, No. 1--2, 335--356 (1995; Zbl 0819.35071)] and constructive feedback stabilization. As application this approach also yields a constructive proof for null controllability, which gives sharing optimal cost \(Ce^{C/T}\) with explicit controls and works perfectly for related nonlinear models such as Navier-Stokes equations [\textit{S. Xiang}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 40, No. 6, 1487--1511 (2023; Zbl 1526.35264)].Boundedness and stability for an indirect signal absorption chemotaxis system with signal-dependent motilityhttps://zbmath.org/1541.350722024-09-27T17:47:02.548271Z"Xu, Lu"https://zbmath.org/authors/?q=ai:xu.lu"Mu, Chunlai"https://zbmath.org/authors/?q=ai:mu.chunlai"Xin, Qiao"https://zbmath.org/authors/?q=ai:xin.qiaoSummary: The paper is concerned with a chemotaxis system with indirect signal absorption and signal-dependent motility
\[
\begin{cases}
u_t = \Delta(\gamma(v)u), \quad & x \in \Omega, t>0,\\
v_t = \Delta v - vw, & x\in \Omega, t>0,\\
\tau w_t = -\delta w+u, & x\in \Omega, t>0,
\end{cases}
\]
under the homogeneous Neumann boundary condition, where \(\Omega\subset R^n\) (\(n \geq 1\)), parameters \(\delta,\tau>0\). \(\gamma(v)\) is a motility function. It is shown that in n-dimensional smooth bounded domains with \(n\leq2\) or \(n\geq3\) and suitable initial data, some appropriate assumptions on \(\gamma(v)\) ensure global existence and stability of a bounded classical solution.Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profilehttps://zbmath.org/1541.350732024-09-27T17:47:02.548271Z"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.51|yang.fan.14|yang.fan.30|yang.fan.33|yang.fan.28|yang.fan.31|yang.fan.34|yang.fan.29|yang.fan.32|yang.fan.2"Yang, Xiongfeng"https://zbmath.org/authors/?q=ai:yang.xiongfengSummary: In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier-Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in \(L^p\) (\( 2 \leq p \leq \infty \)) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.General decay and blow-up of solution for a viscoelastic wave equation with nonlinear degenerate damping and source termshttps://zbmath.org/1541.350742024-09-27T17:47:02.548271Z"Yin, Ziyang"https://zbmath.org/authors/?q=ai:yin.ziyang"Wang, Shubin"https://zbmath.org/authors/?q=ai:wang.shubinSummary: We are concerned with the initial-boundary value problem for a nonlinear viscoelastic problem with the degenerate damping and the nonlinear source term in this paper. Under appropriate assumptions on the initial data, memory kernel and parameters, we establish several results regarding the global solution, blow-up properties and general decay in the case of positive initial energy. We overcome the difficulties brought by positive initial energy by using the potential well theory.Existence, uniqueness and asymptotic behavior of solutions for a nonsmooth producer-grazer system with stoichiometric constraintshttps://zbmath.org/1541.350752024-09-27T17:47:02.548271Z"Zhang, Conghui"https://zbmath.org/authors/?q=ai:zhang.conghui"Zhang, Haifeng"https://zbmath.org/authors/?q=ai:zhang.haifeng"Li, Shanbing"https://zbmath.org/authors/?q=ai:li.shanbing(no abstract)Convergence problem of the generalized Kadomtsev-Petviashvili II equation in anisotropic Sobolev spacehttps://zbmath.org/1541.350762024-09-27T17:47:02.548271Z"Zhang, Qiaoqiao"https://zbmath.org/authors/?q=ai:zhang.qiaoqiao"Yang, Meihua"https://zbmath.org/authors/?q=ai:yang.meihua"Xu, Haoyuan"https://zbmath.org/authors/?q=ai:xu.haoyuan"Yan, Wei"https://zbmath.org/authors/?q=ai:yan.weiSummary: The almost everywhere pointwise and uniform convergences for the generalized KP-II equation are investigated when the initial data is in anisotropic Sobolev space \(H^{s_1, s_2}(\mathbb{R}^2)\). Firstly, we show that the solution \(u(x, y, t)\) converges pointwisely to the initial data \(f(x, y)\in H^{s_1, s_2}(\mathbb{R}^2)\) for a.e. \((x, y) \in\mathbb{R}^2\) when \(s_1 \geq \frac{1}{4}\), \(s_2 \geq \frac{1}{4}\). The proof relies upon the Strichartz estimate and high-low frequency decomposition. Secondly, We prove that \(s_1 \geq \frac{1}{4}\), \(s_2 \geq \frac{1}{4}\) is a necessary condition for the maximal function estimate of the generalized KP-II equation to hold. Finally, by using the Fourier restriction norm method, we establish the nonlinear smoothing estimate to show the uniform convergence of the generalized KP-II equation in \(H^{s_1, s_2}(\mathbb{R}^2)\) with \(s_1 \geq \frac{3}{2} - \frac{\alpha}{4} + \epsilon\), \(s_2 > \frac{1}{2}\) and \(\alpha \geq 4\).Existence of polynomial attractor for a class of extensible beams with nonlocal weak dampinghttps://zbmath.org/1541.350772024-09-27T17:47:02.548271Z"Zhao, Chunxiang"https://zbmath.org/authors/?q=ai:zhao.chunxiang"Zhao, Chunyan"https://zbmath.org/authors/?q=ai:zhao.chunyan"Zhong, Chengkui"https://zbmath.org/authors/?q=ai:zhong.chengkuiSummary: In this paper, we put forward the concept of polynomial attractor and study the connection between the polynomial attractors and the estimate of attractive velocity of bounded sets for infinite-dimensional dynamical systems. Then we prove the existence of polynomial attractor for a class of extensible beams with nonlocal weak damping for the case that the nonlinear term \(f\) has subcritical growth.New energy decay for a nonlinearly damped system of wave equationshttps://zbmath.org/1541.350782024-09-27T17:47:02.548271Z"Zhu, Xiangyu"https://zbmath.org/authors/?q=ai:zhu.xiangyu"Liao, Menglan"https://zbmath.org/authors/?q=ai:liao.menglanSummary: In this paper, a nonlinearly damped system of wave equations is considered. Uniform energy decay was discussed in the previous work [\textit{C. O. Alves} et al., Discrete Contin. Dyn. Syst., Ser. S 2, No. 3, 583--608 (2009; Zbl 1183.35033)] for \(m\), \(r\in [1,5]\) if the space dimension is 3. New energy decay is proposed for \(m\), \(r\in (5,+\infty)\) by choosing appropriate multiplier related to a non-increasing differential function. As an example, a logarithmic energy decay is also presented.
{\copyright} 2024 Wiley-VCH GmbH.Energy decay analysis for porous elastic system with microtemperature: classical vs second spectrum approachhttps://zbmath.org/1541.350792024-09-27T17:47:02.548271Z"Zougheib, Hamza"https://zbmath.org/authors/?q=ai:zougheib.hamza"El Arwadi, Toufic"https://zbmath.org/authors/?q=ai:el-arwadi.toufic"El-Hindi, Mohammad"https://zbmath.org/authors/?q=ai:el-hindi.mohammad"Soufyane, Abdelaziz"https://zbmath.org/authors/?q=ai:soufyane.abdelazizSummary: The stability features of the dissipative porous elastic systems have piqued the interest of several researchers. The desired exponential decay property of the energy is obtained unless the nonphysical equal speed condition is imposed. This work analyzes the porous elastic system with micro-temperature. First, the exponential stability is obtained in case where there is an assumption on physical constants. Then from a second-spectrum viewpoint, the system's global well-posedness is proved using the Faedo-Galerkin method. Later, we prove that the microtemperature effect is enough to get the exponential stability of the solution without any assumption on the physical constants. A numerical scheme is introduced. Finally, we present some numerical results which demonstrates the exponential behavior of the solution.Uniform dynamics of partially damped semilinear Bresse systemshttps://zbmath.org/1541.350802024-09-27T17:47:02.548271Z"Araújo, Rawlilson O."https://zbmath.org/authors/?q=ai:araujo.rawlilson-o"Ma, To Fu"https://zbmath.org/authors/?q=ai:ma.to-fu"Marinho, Sheyla S."https://zbmath.org/authors/?q=ai:marinho.sheyla-s"Seminario-Huertas, Paulo N."https://zbmath.org/authors/?q=ai:seminario-huertas.paulo-nicanor(no abstract)Attractors of Ginzburg-Landau equations with oscillating terms in porous media: homogenization procedurehttps://zbmath.org/1541.350812024-09-27T17:47:02.548271Z"Bekmaganbetov, Kuanysh A."https://zbmath.org/authors/?q=ai:bekmaganbetov.kuanysh-abdrakhmanovich"Chechkin, Gregory A."https://zbmath.org/authors/?q=ai:chechkin.gregory-a"Tolemis, Abylaikhan A."https://zbmath.org/authors/?q=ai:tolemis.abylaikhan-a(no abstract)On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growthhttps://zbmath.org/1541.350822024-09-27T17:47:02.548271Z"Bortolan, Matheus Cheque"https://zbmath.org/authors/?q=ai:bortolan.m-c"Garcìa-Fuentes, Juan"https://zbmath.org/authors/?q=ai:garcia-fuentes.juan"Fernandes, Juliana"https://zbmath.org/authors/?q=ai:fernandes.juliana"Kalita, Piotr"https://zbmath.org/authors/?q=ai:kalita.piotrSummary: We introduce the concept of a \textit{maximal} \(B\)-\textit{attractor} for a semigroup, and find conditions under which a semigroup in a metric space possess a maximal \(B\)-attractor. We also establish a condition under which a maximal \(B\)-attractor is also a maximal attractor, in the sense of \textit{V. V. Chepyzhov} and \textit{A. Yu. Goritskij} [Adv. Sov. Math. 10, 85--128 (1992; Zbl 0768.58029)]. Finally, we apply our result to a parabolic semilinear PDE, where the nonlinearity can be unbounded, as long as it grows linearly with a controlled growth constant, and find a maximal \(B\)-attractor for it. Furthermore, under a Lipschitz condition on the nonlinearity, with small Lipschitz constant, we prove that this maximal \(B\)-attractor is, in fact, a maximal attractor.Design of Sturm global attractors. I: Meanders with three noses, and reversibilityhttps://zbmath.org/1541.350832024-09-27T17:47:02.548271Z"Fiedler, Bernold"https://zbmath.org/authors/?q=ai:fiedler.bernold"Rocha, Carlos"https://zbmath.org/authors/?q=ai:rocha.carlos(no abstract)Attractors for partially damped systems of binary mixtures of solidshttps://zbmath.org/1541.350842024-09-27T17:47:02.548271Z"Freitas, Mirelson M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Apalara, Tijani A."https://zbmath.org/authors/?q=ai:apalara.tijani-abdulaziz"Ramos, Anderson J. A."https://zbmath.org/authors/?q=ai:ramos.anderson-j-a"Costa, Rafael S."https://zbmath.org/authors/?q=ai:costa.rafael-s(no abstract)Existence of global attractors and convergence of solutions for the Cahn-Hilliard equation on manifolds with conical singularitieshttps://zbmath.org/1541.350852024-09-27T17:47:02.548271Z"Lopes, Pedro T. P."https://zbmath.org/authors/?q=ai:lopes.pedro-t-p"Roidos, Nikolaos"https://zbmath.org/authors/?q=ai:roidos.nikolaosSummary: We consider the Cahn-Hilliard equation on manifolds with conical singularities and prove existence of global attractors in higher order Mellin-Sobolev spaces with asymptotics. We also show convergence of solutions in the same spaces to an equilibrium point and provide asymptotic behavior of the equilibrium near the conical tips in terms of the local geometry.The index theory for multivalued dynamical systems with applications to reaction-diffusion equations with discontinuous nonlinearityhttps://zbmath.org/1541.350862024-09-27T17:47:02.548271Z"Moreira, Estefani M."https://zbmath.org/authors/?q=ai:moreira.estefani-m"Valero, José"https://zbmath.org/authors/?q=ai:valero.joseSummary: In this paper, we develop first a theory of existence of index pairs for multivalued semiflows. We apply this result to a reaction-diffusion equation having a discontinuous nonlinearity which gives rise to a differential inclusion governed by the Heaviside function. Second, we prove that, in a suitable regular phase space, the Conley index of the non-zero fixed points of this inclusion is a pointed sphere.The existence and dimension of the attractor for a 3D flow of a non-Newtonian fluid subject to dynamic boundary conditionshttps://zbmath.org/1541.350872024-09-27T17:47:02.548271Z"Pražák, Dalibor"https://zbmath.org/authors/?q=ai:prazak.dalibor"Priyasad, Buddhika"https://zbmath.org/authors/?q=ai:priyasad.buddhika(no abstract)Dynamics of parabolic equations in domains with a small hole. II: Continuity of the attractorshttps://zbmath.org/1541.350882024-09-27T17:47:02.548271Z"Tavares-Lima, E. A."https://zbmath.org/authors/?q=ai:tavares-lima.e-a"Lozada-Cruz, G."https://zbmath.org/authors/?q=ai:lozada-cruz.germanSummary: In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter \(\varepsilon \). In other words, we prove that the family of attractors behaves continuously as \(\varepsilon \to 0\). We will also provide the convergence rates in terms of the parameter.Uniform attractors for wave equations with critical and nonautonomous nonlinearityhttps://zbmath.org/1541.350892024-09-27T17:47:02.548271Z"Zhu, Xiangming"https://zbmath.org/authors/?q=ai:zhu.xiangmingSummary: Existence and structure of uniform attractors for wave equations with a time-dependent and critical nonlinearity are considered in this paper. Firstly, the well-posedness of the equation with an appropriately defined symbol space is presented. Then the existence and structure of uniform attractors are proved by showing the uniformly asymptotical compactness of the processes via energy method.Qualitative results for a generalized 2-component Camassa-Holm system with weak dissipation termhttps://zbmath.org/1541.350902024-09-27T17:47:02.548271Z"Dündar, Nurhan"https://zbmath.org/authors/?q=ai:dundar.nurhanSummary: Our main aim in the current study is to examine the mathematical properties of a generalized 2-component Camassa-Holm system with a weakly dissipative term. Firstly, we acquire the theorem of well-posedness in locally for the generalized system with weak dissipation. Then, we demonstrate that this system can reveal the blow-up phenomenon. Finally, we acquire the theorem of global existence utilizing a method of the Lyapunov function.Construction of \(L^2\) log-log blowup solutions for the mass critical nonlinear Schrödinger equationhttps://zbmath.org/1541.350912024-09-27T17:47:02.548271Z"Fan, Chenjie"https://zbmath.org/authors/?q=ai:fan.chenjie"Mendelson, Dana"https://zbmath.org/authors/?q=ai:mendelson.danaSummary: In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrödinger equation on \(\mathbb{R}^2\) under rough but structured random perturbations at \(L^2(\mathbb{R}^2)\) regularity. In particular, by employing probabilistic methods, we provide a construction of a family of \(L^2(\mathbb{R}^2)\) regularity solutions which do not lie in \(H^s(\mathbb{R}^2)\) for any \(s > 0\), and which blowup according to the log-log dynamics.Lifespan estimates for a special quasilinear time-dependent damped wave equationhttps://zbmath.org/1541.350922024-09-27T17:47:02.548271Z"Girardi, Giovanni"https://zbmath.org/authors/?q=ai:girardi.giovanni"Lucente, Sandra"https://zbmath.org/authors/?q=ai:lucente.sandraSummary: In this paper we consider a quasilinear Cauchy problem for the scale invariant damped wave equation
\[
v_{tt}-\Delta v+\frac{\mu}{(1+t)}v_t+\frac{\mu}{2} \left (\frac{\mu}{2}-1 \right)\frac{v}{(1+t)^2} =\left | \frac{\mu}{2(1+t)}v+v_t\right |{}^p,
\]
with \(\mu \geq 0\), \(v = v(t, x)\) and \(x\in \mathbb{R}^n\). The particular structure of the nonlinear term, guarantees a blow up result and a lifespan estimate, assuming radial initial data having slow decay. In particular the range of admissible exponents \(p\) depends on \(\mu\), \(n\) and the rate of the initial data decay.
For the entire collection see [Zbl 1497.42002].Existence and blow-up study of a quasilinear wave equation with damping and source terms of variable exponents-type acting on the boundaryhttps://zbmath.org/1541.350932024-09-27T17:47:02.548271Z"Kafini, Mohammad"https://zbmath.org/authors/?q=ai:kafini.mohammad-mustafa"Al-Gharabli, Mohammad M."https://zbmath.org/authors/?q=ai:algharabli.mohammad-m"Al-Mahdi, Adel M."https://zbmath.org/authors/?q=ai:al-mahdi.adel-mSummary: In this work, we are concerned with a quasilinear wave equation with nonlinear damping and source terms of variable exponents-type acting in a part of the boundary. Under suitable conditions on the exponents and the initial data, we study the blow-up properties. Firstly, by using Faedo-Galerkin method and Banach-Fixed-Point Theorem, we establish the existence of a weak solution, under suitable assumptions on the variable exponents and the initial data. Secondly, we show a finite time blow-up with lower and upper bound as well. Next, an infinite time blow-up is proved under some conditions in the exponents and the initial data.Finite-time blow-up of weak solutions to a chemotaxis system with gradient dependent chemotactic sensitivityhttps://zbmath.org/1541.350942024-09-27T17:47:02.548271Z"Kohatsu, Shohei"https://zbmath.org/authors/?q=ai:kohatsu.shoheiSummary: This paper deals with the parabolic-elliptic chemotaxis system with gradient dependent chemotactic sensitivity,
\[
\begin{cases}
u_t = \Delta u-\chi \nabla\cdot (u|\nabla v|^{p-2} \nabla v), & x \in \Omega,\ t>0, \\
0 = \Delta v-\mu +u, & x \in \Omega,\ t>0,
\end{cases}
\]
where \(\Omega : = B_R (0) \subset \mathbb{R}^n\) (\(n\geq 2\)) is a ball with some \(R>0\), \(\chi >0\), \(p \in (\frac{n}{n-1}, \infty)\) and \(\mu := \frac{1}{|\Omega|} \int_{\Omega}u_0\), where \(u_0\) is an initial datum of arbitrary size. In the case that \(p \in (1, \frac{n}{n-1})\), \textit{M. Negreanu} and \textit{J. I. Tello} [J. Differ. Equations 265, No. 3, 733--751 (2018; Zbl 1391.35186)] established global existence and uniform boundedness of solutions, whereas when \(p \in (\frac{n}{n-1}, 2)\), \textit{J. I. Tello} [Commun. Partial Differ. Equations 47, No. 2, 307--345 (2022; Zbl 1491.35071)] showed that solutions blow up in finite time under the condition that \(\mu > 6\) and \(\chi\) is large enough. These works imply that the number \(p = \frac{n}{n-1}\) certainly plays the role of a critical blow-up exponent, and it is expected that when \(p>\frac{n}{n-1}\), for arbitrary \(\mu > 0\) the system admits at least one solution which blows up in finite time. The purpose of this paper is to prove that this conjecture is true within a framework of weak solutions with a moment inequality.On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theoryhttps://zbmath.org/1541.350952024-09-27T17:47:02.548271Z"Korpusov, M. O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Shlyapugin, G. I."https://zbmath.org/authors/?q=ai:shlyapugin.g-iSummary: In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities \({\left|\nabla u\right|}^q\), \({\partial }_t{\left|\nabla u\right|}^q,\) and \({{\partial }_t^2\left|\nabla u\right|}^q\) are considered. For the corresponding Cauchy problems, we obtain results on local-in-time unique solvability in the weak sense and on blow-up for a finite time. These three equations have the same critical exponent \(q = 3/2\) since weak solutions behave differently for \(1 < q \leq 3/2\) and for \(q > 3/2\). By the method of nonlinear capacity proposed by S. I. Pokhozhaev, we obtain a priori estimates, which imply the absence of local and global weak solutions.Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populationshttps://zbmath.org/1541.350962024-09-27T17:47:02.548271Z"Lin, Ke"https://zbmath.org/authors/?q=ai:lin.ke"Zeng, Rong"https://zbmath.org/authors/?q=ai:zeng.rongSummary: This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions \(d \geq 3\). We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.Nonexistence of solutions for a higher-order wave equation with delay and variable-exponentshttps://zbmath.org/1541.350972024-09-27T17:47:02.548271Z"Pişkin, Erhan"https://zbmath.org/authors/?q=ai:piskin.erhan"Yüksekkaya, Hazal"https://zbmath.org/authors/?q=ai:yuksekkaya.hazalSummary: In this article, we deal with a higher-order wave equation with delay term and variable exponents. Under suitable conditions, we prove the nonexistence of solutions in a finite time. Generally, the problems with variable exponents arise in many branches in sciences such as nonlinear elasticity theory, electrorheological fluids and image processing. Time delays often appear in many practical problems such as thermal, biological, chemical, physical and economic phenomena.
For the entire collection see [Zbl 1521.76009].Lower and upper bounds for the explosion times of a system of semilinear SPDEshttps://zbmath.org/1541.350982024-09-27T17:47:02.548271Z"Sankar, S."https://zbmath.org/authors/?q=ai:sankar.s-murali|sankar.sriram|sankar.s-uma|sankar.s-ravi|sankar.sridhar|sankar.sagi|sankar.soumya"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-t"Karthikeyan, S."https://zbmath.org/authors/?q=ai:karthikeyan.subramaniyam|karthikeyan.s-s|karthikeyan.s|karthikeyan.shanmugasundaram|karthikeyan.selvarajSummary: In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.Blow-up time analysis of parabolic equations with variable nonlinearitieshttps://zbmath.org/1541.350992024-09-27T17:47:02.548271Z"Soufiane, Benkouider"https://zbmath.org/authors/?q=ai:soufiane.benkouider"Abita, Rahmoune"https://zbmath.org/authors/?q=ai:abita.rahmoune(no abstract)Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa-Holm equationhttps://zbmath.org/1541.351002024-09-27T17:47:02.548271Z"Wan, Zhenyu"https://zbmath.org/authors/?q=ai:wan.zhenyu"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.2|wang.ying.23|wang.ying.35|wang.ying.19|wang.ying.16|wang.ying.36|wang.ying.8|wang.ying.31|wang.ying|wang.ying.42|wang.ying.12|wang.ying.9|wang.ying.38|wang.ying.53"Zhu, Min"https://zbmath.org/authors/?q=ai:zhu.min|zhu.min.4|zhu.min.2|zhu.min.3|zhu.min.1Summary: This paper aims to establish a mechanism for the blow-up on a class of weakly dissipative shallow-water equations with a vavariable dipersion term, which is related to the integrable systems: the Camassa-Holm equation and the Novikov equation. Our blow-up analysis commences with the consideration of two cases. In the first case, the linear dispersion parameter is \(\gamma \in\mathbb{R}\), while in the second case, \(\gamma\) is equal to zero. The approach is to extract the true blow-up component and instead trace its dynamics to ensures the occurrence of wave-breaking in finite time before the other component undergoes degeneration. To address the issue of non-conservation in the previous functional which is caused by weak linear dispersion and the loss of the conservation law \(\mathcal{H}_1[u] = \int_{\mathbb{R}}(u^2 + u_x^2) \operatorname{d}x\) due to the presence of a weakly dissipative term, we propose alternative methods. These methods include making a modified functional \(J(t)\) (see Lemma 3.2) and establishing an energy inequality. Moreover, we investigate the formation of singularities by tracing the whole blow-up quantity. Lastly, we examine the Gevrey regularity and analyticity of solutions to the system in the Gevrey-Sobolev spaces by utilizing the generalized Ovsyannikov theorem and show the continuity of the data-to-solution mapping.The critical mass curve and chemotactic collapse of a two-species chemotaxis system with two chemicalshttps://zbmath.org/1541.351012024-09-27T17:47:02.548271Z"Yu, Hao"https://zbmath.org/authors/?q=ai:yu.hao.2|yu.hao.1|yu.hao|yu.hao.4|yu.hao.3"Xue, Bingqian"https://zbmath.org/authors/?q=ai:xue.bingqian"Hu, YinYin"https://zbmath.org/authors/?q=ai:hu.yinyin"Zhao, Lifen"https://zbmath.org/authors/?q=ai:zhao.lifenSummary: This paper considers the following two-species chemotaxis system with two chemicals
\[
\begin{cases}
u_t = \Delta u - \nabla \cdot (u \nabla v), \quad & x \in \Omega, t > 0, \\
0 = \Delta v - \alpha v + w, & x \in \Omega, t > 0, \\
w_t = \Delta w - \nabla \cdot (w \nabla z), & x \in \Omega, t > 0, \\
0 = \Delta z - \gamma z + u, & x \in \Omega, t > 0
\end{cases}
\]
subject to the homogeneous Neumann boundary condition with \(\alpha, \gamma > 0\), where \(\Omega \subset \mathbb{R}^2\) is a smooth bounded domain. In the previous paper [Nonlinearity 31, No. 2, 502--514 (2018; Zbl 1382.35057)], we proved that the system possesses finite-time blow-up solutions provided
\[
m_u m_w - 2 \pi (m_u + m_w) > 0, \tag{0.1}
\]
where \(m_u\) and \(m_w\) denote the initial masses of \(u\) and \(w\) respectively. In this paper, we first establish the critical mass curve \(m_u m_w - 2 \pi (m_u + m_w) = 0\) by proving that solutions are globally bounded whenever
\[
m_u m_w - 2 \pi (m_u + m_w) < 0, \tag{0.2}
\]
which means the blow-up condition (0.1) is optimal. Based on this, we further show for the blow-up region (0.1) that \((u (x, t) d x, w (x, t) d x)\) forms a delta function singularity at the blow-up point \(x_0\) as \(t \to T_{\max}\) with the collapse mass \((M_{x_0}^u , M_{x_0}^w)\) satisfying
\[
M_{x_0}^u M_{x_0}^w - M_\ast (M_{x_0}^u + M_{x_0}^w) \geq 0,
\]
where \(M_\ast = 4 \pi\) for \(x_0 \in \Omega\) and \(M_\ast = 2 \pi\) for \(x_0 \in \partial \Omega \).Effective interface conditions for a porous medium type problemhttps://zbmath.org/1541.351022024-09-27T17:47:02.548271Z"Ciavolella, Giorgia"https://zbmath.org/authors/?q=ai:ciavolella.giorgia"David, Noemi"https://zbmath.org/authors/?q=ai:david.noemi"Poulain, Alexandre"https://zbmath.org/authors/?q=ai:poulain.alexandreSummary: Motivated by biological applications on tumour invasion through thin membranes, we study a porous medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the \textit{effective interface problem} and the transmission conditions on the limiting zero-thickness surface, formally derived by \textit{M. A. J. Chaplain} et al. [SIAM J. Appl. Math. 79, No. 5, 2011--2031 (2019; Zbl 1428.35619)], which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator, which allows us to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.On the existence of global solutions for the 3D chemorepulsion systemhttps://zbmath.org/1541.351032024-09-27T17:47:02.548271Z"Cieślak, Tomasz"https://zbmath.org/authors/?q=ai:cieslak.tomasz"Fuest, Mario"https://zbmath.org/authors/?q=ai:fuest.mario"Hajduk, Karol"https://zbmath.org/authors/?q=ai:hajduk.karol-w"Sierżęga, Mikołaj"https://zbmath.org/authors/?q=ai:sierzega.mikolajSummary: In this paper, we give sufficient conditions for global-in-time existence of classical solutions for the fully parabolic chemorepulsion system posed on a convex, bounded three-dimensional domain. Our main result establishes global-in-time existence of regular nonnegative solutions provided that \(\nabla \sqrt{u}\in L^4 (0,T;L^2 (\Omega))\). Our method is related to the Bakry-Émery calculation and appears to be new in this context.Global boundedness in an attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanismhttps://zbmath.org/1541.351042024-09-27T17:47:02.548271Z"Wang, Chang-Jian"https://zbmath.org/authors/?q=ai:wang.changjian"Zhu, Jia-Yue"https://zbmath.org/authors/?q=ai:zhu.jia-yueSummary: We study the following chemotaxis system involving nonlinear indirect signal mechanism
\[
\begin{cases}
u_t = \Delta u - \xi \nabla \cdot (u \nabla v) + \chi \nabla \cdot (u\nabla z), & x \in\Omega,\ t>0, \\
v_t = \Delta v-v+w^{\gamma_1}, & x \in \Omega,\ t>0, \\
0 = \Delta w-w+u^{\gamma_2}, & x \in\Omega,\ t>0, \\
0 = \Delta z-z+u^{\gamma_3}, & x \in \Omega,
\end{cases}
\]
under homogeneous Neumann boundary conditions in a bounded and smooth domain \(\Omega \subset \mathbb{R}^n\) (\(n\geq 1\)), where \(\xi, \chi, \gamma_1, \gamma_2, \gamma_3 >0\). With the aid of maximum Sobolev regularity and a priori estimates of \(w\) and \(z\), it has been proved that if either random diffusion or repulsion mechanism dominates the nonlinear indirect attraction mechanism with \(\gamma_1 \gamma_2 <\max \{\frac{2}{n}, \gamma_3\}\), then the system admits a global classical solution, which is bounded in \(\Omega \times (0, \infty)\). Moreover, we also show that under balance situations with \(\gamma_1 \gamma_2 = \max \{\frac{2}{n}, \gamma_3\}\), the existence of global classical solution is determined by the sizes of chemotaxis sensitivity coefficients \(\xi\) and \(\chi\) as well as the initial data \(u_0\). This work extends the previous results, where the production of signal is nonlinear mechanism.Gradient estimates for the CR heat equation on closed Sasakian manifoldshttps://zbmath.org/1541.351052024-09-27T17:47:02.548271Z"Zhao, Biqiang"https://zbmath.org/authors/?q=ai:zhao.biqiangSummary: In this paper, we obtain a CR version Li-Yau type gradient estimate for positive solutions of the CR heat equation on closed Sasakian manifolds. As its applications, we derive the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formula for closed Sasakian manifolds.Oblique derivative problem for parabolic systems with quadratic nonlinearity in gradient. Boundary regularity resultshttps://zbmath.org/1541.351062024-09-27T17:47:02.548271Z"Arkhipova, A. A."https://zbmath.org/authors/?q=ai:arkhipova.arina-a-arkhipovaSummary: We consider a class of quasilinear parabolic systems with strong (quadratic) nonlinearity in the gradient. Under a one-sided condition on the quadratic term, we study the boundary regularity of weak solutions to the oblique derivative problem. We describe conditions that guarantee the local Hölder continuity of weak (possibly unbounded) solutions.Asymptotic regularity for a random walk over ellipsoidshttps://zbmath.org/1541.351072024-09-27T17:47:02.548271Z"Arroyo, Ángel"https://zbmath.org/authors/?q=ai:arroyo.angelFor the entire collection see [Zbl 1497.42002].Iterates of differential operators of Shubin type in anisotropic Roumieu Gelfand-Shilov spaceshttps://zbmath.org/1541.351082024-09-27T17:47:02.548271Z"Bensaid, M'Hamed"https://zbmath.org/authors/?q=ai:bensaid.mhamed"Chaili, Rachid"https://zbmath.org/authors/?q=ai:chaili.rachidFirst contribution on the problem of the iterates was provided by \textit{T. Kotake} and \textit{M. S. Narasimhan} [Bull. Soc. Math. Fr. 90, 449--471 (1962; Zbl 0104.32503)], as a generalization of the hypoellipticity problem in the analytic and Gevrey classes. At present, attention of the scholars is addressed to the context of the Gelfand-Shilov classes for operators globally defined in Euclidean spaces with polynomial coefficients, see for example [\textit{M. Cappiello} et al., J. Anal. Math. 138, No. 2, 857--870 (2019; Zbl 1454.35100)]. The authors of the present paper generalize such results to ultradifferentiable Gelfand-Shilov spaces, of the type of those in [\textit{H. Komatsu}, J. Fac. Sci., Univ. Tokyo, Sect. I A 20, 25--105 (1973; Zbl 0258.46039)]. A very precise theorem is given, showing that ultradifferentiable vectors associated to a globally elliptic operator belong to suitable ultradifferentiable spaces. As a consequence, the authors obtain a result of hypoellipticity in the same framework.
Reviewer: Luigi Rodino (Torino)A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applicationshttps://zbmath.org/1541.351092024-09-27T17:47:02.548271Z"Desvillettes, Laurent"https://zbmath.org/authors/?q=ai:desvillettes.laurent"He, Ling-Bing"https://zbmath.org/authors/?q=ai:he.lingbing"Jiang, Jin-Cheng"https://zbmath.org/authors/?q=ai:jiang.jin-chengSummary: We describe a time-dependent functional involving the relative entropy and \(\dot{H}^1 \) the seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.On the optimal \(L^q\)-regularity for viscous Hamilton-Jacobi equations with subquadratic growth in the gradienthttps://zbmath.org/1541.351102024-09-27T17:47:02.548271Z"Goffi, Alessandro"https://zbmath.org/authors/?q=ai:goffi.alessandroSummary: This paper studies a maximal \(L^q\)-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a \(L^p\) duality method, which exploits the regularity properties of solutions to stationary Fokker-Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold \(q=d(\gamma-1)/\gamma\), \(d\) being the dimension of the ambient space and \(\gamma>1\) the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range \(q>1\).Maximal regularity for the heat equation with various boundary conditions in an infinite layerhttps://zbmath.org/1541.351112024-09-27T17:47:02.548271Z"Kajiwara, Naoto"https://zbmath.org/authors/?q=ai:kajiwara.naoto"Matsui, Aiki"https://zbmath.org/authors/?q=ai:matsui.aikiSummary: This paper treats resolvent \(L_q\) estimate and maximal \(L_p\)-\(L_q\) regularity for the heat equation with various boundary conditions in an infinite layer. We need to consider two boundary conditions on upper boundary and lower boundary. We are able to choose any pair of Dirichlet, Neumann and Robin boundary conditions. We construct the solutions of Fourier multiplier operators and we use a theorem for an integral operator, which derives \(L_q\)-boundedness and \(L_p\)-\(L_q\) boundedness. The key is that the holomorphic symbols can be properly estimated from above.Nonlocal functionals with non-standard growthhttps://zbmath.org/1541.351122024-09-27T17:47:02.548271Z"Kim, Minhyun"https://zbmath.org/authors/?q=ai:kim.minhyunSummary: In this note, we review recent progress on the De Giorgi-Nash-Moser theory for nonlocal functionals with non-standard growth, which include functionals with Orlicz growth, variable exponents, double phase growth, and orthotropic structure. Some open problems are suggested.
For the entire collection see [Zbl 1537.35003].Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilitieshttps://zbmath.org/1541.351132024-09-27T17:47:02.548271Z"Li, Genglin"https://zbmath.org/authors/?q=ai:li.genglin"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michael(no abstract)Weighted \(W^{1, 2}_{p(\cdot)}\)-estimate for fully nonlinear parabolic equations with a relaxed convexityhttps://zbmath.org/1541.351142024-09-27T17:47:02.548271Z"Tian, Hong"https://zbmath.org/authors/?q=ai:tian.hong.1"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhouSummary: We devote this paper to global estimate in weighted variable exponent Sobolev spaces for fully nonlinear parabolic equations under a relaxed convexity condition. It is assumed that the associated variable exponent is log-Hölder continuous, the weight belongs to certain Muckenhoupt class concerning the variable exponent, the leading part of nonlinearity satisfies a relaxed convexity in Hessian and is of VMO condition in space-time variables, and the boundary of underlying domain satisfies \(C^{1, 1}\)-smooth. Our key strategy is to utilize a unified approach based on the generalized versions of Fefferman-Stein theorem of the sharp functions and extrapolation to establish the estimates of \(D^2u\) and \(D_t u\) within the framework of weighted variable exponent Lebesgue spaces.Traveling waves for the porous medium equation in the incompressible limit: asymptotic behavior and nonlinear stabilityhttps://zbmath.org/1541.351152024-09-27T17:47:02.548271Z"Dalibard, Anne-Laure"https://zbmath.org/authors/?q=ai:dalibard.anne-laure"Lopez-Ruiz, Gabriela"https://zbmath.org/authors/?q=ai:lopez-ruiz.gabriela"Perrin, Charlotte"https://zbmath.org/authors/?q=ai:perrin.charlotteSummary: In this study, we analyze the behavior of monotone traveling waves of a one-dimensional porous medium equation modeling mechanical properties of living tissues. We are interested in the asymptotics where the pressure, which governs the diffusion process and limits the creation of new cells, becomes very stiff, and the porous medium equation degenerates towards a free boundary problem of Hele-Shaw type. This is the so-called \textit{incompressible limit}. The solutions of the limit Hele-Shaw problem then couple ``free dynamics'' with zero pressure, and ``incompressible dynamics'' with positive pressure and constant density. In the first part of the work, we provide a refined description of the traveling waves for the porous medium equation in the vicinity of the transition between the free domain and the incompressible domain. The second part of the study is devoted to the analysis of the stability of the traveling waves. We prove that the linearized system enjoys a spectral gap property in suitable weighted \(L^2\) spaces, and we give quantitative estimates on the rate of decay of solutions. The nonlinear terms are treated perturbatively, using an \(L^{\infty}\) control stemming from the maximum principle. As a consequence, we prove that traveling waves are stable under small perturbations. This constitutes the first nonlinear asymptotic stability result concerning smooth fronts of degenerate diffusion equations with a Fisher-KPP reaction term.Existence and stability of traveling wavefronts for a nonlocal delay Belousov-Zhabotinskii systemhttps://zbmath.org/1541.351162024-09-27T17:47:02.548271Z"Du, Ting-Ting"https://zbmath.org/authors/?q=ai:du.tingting"Zhang, Guo-Bao"https://zbmath.org/authors/?q=ai:zhang.guobao.1|zhang.guobao"Hao, Yu-Cai"https://zbmath.org/authors/?q=ai:hao.yu-cai"Shu, Ya-Qin"https://zbmath.org/authors/?q=ai:shu.ya-qin(no abstract)Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delayhttps://zbmath.org/1541.351172024-09-27T17:47:02.548271Z"Huang, Rui"https://zbmath.org/authors/?q=ai:huang.rui"Wang, Zhuangzhuang"https://zbmath.org/authors/?q=ai:wang.zhuangzhuang"Xu, Tianyuan"https://zbmath.org/authors/?q=ai:xu.tianyuan.1(no abstract)Front propagation and global bifurcations in a multivariable reaction-diffusion modelhttps://zbmath.org/1541.351182024-09-27T17:47:02.548271Z"Knobloch, Edgar"https://zbmath.org/authors/?q=ai:knobloch.edgar"Yochelis, Arik"https://zbmath.org/authors/?q=ai:yochelis.arik(no abstract)Multivortex traveling waves for the Schrödinger map equationhttps://zbmath.org/1541.351192024-09-27T17:47:02.548271Z"Tianpei, Guo"https://zbmath.org/authors/?q=ai:tianpei.guoSummary: We construct traveling wave solutions for the Schrödinger map equation in \(\mathbb{R}^2\). These solutions have \(n(n + 1)/2\) pairs of degree \(\pm 1\) vortices. The locations of those vortices are symmetric in the plane and determined by the roots of a special class of Adler-Moser polynomials. With a few modifications, a similar construction allows for the creation of traveling wave solutions of the Schrödinger map equation in \(\mathbb{R}^3\). These solutions have the shape of \(2n + 1\) vortex rings, whose locations are given by a sequence of polynomials with rational coefficients and are far away from each other.
{\copyright 2024 American Institute of Physics}Single-traveling-wave and double-wave polynomial solutions of the generalized (3 + 1)-dimensional modified Kadomtsev-Petviashvili equations with variable coefficientshttps://zbmath.org/1541.351202024-09-27T17:47:02.548271Z"Wang, Lingyu"https://zbmath.org/authors/?q=ai:wang.lingyu"Gao, Ben"https://zbmath.org/authors/?q=ai:gao.benSummary: In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev-Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently.Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidenceshttps://zbmath.org/1541.351212024-09-27T17:47:02.548271Z"Wu, Weixin"https://zbmath.org/authors/?q=ai:wu.weixin"Zhang, Wenhui"https://zbmath.org/authors/?q=ai:zhang.wenhuiSummary: This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed \(c^\ast\) and the basic reproduction number \(R_0\) are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when \(\mathcal{R}_0>1\), for every speed \(c>c^\ast\) there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any \(0<c< c^*\) when \(\mathcal{R}_0>1\) or \(c>0\) when \(\mathcal{R}_0<1\). Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.The dynamics of traveling wavefronts for a model describing host tissue degradation by bacteriahttps://zbmath.org/1541.351222024-09-27T17:47:02.548271Z"Yang, Xing-Xing"https://zbmath.org/authors/?q=ai:yang.xingxing"Zhang, Guo-Bao"https://zbmath.org/authors/?q=ai:zhang.guobao.1|zhang.guobao"Tian, Ge"https://zbmath.org/authors/?q=ai:tian.ge.1|tian.geSummary: In this paper, we mainly investigate the dynamics of traveling wavefronts for a model describing host tissue degradation by bacteria. We first establish the existence of spreading speed, and show that the spreading speed coincides with the minimal wave speed of traveling wavefronts. Moreover, a lower bound estimate of the spreading speed is given. Then, we prove that the traveling wavefronts with large speeds are globally exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as \(x\to -\infty\), but the initial perturbation can be arbitrarily large in other locations. The adopted methods are the weighted energy and the squeezing technique.Symmetry breaking of solitons in the PT-symmetric nonlinear Schrödinger equation with the cubic-quintic competing saturable nonlinearityhttps://zbmath.org/1541.351232024-09-27T17:47:02.548271Z"Bo, Wen-Bo"https://zbmath.org/authors/?q=ai:bo.wen-bo"Wang, Ru-Ru"https://zbmath.org/authors/?q=ai:wang.ru-ru"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.188"Wang, Yue-Yue"https://zbmath.org/authors/?q=ai:wang.yueyueSummary: The symmetry breaking of solitons in the nonlinear Schrödinger equation with cubic-quintic competing nonlinearity and parity-time symmetric potential is studied. At first, a new asymmetric branch separates from the fundamental symmetric soliton at the first power critical point, and then, the asymmetric branch passes through the branch of the fundamental symmetric soliton and finally merges into the branch of the fundamental symmetric soliton at the second power critical point, while the power of the soliton increases. This leads to the symmetry breaking and double-loop bifurcation of fundamental symmetric solitons. From the power-propagation constant curves of solitons, symmetric fundamental and tripole solitons, asymmetric solitons can also exist. The stability of symmetric fundamental solitons, asymmetric solitons, and symmetric tripole solitons is discussed by the linear stability analysis and direct simulation. Results indicate that symmetric fundamental solitons and symmetric tripole solitons tend to be stable with the increase in the soliton power. Asymmetric solitons are unstable in both high and low power regions. Moreover, with the increase in saturable nonlinearity, the stability region of fundamental symmetric solitons and symmetric tripole solitons becomes wider.
{\copyright 2022 American Institute of Physics}The nonlinear vibration and dispersive wave systems with cross-kink and solitary wave solutionshttps://zbmath.org/1541.351242024-09-27T17:47:02.548271Z"Li, Ruijuan"https://zbmath.org/authors/?q=ai:li.ruijuan"Manafian, Jalil"https://zbmath.org/authors/?q=ai:manafian-heris.jalil"Lafta, Holya A."https://zbmath.org/authors/?q=ai:lafta.holya-a"Kareem, Hawraa A."https://zbmath.org/authors/?q=ai:kareem.hawraa-a"Uktamov, Khusniddin Fakhriddinovich"https://zbmath.org/authors/?q=ai:uktamov.khusniddin-fakhriddinovich"Abotaleb, Mostafa"https://zbmath.org/authors/?q=ai:abotaleb.mostafaSummary: This paper investigates the cross-kink and solitary wave solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, cross-kink and solitary wave solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole-Hopf algorithm to get the exact solutions of the \((2+1)\)-dimensional modified dispersive water-wave system. The analytical treatment of cross-kink, solitary wave solutions is studied and plotted in three forms of two, density and three-D plots. A nonlinear vibration system will be investigated. Employing appropriate mathematical assumptions, the novel kinds of the cross-kink and solitary wave solutions are derived and constructed in view of the combination of kink, periodic and soliton for cross-kink and also a combination of two kinks in terms of exponential functions for solitary of the governing equation. To achieve this, the illustrative example of the \((2+1)\)-D modified dispersive water-wave system is furnished to exhibit the feasibility and reliability of the procedure utilized in this research. The trajectory solutions of the traveling waves are presented explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested technique with the other existing schemes, the results present that the execution of this technique is concise, simple and straightforward.Soliton solutions of dual-mode Kawahara equation via Lie symmetry analysishttps://zbmath.org/1541.351252024-09-27T17:47:02.548271Z"Malik, Sandeep"https://zbmath.org/authors/?q=ai:malik.sandeep"Kumar, Sachin"https://zbmath.org/authors/?q=ai:kumar.sachin.1Summary: In this article, we investigate a newly proposed dual-mode Kawahara equation. Our main aim in this paper is to find out the soliton and periodic solutions of the Kawahara equation. Initially, we reduce the governing equation into an ordinary differential equation by applying the Lie symmetry analysis. Further, we derive the soliton and periodic solutions via three integration methods, namely sech-csch scheme, exp-expansion method, and modified F-expansion method.
For the entire collection see [Zbl 1521.76009].Soliton solutions of (2+1)-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation by using Lie symmetry methodhttps://zbmath.org/1541.351262024-09-27T17:47:02.548271Z"Mishra, Shivam Kumar"https://zbmath.org/authors/?q=ai:mishra.shivam-kumarSummary: In the present article, the Lie transformation method has been used to find out the group-invariant solutions of (2+1)-dimensional modified Calogero-Bogoyavlenskii-Schiff(mCBS) equation. The equation has been reduced to ordinary differential equations (ODEs) using the method. The method plays a significant role in the reduction of the number of independent variables in the system by one in each proceeding stage and, eventually, forms an ODE whose exact solutions can be established. Moreover, solutions derived here contain some arbitrary constants and functions. These solutions are mainly multisoliton, single soliton, periodic or quasi-periodic and evolutionary wave types. Finally, with the adjustments in these arbitrary parameters and functions, some graphs have been plotted.
For the entire collection see [Zbl 1521.76009].Fast method for 2D Dirichlet problem for circular multiply connected domainshttps://zbmath.org/1541.351272024-09-27T17:47:02.548271Z"Bar, Olaf"https://zbmath.org/authors/?q=ai:bar.olaf"Wójcik, Krzysztof"https://zbmath.org/authors/?q=ai:wojcik.krzysztofSummary: This paper is devoted the optimization of application to determine the flux around closely spaced nonoverlapping disks on the conductive plane. This method is based on successive approximations applied to the functional equations. This paper is concerned on influence of checking diagrams on convergence fast Poincaré series method. This can be used to solve Laplace's equation on a conductive plane with nonoverlapping inclusions. The initial stream is composed from set of two-point functions which are dependent on the graph which represents connection between nearest neighbours circles.
For the entire collection see [Zbl 1497.42002].On the summability and convergence of formal solutions of linear \(q\)-difference-differential equations with constant coefficientshttps://zbmath.org/1541.351282024-09-27T17:47:02.548271Z"Ichinobe, Kunio"https://zbmath.org/authors/?q=ai:ichinobe.kunio"Michalik, Sławomir"https://zbmath.org/authors/?q=ai:michalik.slawomirIn this article, the authors are interested in the Cauchy problem for homogeneous linear \(q\)-difference-differential equations of the form
\[
\begin{cases}
P(D_{q,t},\partial_z)u=0\\
D_{q,t}^ju(0,z)=\varphi_j(z)\text{ for }j=0,\dots,p-1
\end{cases}
\]
where \(P(D_{q,t},\partial_z)\) is a general linear \(q\)-difference-differential operator with constant coefficients of order \(p\) with respect to the \(q\)-difference operator \(D_{q,t}\) defined by \[D_{q,t}u(t,z)=\dfrac{u(qt,z)-u(t,z)}{qt-t},\quad q\in[0,1[,\] and where the initial data \(\varphi_j(z)\) are holomorphic functions in a neighborhood of the origin \(0\in\mathbb{C}\) for all \(j=0,\dots,p-1\).
They characterize convergent, \(k\)-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the initial data \(\varphi_j(z)\). They also introduce and characterize sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.
Reviewer: Pascal Remy (Carrières-sur-Seine)Global weak solutions in a singular taxis-type system with signal consumptionhttps://zbmath.org/1541.351292024-09-27T17:47:02.548271Z"Chen, Zhen"https://zbmath.org/authors/?q=ai:chen.zhen.4|chen.zhen.1|chen.zhen.5|chen.zhen.2|chen.zhen"Li, Genglin"https://zbmath.org/authors/?q=ai:li.genglinSummary: Relevant to modeling starvation-driven species dispersal is the chemotaxis-consumption system, featuring singular signal-dependent motilities, given by
\[
\begin{cases}
u_t = \Delta (u^m v^{- \alpha}), \\
v_t = \Delta v - u v,
\end{cases}
\]
which is considered under homogeneous boundary conditions in smoothly bounded domains \(\Omega \subset \mathbb{R}^n\), \(n \geq 1\), with \(m > 1\) and \(\alpha > 0\).
In the context of concurrent strengthening of diffusion and cross-diffusion in the first equation of this system, regulated by the motility function \(v^{- \alpha}\) and the porous-medium-type diffusion concerning species, with both the diffusion and cross-diffusion exhibiting potential singularities near \(\{ v = 0 \} \), it is shown that for all sufficiently regular initial data, when the species diffusion partially conforms to mild porous-medium-type behavior (i.e., \( \max \{ 1, \frac{ n - 2}{ 4} \} < m \leq \frac{ n}{ 2}\)) and the motility function \(v^{- \alpha}\) displays appropriately strong singularities (quantified by \(\alpha > \frac{ n - 2 m}{ 4 m - n + 2})\), the system admits globally defined weak solutions, whereas in situations characterized by pronounced porous-medium-type diffusion in the species (i.e., \( m > \frac{ n}{ 2}\)), not only can global weak solutions be constructed, but they are continuous and locally bounded, even in the presence of arbitrarily strong singular behavior in the motility function \(v^{- \alpha}\) in the vicinity of \(\{ v = 0 \} \).Renormalized solutions to a parabolic equation with mixed boundary constraintshttps://zbmath.org/1541.351302024-09-27T17:47:02.548271Z"Do, Tan Duc"https://zbmath.org/authors/?q=ai:do.tan-duc"Truong, Le Xuan"https://zbmath.org/authors/?q=ai:truong.le-xuan"Trong, Nguyen Ngoc"https://zbmath.org/authors/?q=ai:trong.nguyen-ngocSummary: We establish the existence and uniqueness of a renormalized solution to the parabolic equation
\[
\frac{\partial b(u)}{\partial t}-\operatorname{div}(a(x,t,u,\nabla u)) = f \quad \text{in } \Omega\times (0,T)
\]
subject to a mixed boundary condition. Here \(b(u)\) is a real function of \(u, -\operatorname{div}(a(x,t,u,\nabla u))\) is of Leray-Lions type and \(f\) is an \(L^1\)-function. Then we compare the renormalized solution to two other notions of solution: distributional solution and weak solution.Timoshenko systems with Cattaneo law and partial Kelvin-Voigt damping: well-posedness and stabilityhttps://zbmath.org/1541.351312024-09-27T17:47:02.548271Z"Enyi, Cyril Dennis"https://zbmath.org/authors/?q=ai:enyi.cyril-dennis(no abstract)Ultra-parabolic Kolmogorov-type equation with multiple impulsive sourceshttps://zbmath.org/1541.351322024-09-27T17:47:02.548271Z"Kuznetsov, Ivan"https://zbmath.org/authors/?q=ai:kuznetsov.ivan-v"Sazhenkov, Sergey"https://zbmath.org/authors/?q=ai:sazhenkov.sergey-alexandrovichSummary: Existence and uniqueness of entropy solutions of the Cauchy-Dirichlet problem for the non-autonomous ultra-parabolic equation with partial diffusivity and multiple impulsive sources is established. The limiting passage from the equation incorporating a single distributed source to the multi-impulsive equation is fulfilled, as the distributed source collapses to a parameterized multi-atomic Dirac delta measure.
For the entire collection see [Zbl 1497.42002].The existence of continuous weak solutions with compact support for degenerate parabolic systems with exchange-type sourcehttps://zbmath.org/1541.351332024-09-27T17:47:02.548271Z"Yang, Tianjie"https://zbmath.org/authors/?q=ai:yang.tianjie"Yuan, Guangwei"https://zbmath.org/authors/?q=ai:yuan.guangwei(no abstract)Global existence of weak solutions for the 3D incompressible Keller-Segel-Navier-Stokes equations with partial diffusionhttps://zbmath.org/1541.351342024-09-27T17:47:02.548271Z"Zhao, Jijie"https://zbmath.org/authors/?q=ai:zhao.jijie"Chen, Xiaoyu"https://zbmath.org/authors/?q=ai:chen.xiaoyu"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian.19(no abstract)A hyperbolic-elliptic-parabolic PDE model of chemotactic E. coli colonieshttps://zbmath.org/1541.351352024-09-27T17:47:02.548271Z"Guo, Haojie"https://zbmath.org/authors/?q=ai:guo.haojie"Meng, Qiu"https://zbmath.org/authors/?q=ai:meng.qiuSummary: This paper studies the hyperbolic-elliptic-parabolic system
\[
\begin{cases} u_t = -\nabla \cdot (u\nabla c) + g(u)nu - b(n)u^{1+\theta}, & (x, t) \in \Omega \times \mathbb{R}^+, \\
0 = \Delta c+\alpha u - \beta c, & (x,t) \in\Omega \times \mathbb{R}^+, \\
n_t = \Delta n-\gamma g(u)nu, & (x,t) \in \Omega \times \mathbb{R}^+, \\
\frac{\partial c}{\partial\nu} = \frac{\partial n}{\partial\nu} = 0, & (x, t) \in \partial \Omega \times \mathbb{R}^+, \\
u (x, 0) = u_0 (x), n(x, 0) = n_0 (x), & x \in \Omega
\end{cases}
\]
in a bounded domain \(\Omega\subset\mathbb{R}^N (N \geq 1)\) with smooth boundary, where \(g(s)\) and \(b(s)\) are nonnegative smooth functions, \(\alpha, \beta, \gamma >0\) and \(\theta \in [0, \infty)\). We prove that if \(\theta > 1\) or \(\theta =1\) with \(\lim_{s \to +\infty} b(s) = b_0 >\alpha\), the system admits a unique global strong solution, while if \(\theta \in [0, 1)\), the system exhibits a finite-time explosion phenomenon in the sense of \(L^{\infty}\) norm for some large \(\| u_0\|_{L^p (\Omega)}\) with \(p>N\). Here the convexity of the domain \(\Omega\) is not required.Gaussian upper bounds for the heat kernel on evolving manifoldshttps://zbmath.org/1541.351362024-09-27T17:47:02.548271Z"Buzano, Reto"https://zbmath.org/authors/?q=ai:buzano.reto"Yudowitz, Louis"https://zbmath.org/authors/?q=ai:yudowitz.louisAuthors' abstract: In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method that was previously used by \textit{E. B. Davies} [Am. J. Math. 109, 319--333 (1987; Zbl 0659.35009)] in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.
Reviewer: Abimbola Abolarinwa (Lagos)A particle method for non-local advection-selection-mutation equationshttps://zbmath.org/1541.351372024-09-27T17:47:02.548271Z"Alvarez, Frank Ernesto"https://zbmath.org/authors/?q=ai:alvarez.frank-ernesto"Guilberteau, Jules"https://zbmath.org/authors/?q=ai:guilberteau.julesSummary: The well-posedness of a non-local advection-selection-mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such problem by a regularized sum of weighted Dirac masses whose characteristics solve a suitably defined ODE system. The convergence of the particle method over any finite interval is shown and an explicit rate of convergence is given. Furthermore, we investigate the asymptotic-preserving properties of the method in large times, providing sufficient conditions for it to hold true as well as examples and counter-examples. Finally, we illustrate the method in two cases taken from the literature.Prediction-correction pedestrian flow by means of minimum flow problemhttps://zbmath.org/1541.351382024-09-27T17:47:02.548271Z"Ennaji, Hamza"https://zbmath.org/authors/?q=ai:ennaji.hamza"Igbida, Noureddine"https://zbmath.org/authors/?q=ai:igbida.noureddine"Jradi, Ghadir"https://zbmath.org/authors/?q=ai:jradi.ghadirSummary: We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation \(\|\nabla\varphi\|=f\), with a positive continuous function \(f\) connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle-Pock's primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.The elasticity complex: compact embeddings and regular decompositionshttps://zbmath.org/1541.351392024-09-27T17:47:02.548271Z"Pauly, Dirk"https://zbmath.org/authors/?q=ai:pauly.dirk"Zulehner, Walter"https://zbmath.org/authors/?q=ai:zulehner.walter(no abstract)Stability of periodic peakons for a generalized-\(\mu\) Camassa-Holm equation with quartic nonlinearitieshttps://zbmath.org/1541.351402024-09-27T17:47:02.548271Z"Li, Zhigang"https://zbmath.org/authors/?q=ai:li.zhigang(no abstract)The Wentzell Laplacian via forms and the approximative tracehttps://zbmath.org/1541.351412024-09-27T17:47:02.548271Z"Arendt, Wolfgang"https://zbmath.org/authors/?q=ai:arendt.wolfgang"Sauter, Manfred"https://zbmath.org/authors/?q=ai:sauter.manfredSummary: We use form methods to define suitable realisations of the Laplacian on a domain \(\Omega\) with Wentzell boundary conditions, i.e. such that \(\partial_{\mathrm{n}} u + \beta u + \Delta u = 0\) holds in a suitable sense on the boundary of \(\Omega \). For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a unified treatment that in part allows irregular and even fractal domains. Moreover, we admit \(\beta\) to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.Stability for the Helmholtz equation in deterministic and random periodic structureshttps://zbmath.org/1541.351422024-09-27T17:47:02.548271Z"Bao, Gang"https://zbmath.org/authors/?q=ai:bao.gang"Lin, Yiwen"https://zbmath.org/authors/?q=ai:lin.yiwen"Xu, Xiang"https://zbmath.org/authors/?q=ai:xu.xiangSummary: Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the stability estimate for the Helmholtz equation in a deterministic periodic structure is established. For the stochastic case, by introducing a variable transform, the variational formulation of the scattering problem in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deterministic case with regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's Theorem further yield the stability result for the scattering problem by random periodic structures. Both stability estimates are explicit with respect to the wavenumber.On the three ball theorem for solutions of the Helmholtz equationhttps://zbmath.org/1541.351432024-09-27T17:47:02.548271Z"Berge, Stine Marie"https://zbmath.org/authors/?q=ai:berge.stine-marie"Malinnikova, Eugenia"https://zbmath.org/authors/?q=ai:malinnikova.eugeniaSummary: Let \(u_k\) be a solution of the Helmholtz equation with the wave number \(k\), \(\varDelta u_k+k^2 u_k=0\), on (a small ball in) either \({\mathbb{R}}^n, {\mathbb{S}}^n \), or \({\mathbb{H}}^n \). For a fixed point \(p\), we define \(M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|\). The following three ball inequality \(M_{u_k}(2r)\le C(k,r,\alpha )M_{u_k}(r)^{\alpha }M_{u_k}(4r)^{1-\alpha }\) is well known, it holds for some \(\alpha \in (0,1)\) and \(C(k,r,\alpha )>0\) independent of \(u_k\). We show that the constant \(C(k,r,\alpha )\) grows exponentially in \(k\) (when \(r\) is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.Critical points of solutions to exterior boundary problemshttps://zbmath.org/1541.351442024-09-27T17:47:02.548271Z"Deng, Haiyun"https://zbmath.org/authors/?q=ai:deng.haiyun"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang.6"Liu, Hairong"https://zbmath.org/authors/?q=ai:liu.hairongSummary: In this article, we mainly study the critical points of solutions to the Laplace equation with Dirichlet boundary conditions in an exterior domain in \(\mathbb{R}^2\). Based on the fine analysis about the structures of connected components of the super-level sets \(\{x \in \mathbb{R}^2\setminus \Omega: u(x) > t\}\) and sub-level sets \(\{x \in \mathbb{R}^2 \setminus \Omega : u(x) < t\}\) for some \(t\), we get the geometric distributions of interior critical point sets of solutions. Exactly, when \(\Omega\) is a smooth bounded simply connected domain, \(u|_{\partial \Omega} = \psi (x)\), \(\lim_{|x| \to \infty} u(x) = - \infty\) and \(\psi (x)\) has \(K\) local maximal points on \(\partial \Omega\), we deduce that \(\sum_{i=1}^l m_i \leq K\), where \(m_1, \dots, m_l\); are the multiplicities of interior critical points \(x_1,\dots, x_l\); of solution \(u\) respectively. In addition, when \(\psi (x)\) has only \(K\) global maximal points and \(K\) equal local minima relative to \(\mathbb{R}^2\setminus \Omega\) on \(\partial \Omega\), we have that \(\sum_{i=1}^l m_i=K\). Moreover, when \(\Omega\) is a domain consisting of \(l\) disjoint smooth bounded simply connected domains, we deduce that \(\sum_{x_i\in \Omega} m_i + \frac{1}{2} \sum_{x_j \in \partial \Omega} m_j =l-1\), and the critical points are contained in the convex hull of the \(l\) simply connected domains.The Mullins-Sekerka problem via the method of potentialshttps://zbmath.org/1541.351452024-09-27T17:47:02.548271Z"Escher, Joachim"https://zbmath.org/authors/?q=ai:escher.joachim"Matioc, Anca-Voichita"https://zbmath.org/authors/?q=ai:matioc.anca-voichita"Matioc, Bogdan-Vasile"https://zbmath.org/authors/?q=ai:matioc.bogdan-vasileSummary: It is shown that the two-dimensional Mullins-Sekerka problem is well-posed in all subcritical Sobolev spaces \(H^r (\mathbb{R})\) with \(r \in (3/2,2)\). This is the first result, where this issue is established in an unbounded geometry. The novelty of our approach is the use of the potential theory to formulate the model as an evolution problem with nonlinearities expressed by singular integral operators.
{\copyright} 2024 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.On strong solvability of one nonlocal boundary value problem for Laplace equation in rectanglehttps://zbmath.org/1541.351462024-09-27T17:47:02.548271Z"Gasymov, Telman"https://zbmath.org/authors/?q=ai:gasymov.telman-benser-oglu"Akhmadli, Baharchin"https://zbmath.org/authors/?q=ai:akhmadli.baharchin"Ildiz, Ümit"https://zbmath.org/authors/?q=ai:ildiz.umitSummary: One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in this work. The concept of a strong solution to this problem is introduced. The correct solvability of this problem in the Sobolev spaces generated by the weighted mixed norm is proved by the Fourier method. In a classic statement, this problem has been earlier considered by \textit{E. I. Moiseev} [Differ. Equations 35, No. 8, 1105--1112 (1999); translation from Differ. Uravn. 35, No. 8, 1094--1100 (1999; Zbl 0973.35085)]. A similar problem has been treated by \textit{M. E. Lerner} and \textit{O. A. Repin} [Differ. Equations 35, No. 8, 1098--1104 (1999); translation from Differ. Uravn. 35, No. 8, 1087--1093 (1999; Zbl 0976.35028)].Domain perturbation for the solution of a periodic Dirichlet problemhttps://zbmath.org/1541.351472024-09-27T17:47:02.548271Z"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We prove that the solution of the periodic Dirichlet problem for the Laplace equation depends real analytically on a suitable parametrization of the shape of the domain, on the periodicity parameters, and on the Dirichlet datum.
For the entire collection see [Zbl 1497.42002].A uniform resolvent estimate for a Helmholtz equation with some large perturbations in an exterior domainhttps://zbmath.org/1541.351482024-09-27T17:47:02.548271Z"Nakazawa, Hideo"https://zbmath.org/authors/?q=ai:nakazawa.hideoSummary: We derive a uniform resolvent estimate for a stationary dissipative wave equation without smallness conditions. Existing results required a smallness condition for the coefficient of the dissipation. This paper removes the assumption of the smallness. Our proof is based on an energy estimate for stationary problems.
For the entire collection see [Zbl 1497.42002].A generalization of the formulas of d'Alembert and Poissonhttps://zbmath.org/1541.351492024-09-27T17:47:02.548271Z"Salekhov, G. S."https://zbmath.org/authors/?q=ai:salekhov.g-s(no abstract)Green functions for stationary Stokes systems with conormal derivative boundary condition in two dimensionshttps://zbmath.org/1541.351502024-09-27T17:47:02.548271Z"Choi, Jongkeun"https://zbmath.org/authors/?q=ai:choi.jongkeun"Kim, Doyoon"https://zbmath.org/authors/?q=ai:kim.doyoonSummary: We construct Green functions of conormal derivative problems for the stationary Stokes system with measurable coefficients in a two-dimensional Reifenberg flat domain.
{\copyright} 2023 Wiley-VCH GmbH.Multi-peak positive solutions for a logarithmic Schrödinger equation via variational methodshttps://zbmath.org/1541.351512024-09-27T17:47:02.548271Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Ji, Chao"https://zbmath.org/authors/?q=ai:ji.chaoSummary: In this paper, using the variational methods, we show the existence and multiplicity of multi-peak positive solutions for the following logarithmic Schrödinger equation:
\[
\begin{cases}
- \epsilon^2 \Delta u + V(x)u = u \log u^2, \quad \text{in } \mathbb{R}^N, \\
u \in H^1 (\mathbb{R}^N),
\end{cases}
\]
where \(\epsilon > 0\), \(N \geq 2\) and \(V: \mathbb{R}^N \to \mathbb{R}\) is a multi-well potential.Bounded Palais-Smale sequences with Morse type information for some constrained functionalshttps://zbmath.org/1541.351522024-09-27T17:47:02.548271Z"Borthwick, Jack"https://zbmath.org/authors/?q=ai:borthwick.jack-a"Chang, Xiaojun"https://zbmath.org/authors/?q=ai:chang.xiaojun"Jeanjean, Louis"https://zbmath.org/authors/?q=ai:jeanjean.louis"Soave, Nicola"https://zbmath.org/authors/?q=ai:soave.nicolaThe authors investigate the existence of bounded Palais-Smale sequences which have Morse index type information for the functionals with a minimax geometry restricted to a constraint.
Reviewer: Rodica Luca (Iaşi)Quasilinear Schrödinger equations for the Heisenberg ferromagnetic spin chainhttps://zbmath.org/1541.351532024-09-27T17:47:02.548271Z"Cheng, Yongkuan"https://zbmath.org/authors/?q=ai:cheng.yongkuan"Shen, Yaotian"https://zbmath.org/authors/?q=ai:shen.yaotianSummary: In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain
\[
-\Delta u+V(x)u-\frac{u}{\sqrt{1-u^2}} \Delta \sqrt{1-u^2}=\lambda |u|^{p-2} u, x \in \mathbb{R}^N,
\]
where \(2 \leq p<2^*\), \(N \geq 3\). By the Ekeland variational principle, the cut off technique, the change of variables and the \(L^{\infty}\) estimate, we study the existence of positive solutions. Here, we construct the \(L^{\infty}\) estimate of the solution in an entirely different way. Particularly, all the constants in the expression of this estimate are so well known.Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentialshttps://zbmath.org/1541.351542024-09-27T17:47:02.548271Z"Dicke, Alexander"https://zbmath.org/authors/?q=ai:dicke.alexander"Seelmann, Albrecht"https://zbmath.org/authors/?q=ai:seelmann.albrecht"Veselić, Ivan"https://zbmath.org/authors/?q=ai:veselic.ivanSummary: We prove a spectral inequality (a specific type of uncertainty relation) for Schrödinger operators with confinement potentials, in particular of Shubin-type. The sensor sets are allowed to decay exponentially, where the precise allowed decay rate depends on the potential. The proof uses an interpolation inequality derived by Carleman estimates, quantitative weighted \(L^2\)-estimates and an \(H^1\)-concentration estimate, all of them for functions in a spectral subspace of the operator.Normalized solutions to Schrödinger equations with critical exponent and mixed nonlocal nonlinearitieshttps://zbmath.org/1541.351552024-09-27T17:47:02.548271Z"Ding, Yanheng"https://zbmath.org/authors/?q=ai:ding.yanheng"Wang, Hua-Yang"https://zbmath.org/authors/?q=ai:wang.huayangSummary: We study the existence and nonexistence of normalized solutions \((u_a, \lambda_a)\in H^1 (\mathbb{R}^N) \times \mathbb{R}\) to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities:
\[
\begin{cases}
-\Delta u=\lambda u+ (I_{\alpha} * \vert u \vert^p) \vert u \vert^{p-2}u+\mu (I_{\alpha} * \vert u \vert^q) \vert u \vert^{q-2}u\quad \text{in }\mathbb{R}^N, \\
\int_{\mathbb{R}^N} \vert u \vert^2 \mathrm{d} x=a^2 >0,
\end{cases}
\]
where \(N\geq 3\), \(\alpha \in (0,N)\), \(\mu \in \mathbb{R}\), \(\frac{N+\alpha}{N}< q< p \leq \frac{N+\alpha}{N-2}\), and \(I_{\alpha}\) is the Riesz potential. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to nonlocal Schrödiger equations with a fixed \(L^2\)-norm \(\Vert u\Vert_2 =a>0\). The leading term is \(L^2\)-supercritical, that is, \(p\in (\frac{N+\alpha +2}{N},\frac{N+\alpha}{N-2}]\), where the Hardy-Littlewood-Sobolev critical exponent \(p=\frac{N+\alpha}{N-2}\) appears. We first prove that there exist two normalized solutions if \(q\in (\frac{N+\alpha}{N},\frac{N+\alpha +2}{N})\) with \(\mu >0\) small, that is, one is at the negative energy level while the other one is at the positive energy level. For \(q=\frac{N+\alpha +2}{N}\), we show that there is a normalized ground state for \(0<\mu <\tilde{\mu}\) and there exist no ground states for \(\mu >\tilde{\mu}\), where \(\tilde{\mu}\) is a sharp positive constant. If \(q\in (\frac{N+\alpha +2}{N},\frac{N+\alpha}{N-2})\), we deduce that there exists a normalized ground state for any \(\mu >0\). We also obtain some existence and nonexistence results for the case \(\mu <0\) and \(q\in (\frac{N+\alpha}{N},\frac{N+\alpha +2}{N}]\). Besides, we analyze the asymptotic behavior of normalized ground states as \(\mu \rightarrow 0^+\).Pointwise convergence for the Schrödinger equation [after Xiumin Du and Ruixiang Zhang]https://zbmath.org/1541.351562024-09-27T17:47:02.548271Z"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathanIn this nice exposition, the author is interested in the Schrödinger equation on \(\mathbb R^n\) and in particular in the Carleson problem, that is, in determining the minimum degree of regularity, measured in terms of the exponent of the Sobolev space, for which convergence is guaranteed almost everywhere. Pointwise convergence is typically demonstrated via analysis of the maximal Schrödinger operator, an object of interest in itself. The maximal operator can in turn be studied using the fractal energy estimates for Schrödinger solutions. Through these connections, progress on the Carleson problem has led to new developments on a surprising number of different problems, such as the Falconer distance problem and the Fourier restriction conjecture. In this paper, the history of this fascinating problem is exposed clearly and in depth
For the entire collection see [Zbl 1531.00041].
Reviewer: Vincenzo Vespri (Firenze)Nonradial solutions of quasilinear Schrödinger equations with general nonlinearityhttps://zbmath.org/1541.351572024-09-27T17:47:02.548271Z"Jing, Yongtao"https://zbmath.org/authors/?q=ai:jing.yongtao"Liu, Haidong"https://zbmath.org/authors/?q=ai:liu.haidong.1"Liu, Zhaoli"https://zbmath.org/authors/?q=ai:liu.zhaoliSummary: Consider the quasilinear Schrödinger equation
\[
-\Delta u+V(x)u-\frac12 \Delta(u^2)u = h(u)+\mu l(u),\quad u\in H^1(\mathbb{R}^N),
\]
where \(V(x)\) is a radial potential allowed to be singular at \(x = 0\), \(h\) is an odd nonlinearity of the Berestycki-Lions type, \( \mu\in\mathbb{R}\) is a small parameter and \(l\) is a general odd function. While most works in the literature are restricted to radial solutions, we develop a new variational approach to derive the existence of multiple \textit{nonradial} solutions by proposing a nonlocal perturbation process.Multiplicity of positive solutions to a class of Schrödinger-type singular problemshttps://zbmath.org/1541.351582024-09-27T17:47:02.548271Z"Ko, Eunkyung"https://zbmath.org/authors/?q=ai:ko.eunkyung"Lee, Eun Kyoung"https://zbmath.org/authors/?q=ai:lee.eunkyoung"Shivaji, R."https://zbmath.org/authors/?q=ai:shivaji.ratnasinghamSummary: We establish a multiplicity result for positive solutions to the Schrödinger-type singular problem: \( -\Delta u+V(x)u = \lambda \frac{ f(u)}{u^\beta}\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq1\) with a smooth boundary \(\partial \Omega\), \(0 \leq \beta <1\), \(\lambda\) is a positive parameter, \( V \in L^\infty(\Omega)\) and \(f:[0, \infty) \rightarrow (0, \infty)\) is a continuous function. In particular, when \(g\) is sublinear at \(\infty\) where \(g(s): = \frac{f(s)}{s^\beta} \), we discuss the existence of at least three positive solutions for a certain range of \(\lambda \). The proofs are mainly based on the sub and supersolution method.Gradient estimates for fundamental solutions of a Schrödinger operator on stratified Lie groupshttps://zbmath.org/1541.351592024-09-27T17:47:02.548271Z"Lin, Qingze"https://zbmath.org/authors/?q=ai:lin.qingze"Xie, Huayou"https://zbmath.org/authors/?q=ai:xie.huayouSummary: Let \(\mathcal{L}=-\Delta_{\mathbb{G}}+\Upsilon\) be a Schrödinger operator with a nonnegative potential \(\Upsilon\) belonging to the reverse Hölder class \(B_{Q/2}\), where \(Q\) is the homogeneous dimension of the stratified Lie group \(\mathbb{G}\). Inspired by \textit{Z. Shen}'s pioneer work [J. Funct. Anal. 167, No. 2, 521--564 (1999; Zbl 0936.35051)] and \textit{H.-Q. Li}'s work [J. Funct. Anal. 161, No. 1, 152--218 (1999; Zbl 0929.22005)], we study fundamental solutions of the Schrödinger operator \(\mathcal{L}\) on the stratified Lie group \(\mathbb{G}\) in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \(\mathcal{L}\) on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \(\mathcal{L}\)-harmonic functions on \(\mathbb{G}\).Normalized solutions for Schrödinger equations with potentials and general nonlinearitieshttps://zbmath.org/1541.351602024-09-27T17:47:02.548271Z"Liu, Yanyan"https://zbmath.org/authors/?q=ai:liu.yanyan"Zhao, Leiga"https://zbmath.org/authors/?q=ai:zhao.leigaSummary: In this paper, we are concerned with the nonlinear Schrödinger equation
\[
-\Delta u + V(x)u + \lambda u = g(u)\text{ in }\mathbb{R}^N,\; \lambda\in\mathbb{R},
\]
with prescribed \(L^2\)-norm \(\int_{\mathbb{R}^N}u^2dx = \rho^2\) and \(\lim_{|x|\rightarrow+\infty}V(x) =: V_\infty \leq +\infty\) under general assumptions on \(g\) which allows at least mass critical growth. For the case of \(V_\infty < \infty\), including singular potential, the sufficient conditions are given for the existence of a ground state solution by developing the minimization methods with constraints proposed in [\textit{B. Bieganowski} and \textit{J. Mederski}, J. Funct. Anal. 280, No. 11, Article ID 108989, 26 p. (2021; Zbl 1465.35151)] and a delicate analysis of estimates on the least energy comparing with the limiting functional. While for the trapping case \(V_\infty = \infty\), the existence of a ground state solution as well as a second solution of mountain pass type is established.Normalized solutions to Schrödinger equations in the strongly sublinear regimehttps://zbmath.org/1541.351612024-09-27T17:47:02.548271Z"Mederski, Jarosław"https://zbmath.org/authors/?q=ai:mederski.jaroslaw"Schino, Jacopo"https://zbmath.org/authors/?q=ai:schino.jacopoSummary: We look for solutions to the Schrödinger equation
\[
-\Delta u + \lambda u = g(u) \quad \text{in }\mathbb{R}^N
\]
coupled with the mass constraint \(\int_{\mathbb{R}^N}|u|^2dx = \rho^2\), with \(N \geq 2\). The behaviour of \(g\) at the origin is allowed to be strongly sublinear, i.e., \(\lim_{s\rightarrow 0}g(s)/s = -\infty\), which includes the case
\[
g(s) = \alpha s\ln s^2 + \mu|s|^{p-2}s
\]
with \(\alpha > 0\) and \(\mu\in\mathbb{R}\), \(2 < p \leq 2^\ast\) properly chosen. We consider a family of approximating problems that can be set in \(H^1(\mathbb{R}^N)\) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about \(g\) that allow us to work in a suitable subspace of \(H^1(\mathbb{R}^N)\), we prove the existence of infinitely, many solutions.Pointwise eigenvector estimates by landscape functions: some variations on the Filoche-Mayboroda-van den Berg boundhttps://zbmath.org/1541.351622024-09-27T17:47:02.548271Z"Mugnolo, Delio"https://zbmath.org/authors/?q=ai:mugnolo.delioSummary: Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue -- much in the spirit of earlier results by Donsker-Varadhan and Bañuelos-Carrol -- as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including \(p\)-Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments.
{\copyright} 2023 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH.On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithmshttps://zbmath.org/1541.351632024-09-27T17:47:02.548271Z"Nickl, Richard"https://zbmath.org/authors/?q=ai:nickl.richard"Wang, Sven"https://zbmath.org/authors/?q=ai:wang.svenSummary: The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevin-type MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data.
Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local `geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schrödinger equation.The Schrödinger equation and the two-slit experiment of quantum mechanicshttps://zbmath.org/1541.351642024-09-27T17:47:02.548271Z"Webb, Glenn"https://zbmath.org/authors/?q=ai:webb.glenn-fSummary: The one-dimensional time-dependent complex-valued Schrödinger equation is used to describe the interference diffraction patterns of two-slit experiments in quantum mechanics. The properties of the diffraction patterns are related to the initial data of the Schrödinger equation. The Schrödinger equation solutions are compared to Fraunhofer diffraction formulas in geometric optics, and to the solutions of nonlocal advection diffusion equations.Constraint minimization problem of the nonlinear Schrödinger equation with the Anderson Hamiltonianhttps://zbmath.org/1541.351652024-09-27T17:47:02.548271Z"Zhang, Qi"https://zbmath.org/authors/?q=ai:zhang.qi.1"Duan, Jinqiao"https://zbmath.org/authors/?q=ai:duan.jinqiaoSummary: We consider the two-dimensional nonlinear Schrödinger equation with the Anderson hamiltonian, which given by the Laplacian plus a white noise potential. After establishing the energy space through the paracontrolled distribution framework, we prove the existence of the minimizer as the least energy solution through studying the minimization problem of the corresponding energy functional subject to \(L^2\) constraints. Subsequently, we study the regularity of the minimizer, which is a ground state solution of the nonlinear Schrödinger equation. Finally, we derive a tail estimate for the distribution of the principal eigenvalue corresponding to the ground state solution by energy estimates.Normalized solutions of the Schrödinger equation with potentialhttps://zbmath.org/1541.351662024-09-27T17:47:02.548271Z"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xin"Zou, Wenming"https://zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, for dimension \(N \geq 2\) and prescribed mass \(m>0\), we consider the following nonlinear scalar field equation with \(L^2\) constraint:
\[
\begin{cases}
-\Delta u + V(x)u + \lambda u=g(u) \qquad \text{ in } \mathbb{R}^N, \\
\int_{\mathbb{R}^N} u^2=m,
\end{cases}
\]
where \(\lambda \in \mathbb{R}\) is a Lagrange multiplier, \(V(x) \in C^1 (\mathbb{R}^N, \mathbb{R})\). In particular, \(g(x) \in C (\mathbb{R}, \mathbb{R})\) satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions \(V(x), g(x)\) by the mountain pass argument.
{\copyright} 2023 Wiley-VCH GmbH.Local near-field scattering data enables unique reconstruction of rough electric potentialshttps://zbmath.org/1541.351672024-09-27T17:47:02.548271Z"Cañizares, Manuel"https://zbmath.org/authors/?q=ai:canizares.manuelSummary: The focus of this paper is the study of the inverse point-source scattering problem, specifically in relation to a certain class of electric potentials. Our research provides a novel uniqueness result for the inverse problem with local data, obtained from the near field pattern. Our work improves the work of \textit{P. Caro} and \textit{A. Garcia} [Commun. Math. Phys. 379, No. 2, 543--587 (2020; Zbl 1450.35204)], who investigated both the direct problem and the inverse problem with global near field data for \textit{critically singular} and \(\delta\)-\textit{shell} potentials. The primary contribution of our research is the introduction of a Runge approximation result for the near field data on the scattering problem which, in combination with an interior regularity argument, enables us to establish a uniqueness result for the inverse problem with local data. Additionaly, we manage to consider a slightly wider class of potentials.
{{\copyright} 2024 IOP Publishing Ltd}Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theoryhttps://zbmath.org/1541.351682024-09-27T17:47:02.548271Z"Davey, Blair"https://zbmath.org/authors/?q=ai:davey.blair"Smit Vega Garcia, Mariana"https://zbmath.org/authors/?q=ai:smit-vega-garcia.marianaThe authors consider the parabolic equation
\[
\operatorname{div}(A\nabla u)+\partial_t u=0,
\]
where \(A=G G^T\) is a non-constant elliptic matrix (and \(G\) is a the Jacobian matrix of a diffeomorphism in \(\mathbb R^d\)), and derive various results, namely a parabolic Carleman estimate, an Almgren-type theorem, and a monotonicity result for Alt-Caffarelli-Friendman-type energies. These results extend and complement the analogous obtained in [\textit{B. Davey}, Arch. Ration. Mech. Anal. 228, No. 1, 159--196 (2018; Zbl 1390.35113)] for the constant coefficient case. The innovation here is the use of transformations that connect so-called parabolic functions \(u = u(x, t)\) defined on \(\mathbb R^d \times (0, T )\) to a sequence of elliptic functions \(v_n = v_n(y)\) defined in \(\mathbb R^{d\times n}\) for \(n \in \mathbb N\), so that all the results are obtained through elliptic arguments.
Reviewer: Davide Buoso (Alessandria)Characterizations of generalized Hardy and BMO spaces via square functionshttps://zbmath.org/1541.351692024-09-27T17:47:02.548271Z"Do, Tan Duc"https://zbmath.org/authors/?q=ai:do.tan-ducSummary: Let \(d \in \{3, 4, 5, \dots\}\). Consider \(L = - \frac{1}{w} \operatorname{div}(A \nabla u) + \mu\) over its maximal domain in \(L_w^2 (\mathbb{R}^d)\). Under certain conditions on the weight \(w\), the coefficient matrix \(A\), and the positive Radon measure \(\mu\), we characterize the Hardy space \(H_L^1 (\mathbb{R}^d)\) and BMO space \(\mathrm{BMO}_L (\mathbb{R}^d)\) associated with \(L\) using square functions. Other results include \(H_L^1 (\mathbb{R}^d) \equiv H_{L, \max}^1 (\mathbb{R}^d)\) and \((H_L^1 (\mathbb{R}^d))^\ast \equiv \mathrm{BMO}_L (\mathbb{R}^d)\) as norm spaces, where \(H_{L, \max}^1 (\mathbb{R}^d)\) denotes the maximal Hardy space associated with \(L\).Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in \(\mathbb{R}^2\)https://zbmath.org/1541.351702024-09-27T17:47:02.548271Z"Hashizume, Masato"https://zbmath.org/authors/?q=ai:hashizume.masatoSummary: In this paper, we investigate variational problems in \(\mathbb{R}^2\) with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.A Liouville-type theorem in conformally invariant equationshttps://zbmath.org/1541.351712024-09-27T17:47:02.548271Z"Li, Mingxiang"https://zbmath.org/authors/?q=ai:li.mingxiang|li.mingxiang.1Summary: Given a smooth function \(K(x)\) satisfying a polynomially cone condition and \(x \cdot \nabla K \leq 0\), we prove that there is no solution \(u \in C^\infty (\mathbb{R}^2)\) of the equation
\[
-\Delta u=K(x)e^{2u} \quad \text{ on } \mathbb{R}^2
\]
with \(u \leq C\) and \(\int_{\mathbb{R}^2} |K(x)|e^{2u} \mathrm{d}x < +\infty\). As a consequence, there is no such solution if \(K(x)\) is a non-constant polynomial with \(x \cdot \nabla K \leq 0\). The latter result already includes a result of \textit{M. Struwe} [J. Eur. Math. Soc. (JEMS) 22, No. 10, 3223--3262 (2020; Zbl 1458.35151)] as a particular case. Higher order cases are set up with additional assumption on the behavior of \(\Delta u\) near infinity.Classification of radial solutions for fully nonlinear equations with Hardy potentialhttps://zbmath.org/1541.351722024-09-27T17:47:02.548271Z"Maia, Liliane"https://zbmath.org/authors/?q=ai:maia.liliane-a"Nornberg, Gabrielle"https://zbmath.org/authors/?q=ai:nornberg.gabrielle"Pacella, Filomena"https://zbmath.org/authors/?q=ai:pacella.filomenaSummary: We study existence, nonexistence and regularity of positive radial solutions for a class of nonlinear equations driven by Pucci extremal operators, power nonlinearity and Hardy weight. We classify both regular continuous nondifferentiable and singular solutions defined in radial domains, punctured or not. We also obtain critical threshold exponents for the solvability in the exterior of a ball, as well as uniqueness and symmetry in annuli. Our results are based on the behavior of the trajectories described through suitable dynamical systems on the plane, in addition to energy monotonicity and asymptotic analysis.Lipschitz bounds for integral functionals with \((p,q)\)-growth conditionshttps://zbmath.org/1541.351732024-09-27T17:47:02.548271Z"Bella, Peter"https://zbmath.org/authors/?q=ai:bella.peter"Schäffner, Mathias"https://zbmath.org/authors/?q=ai:schaffner.mathiasThe authors study local regularity properties of local minimizers of scalar integral functionals where the convex integrand satisfies controlled \((p, q)\)-growth conditions. Lipschitz continuity under sharp assumptions on the forcing term and improved assumptions on the growth conditions on the integrand with respect to the existing literature are proved. As an interesting by-product, an \(L^{\infty}\)-\(L^2\)-estimate for solutions of linear uniformly elliptic equations in divergence form is established. This interesting estimate is optimal with respect to the ellipticity ratio of the coefficients.
Reviewer: Vincenzo Vespri (Firenze)An energy model for harmonic functions with junctionshttps://zbmath.org/1541.351742024-09-27T17:47:02.548271Z"De Silva, D."https://zbmath.org/authors/?q=ai:de-silva.daniela"Savin, O."https://zbmath.org/authors/?q=ai:savin.ovidiu-vSummary: We consider an energy model for harmonic graphs with junctions and study the regularity properties of minimizers and their free boundaries.Unique solvability of the Zaremba problem for linear second order elliptic equations with drifthttps://zbmath.org/1541.351752024-09-27T17:47:02.548271Z"Alkhutov, Yu. A."https://zbmath.org/authors/?q=ai:alkhutov.yuriy-alexandrovich"Chechkin, G. A."https://zbmath.org/authors/?q=ai:chechkin.gregory-aSummary: We establish the unique solvability of the Zaremba problem for linear second order elliptic equations in divergence form with measurable coefficients and lower order terms.Positive solutions, positive radial solutions and uniqueness results for some nonlocal elliptic problemshttps://zbmath.org/1541.351762024-09-27T17:47:02.548271Z"Bellamouchi, Chahinez"https://zbmath.org/authors/?q=ai:bellamouchi.chahinez"Zaouche, Elmehdi"https://zbmath.org/authors/?q=ai:zaouche.elmehdiSummary: In this paper, we prove existence of a positive solution to a one-dimensional nonlocal elliptic problem and existence of a positive radial solution to a multidimensional nonlocal elliptic problem under weak conditions on the reaction terms and the diffusion coefficients. We use Krasnoselskii's fixed point theorem. Uniqueness results are also given.On the contraction properties for weak solutions to linear elliptic equations with \(L^2\)-drifts of negative divergencehttps://zbmath.org/1541.351772024-09-27T17:47:02.548271Z"Lee, Haesung"https://zbmath.org/authors/?q=ai:lee.haesungSummary: We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with \(L^2\)-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the \(L^r\)-contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian \(C_0\)-resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an \(L^1\)-stability result through an extended version of the \(L^1\)-contraction property.Robin problems for elliptic equations with singular drifts on Lipschitz domainshttps://zbmath.org/1541.351782024-09-27T17:47:02.548271Z"Ma, Wenxian"https://zbmath.org/authors/?q=ai:ma.wenxian"Yang, Sibei"https://zbmath.org/authors/?q=ai:yang.sibeiSummary: Let \(n \geq 2\) and \(\Omega \subset \mathbb{R}^n\) be a bounded Lipschitz domain. Assume that \(\mathbf{b} \in L^{n*}(\Omega; \mathbb{R}^n)\) and \(\gamma\) is a non-negative function on \(\partial \Omega\) satisfying some mild assumptions, where \(n^* := n\) when \(n \geq 3\) and \(n^* \in (2, \infty)\) when \(n=2\). In this article, we establish the unique solvability of the Robin problems
\[
\begin{cases}
\quad\;\; -\Delta u + \operatorname{div}(u\mathbf{b}) = f & \text{ in } \Omega, \\
(\nabla u-u\mathbf{b}) \cdot \boldsymbol{\nu} + \gamma u = u_R & \text{ on } \partial \Omega
\end{cases}
\]
and
\[
\begin{cases}
-\Delta v - \mathbf{b} \cdot \nabla v = g & \text{ in } \Omega, \\
\quad\, \nabla v \cdot \boldsymbol{\nu} + \gamma v = v_R & \text{ on } \partial \Omega
\end{cases}
\]
in the Bessel potential space \(L^p_\alpha (\Omega)\), where \(\alpha \in (0, 2)\) and \(p \in (1, \infty)\) satisfy some restraint conditions, and \(\boldsymbol{\nu}\) denotes the outward unit normal to the boundary \(\partial \Omega\). The results obtained in this article extend the corresponding results established by \textit{H. Kim} and \textit{H. Kwon} [Trans. Am. Math. Soc. 375, No. 9, 6537--6574 (2022; Zbl 1498.35204)] for the Dirichlet and the Neumann problems to the case of the Robin problem.The anisotropic Calderón problem for high fixed frequencyhttps://zbmath.org/1541.351792024-09-27T17:47:02.548271Z"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-a"Wang, Yiran"https://zbmath.org/authors/?q=ai:wang.yiranSummary: We consider Schrödinger operators at a fixed high frequency on simply connected compact Riemannian manifolds with nonpositive sectional curvatures and smooth strictly convex boundaries. We prove that the Dirichlet-to-Neumann map uniquely determines the potential.New multiplicity results in prescribing \(Q\)-curvature on standard sphereshttps://zbmath.org/1541.351802024-09-27T17:47:02.548271Z"Ben Ayed, Mohamed"https://zbmath.org/authors/?q=ai:ben-ayed.mohamed"El Mehdi, Khalil"https://zbmath.org/authors/?q=ai:el-mehdi.khalil-oSummary: In this paper, we study the problem of prescribing \(Q\)-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function \(K\) so that it is the \(Q\)-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function \(K\).Non-degeneracy of \(\mathcal{O}(3)\) invariant solutions for higher order prescribed curvature problem and applicationshttps://zbmath.org/1541.351812024-09-27T17:47:02.548271Z"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan.1|gao.yuan.3|gao.yuan"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Hu, Yichen"https://zbmath.org/authors/?q=ai:hu.yichenSummary: We consider the following prescribed curvature problem involving polyharmonic operators on \(\mathbb{S}^N\)
\[
D^m \tilde{u} = \widetilde{K}(y)\tilde{u}^{m^* -1}, \quad \tilde{u}>0 \text{ in }\mathbb{S}^N, \quad\tilde{u} \in H^m (\mathbb{S}^N),
\]
where \(\widetilde{K}(y)>0\) is a radial function, \(m^* =\frac{2N}{N-2m}, m\geq 1\) is an integer and \(D^m\) is \(2m\)-order differential operator given by
\[
D^m =\prod_{i=1}^m \bigg( -\Delta_g +\frac{1}{4}(N-2i)(N+2i-2)\bigg).
\]
Here \(\Delta_g\) is the Laplace-Beltrami operator on \(\mathbb{S}^N\), and \(\mathbb{S}^N\) is the unit sphere with Riemann metric \(g\). We are concerned with the solutions which are invariant under some non-trivial sub-group of \(\mathcal{O}(3)\) to the above problem. We first prove a non-degeneracy result for this kind of \(\mathcal{O}(3)\) invariant solutions. As an application, we consider an eigenvalue problem, we investigate the properties of the eigenvalues and obtain the Morse index estimate of the \(\mathcal{O}(3)\) invariant solutions. Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of the Green function.Sharp higher order Adams' inequality with exact growth condition on weighted Sobolev spaceshttps://zbmath.org/1541.351822024-09-27T17:47:02.548271Z"do Ó, João Marcos"https://zbmath.org/authors/?q=ai:do-o.joao-m-bezerra"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Ponciano, Raoní"https://zbmath.org/authors/?q=ai:ponciano.raoniSummary: This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory.Sign-changing bubble tower solutions for a Paneitz-type problemhttps://zbmath.org/1541.351832024-09-27T17:47:02.548271Z"Chen, Wenjing"https://zbmath.org/authors/?q=ai:chen.wenjing"Huang, Xiaomeng"https://zbmath.org/authors/?q=ai:huang.xiaomengSummary: This paper is concerned with the following biharmonic problem
\[
\begin{cases}
\Delta^2u=|u|^{\frac{8}{N-4}}u & \text{ in }\Omega\backslash\overline{B(\xi_0,\varepsilon)},\\
u=\Delta u=0 & \text{ on }\partial\left(\Omega\backslash\overline{B(\xi_0,\varepsilon)}\right),
\end{cases}\tag{0.1}
\]
where \(\Omega\) is an open bounded domain in \(\mathbb{R}^N\), \(N\geqslant 5\), and \(B(\xi_0,\varepsilon)\) is a ball centered at \(\xi_0\) with radius \(\varepsilon,\varepsilon\) is a small positive parameter. We obtain the existence of solutions for problem (0.1), which is an arbitrary large number of sign-changing solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the centre of the hole.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Qualitative properties of weighted fourth order elliptic problemshttps://zbmath.org/1541.351842024-09-27T17:47:02.548271Z"Guo, Zongming"https://zbmath.org/authors/?q=ai:guo.zongming"Wan, Fangshu"https://zbmath.org/authors/?q=ai:wan.fangshuSummary: Qualitative properties of nonnegative solutions of weighted fourth order elliptic problems arising from the equations on singular manifolds with conical metrics are studied. The weights may be singular in the domains. We obtain the uniqueness of positive radial solution of the Navier boundary value problem in the ball \(B\) and its exact regularity at the singular point \(x = 0 \), which determines the exact regularity of the unique positive radial solution in \(B \). We also establish the Liouville type results for the equation on \(\mathbb{R}^N \backslash \{0\} \).Global bifurcation of positive solutions for a class of superlinear biharmonic equationshttps://zbmath.org/1541.351852024-09-27T17:47:02.548271Z"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantang"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyun"Zhao, Jiao"https://zbmath.org/authors/?q=ai:zhao.jiaoSummary: We are concerned with semilinear biharmonic equations of the form
\[
\begin{cases}
\Delta^2 u = \lambda f (u) & \text{in } B, \\
u = \frac{\partial u}{\partial \nu} = 0 & \text{on } \partial B,
\end{cases}
\]
where \(B\) denotes the unit ball in \(\mathbb{R}^N\), \(N \geq 1\), \(\lambda > 0\) is a parameter, \(\partial / \partial \nu\) is the outward normal derivative, \(f\) : \(\mathbb{R} \to (0, + \infty)\) is a continuous function that has super-linear growth at infinity. We use bifurcation theory, combined with an approximation scheme to establish the existence of an unbounded branch of positive radial solutions, which is bounded in positive \(\lambda \)-direction. If in addition, \(f\) satisfies certain subcritical condition, we show that the branch must bifurcate from infinity at \(\lambda = 0\).On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theoremhttps://zbmath.org/1541.351862024-09-27T17:47:02.548271Z"Unal, Cihan"https://zbmath.org/authors/?q=ai:unal.cihanSummary: In this work, we consider a special nondegenerate equation with two weights. We investigate multiplicity result of this biharmonic equation. Mainly, our purpose is to obtain this result using an alternative Ricceri's theorem. Moreover, we give some compact embeddings in variable exponent Sobolev spaces with second order to prove the main idea.On first order elliptic systemshttps://zbmath.org/1541.351872024-09-27T17:47:02.548271Z"Ndjinga, Michaël"https://zbmath.org/authors/?q=ai:ndjinga.michael"Ngwamou, Sédrick Kameni"https://zbmath.org/authors/?q=ai:ngwamou.sedrick-kameniSummary: The aim of this paper is the study of first-order stationary systems of PDEs of the form \(\sum_k A_k \partial_k U + KU=0\) with \(K \ngtr 0\) on \(\Omega = \mathbb{R}^d\) and \(\Omega \subset \mathbb{R}^d\) bounded. We prove that the classical assumption \(K>0\) is not necessary for the well-posedness of the system and is violated in the particular case of the first-order Poisson problem. In the case \(\Omega = \mathbb{R}^d\), we use Fourier analysis for the existence and uniqueness of solutions. For \(\Omega \subset \mathbb{R}^d\) bounded, we use a complex analog of the \textit{Banach}-\textit{Nečas}-\textit{Babuška} theorem to obtain the existence and uniqueness of a solution in a setting that encompasses both Friedrichs' systems and the first order reduction of the Poisson problem. The techniques used to prove the classical inf-sup conditions are inspired by harmonic analysis arguments that are consistent with the case \(\Omega = \mathbb{R}^d\). In order to illustrate our approach, we study in detail the reduction of the Poisson equation to a first-order system.Normalized solutions for Schrödinger systems in dimension twohttps://zbmath.org/1541.351882024-09-27T17:47:02.548271Z"Deng, Shengbing"https://zbmath.org/authors/?q=ai:deng.shengbing"Yu, Junwei"https://zbmath.org/authors/?q=ai:yu.junweiSummary: In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with exponential growth
\[\begin{cases}
- \Delta u + \lambda_1 u = H_u (u, v), \qquad \text{in } \mathbb{R}^2, \\
- \Delta v + \lambda_2 v = H_v (u, v), \qquad \text{in } \mathbb{R}^2, \\
\int\limits_{\mathbb{R}^2} | u |^2 d x = a^2, \quad \int\limits_{\mathbb{R}^2} | v |^2 d x = b^2,
\end{cases}\]
where \(a, b > 0\) are prescribed, \(\lambda_1, \lambda_2 \in \mathbb{R}\) and the functions \(H_u\), \(H_v\) are partial derivatives of a Carathéodory function \(H\) with \(H_u\), \(H_v\) satisfying exponential growth in \(\mathbb{R}^2\). Our main result is totally new for the Schrödinger system in \(\mathbb{R}^2\). Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.Normalized solutions of linear and nonlinear coupled Choquard systems with potentialshttps://zbmath.org/1541.351892024-09-27T17:47:02.548271Z"Guo, Zhenyu"https://zbmath.org/authors/?q=ai:guo.zhenyu.1|guo.zhenyu"Jin, Wenyan"https://zbmath.org/authors/?q=ai:jin.wenyanSummary: In this paper, we study Choquard systems with linear and nonlinear couplings with different potentials under the \(L^2\)-constraint. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for \(L^2\)-subcritical case when the dimension is greater than or equal to 2 without potentials. In addition, a positive solution with prescribed \(L^2\)-constraint under some appropriate assumptions with the potentials was obtained. The proof is based on the refined energy estimates.Total masses of solutions to general Toda systems with singular sourceshttps://zbmath.org/1541.351902024-09-27T17:47:02.548271Z"Karmakar, Debabrata"https://zbmath.org/authors/?q=ai:karmakar.debabrata"Lin, Chang-Shou"https://zbmath.org/authors/?q=ai:lin.chang-shou"Nie, Zhaohu"https://zbmath.org/authors/?q=ai:nie.zhaohu"Wei, Juncheng"https://zbmath.org/authors/?q=ai:wei.junchengThe authors consider the following Toda system on \(\mathbb{R}^2\)
\[
\left\{\begin{array}{ll} \Delta u_i+4\sum_{j=1}^na_{ij}e^{u_j}=4\pi \gamma_i\delta_0, \ \ &\gamma_i>-1,\\
\displaystyle{\int_{\mathbb{R}^2}e^{u_i}dx}<\infty, &1\leq i\leq n, \end{array}\right.\tag{1}
\]
where \((a_{ij})\) is the Cartan matrix of rank \(n\) associated to a complex simple Lie algebra \(\mathfrak{g}\), and \(\delta_0\) is the Dirac measure at \(0\). A solution of problem \((1)\) is a function \(u=(u_1,\dots,u_n)\in C^2(\mathbb{R}^2\setminus \{0\},\mathbb{R}^n)\) satisfying the equation \(\Delta u_i+4\sum_{j=1}^na_{ij}e^{u_j}=0\) in \(\mathbb{R}^2\setminus \{0\}\), and \(u_i(x)=2\gamma_i\log|x|+O(1)\) near \(0\), for \(i=1,\dots,n\).
The main result of the paper concerns the asymptotic behavior of the solutions. The authors prove that, if \(u=(u_1,\dots,u_n)\) is a solution to system \((1)\), then, setting
\[
U_i=\sum_{j=1}^na^{ij}u_j, \ \ \ \text{and} \ \ \ \gamma^i=\sum_{j=1}^na^{ij}\gamma_j, \ \ \ i=1,\dots,n,
\]
where \((a^{ij})\) is the inverse of \((a_{ij})\), one has
\[
U_i(z)=2(\gamma^i-\langle\omega_i-\kappa \omega_i,\omega_0\rangle)\log|z|+O(1) \ \ \ \text{as} \ \ z\rightarrow \infty,
\]
and
\[
\sigma_i(u)=2\langle\omega_i-\kappa \omega_i,\omega_0\rangle.
\]
Here,
\[
\sigma_i(u)=\frac{4}{2\pi}\int_{\mathbb{R}^2}e^{u_i}dx, \ \ \ i=1,\dots,n,
\]
is the total mass of \(u_i\), \(\kappa\) is the longest element of the Weyl group of the Lie algebra \(\mathfrak{g}\), \(\langle \cdot,\cdot \rangle\) is the pairing between the real Cartan subalgebra \(\mathfrak{h}_0\) and its dual \(\mathfrak{h}_0'\), \(\omega_i\in \mathfrak{h}_0'\) is the \(i\)-th fundamental weight, and \(\omega_0\in \mathfrak{h}_0\). The total mass \(\sigma_i(u)\) turns out to be an even integer if \(\gamma_i\in \mathbb{Z}_{\geq 0}\).
The above theorem generalizes previous results obtained in the case of Toda systems of types \(A\), \(G_2\) and \(B\), \(C\).
Reviewer: Giovanni Anello (Messina)Normalized solutions of two-component nonlinear Schrödinger equations with linear coupleshttps://zbmath.org/1541.351912024-09-27T17:47:02.548271Z"Li, J. M."https://zbmath.org/authors/?q=ai:li.jiaming|li.jinmian|li.jianmei|li.jingming|li.jiamei|li.junmin|li.jimeng|li.junming|li.jiu-ming|li.jingmei|li.jiamin|li.jiangmeng|li.junmei|li.jinmei|li.jimei|li.jianmin|li.jie-ming|li.jin-man|li.jinming|li.jinming.1|li.jiemei|li.jiaomei|li.jiming|li.jiemin|li.jing-min|li.jingmao|li.jiameng|li.jianming|li.jian-meng|li.jimin"Shen, Z. F."https://zbmath.org/authors/?q=ai:shen.zhifei|shen.zhaofen|shen.zifan|shen.zhifang|shen.zhoufeng|shen.zifeiSummary: In this paper, we focus on the following nonlinear Schrödinger equations with linear couples
\[\begin{cases}
-\Delta u + V_1(x)u + \lambda_1 u = \mu_1 \int_{\mathbb{R}^3} \frac{|u(y)|^p}{|x-y|} \mathrm{d} y |u|^{p-2}u + \beta v \quad & \text{in } \mathbb{R}^3, \\
-\Delta v + V_2(x)v + \lambda_2v = \mu_2 \int_{\mathbb{R}^3} \frac{|v(y)|^q}{|x-y|} \mathrm{d}y |v|^{q-2} v + \beta u \quad & \text{in } \mathbb{R}^3, \\
\int_{\mathbb{R}^3} |u|^2 \mathrm{d} x=a, \quad \int_{\mathbb{R}^3} |v|^2 \mathrm{d}x = b,
\end{cases}\]
where \(\frac{5}{3}<p\), \(q<\frac{7}{3}\), \(\mu_1, \mu_2 > 0\), \(a,b\geq 0\), \(\beta \in \mathbb{R}\setminus \{0\}\), \(\lambda_1,\lambda_2\in \mathbb{R}\) are Lagrange multipliers and \(V_1(x),V_2(x):\mathbb{R}^3\to \mathbb{R}\) are trapping potentials. We prove the existence of the solutions with prescribed \(L^2(\mathbb{R})\) -norm with trivial trapping potentials and nontrivial trapping potentials by applying the rearrangement inequalities.Normalized solutions for a Schrödinger system with critical Sobolev growth in \(\mathbb{R}^3\)https://zbmath.org/1541.351922024-09-27T17:47:02.548271Z"Liu, Mei-Qi"https://zbmath.org/authors/?q=ai:liu.meiqi"Zou, Wenming"https://zbmath.org/authors/?q=ai:zou.wenmingSummary: We study the following critical Schrödinger system in \(\mathbb{R}^3\):
\[
\begin{cases}
-\Delta u+\lambda_1 u=|u|^4 u+\mu_1 |u|^{p-2} u+\alpha \nu |u|^{\alpha -2} u|v|^{\beta}, \\
-\Delta v+\lambda_2 v=|v|^4 v+\mu_2|v|^{p-2}v+\beta \nu |u|^{\alpha} |v|^{\beta -2} v, \\
\int_{\mathbb{R}^3} u^2 dx=a^2 \text{ and } \int_{\mathbb{R}^3} v^2d x=b^2, \; u,v \in H^1 (\mathbb{R}^3),
\end{cases}
\]
where \(\alpha, \beta >1\), \(\alpha +\beta =2^*=6\), \(p \in (2,6)\), \(\nu >0\), and \(\mu_1, \mu_2, \lambda_1, \lambda_2 \in \mathbb{R}\). Any \((u,v)\) solving such system (for some \(\lambda_1,\lambda_2\)) is called the normalized solution in the literature, where the normalization is settled in \(L^2 (\mathbb{R}^3)\). We show that this system has a positive ground state for \(p \in (2, \frac{10}{3})\) in the case of \(\mu_1, \mu_2>0\). For the case of \(2<p<6\) and \(\mu_1, \mu_2<0\), we obtain the non-existence results.
{\copyright} 2023 Wiley-VCH GmbH.Existence and asymptotic behavior of ground states for linearly coupled systems involving exponential growthhttps://zbmath.org/1541.351932024-09-27T17:47:02.548271Z"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"de Albuquerque, José Carlos"https://zbmath.org/authors/?q=ai:de-albuquerque.jose-carlos"dos Santos, Edjane O."https://zbmath.org/authors/?q=ai:dos-santos.edjane-oSummary: In this paper we study the following class of linearly coupled systems in the plane:
\[
\begin{cases}
-\Delta u + u = f_1(u) + \lambda v, \quad \text{in} \quad \mathbb{R}^2, \\
-\Delta v + v = f_2(v) + \lambda u, \quad \text{in} \quad \mathbb{R}^2, \
\end{cases}
\]
where \(f_1, f_2\) are continuous functions with critical exponential growth in the sense of Trudinger-Moser inequality and \(0<\lambda <1\) is a parameter. First, for any \(\lambda \in (0,1)\), by using minimization arguments and minimax estimates we prove the existence of a positive ground state solution. Moreover, we study the asymptotic behavior of these solutions when \(\lambda \to 0^+\). This class of systems can model phenomena in nonlinear optics and in plasma physics.Hamiltonian systems involving exponential growth in \(\mathbb{R}^2\) with general nonlinearitieshttps://zbmath.org/1541.351942024-09-27T17:47:02.548271Z"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"Souza, Manassés de"https://zbmath.org/authors/?q=ai:souza.manasses-de"Menezes, Marta"https://zbmath.org/authors/?q=ai:menezes.martaSummary: In this work, we establish the existence of ground state solution for Hamiltonian systems of the form
\[
\begin{cases}
-\Delta u + V(x)u = H_v(x, u, v), & x\in\mathbb{R}^2, \\
-\Delta v + V(x)v = H_u(x, u, v), & x\in\mathbb{R}^2,
\end{cases}
\]
where \(V \in C(\mathbb{R}^2, (0, \infty))\) and \(H \in C^1(\mathbb{R}^2\times\mathbb{R}^2, \mathbb{R})\) is allowed to have an exponential growth with respect to the Trudinger-Moser inequality. We study the case where \(V\) and \(H\) are periodic or asymptotically periodic. In the proof of the main results, we have used a reduction method involving the generalized Nehari manifold and also a linking theorem. In our approach, as we deal with general nonlinearities, it was necessary to obtain a new version of the Trudinger-Moser inequality.Nonexistence of ground state sign-changing solutions for autonomous Schrödinger-Poisson system with critical growthhttps://zbmath.org/1541.351952024-09-27T17:47:02.548271Z"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.38|wang.ying.9|wang.ying.53|wang.ying.2|wang.ying.23|wang.ying.35|wang.ying.19|wang.ying.16|wang.ying.36|wang.ying.8|wang.ying.31|wang.ying|wang.ying.42|wang.ying.12"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rong.1|yuan.rong(no abstract)Young measure theory for steady problems in Orlicz-Sobolev spaceshttps://zbmath.org/1541.351962024-09-27T17:47:02.548271Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: In this paper, we study the existence of weak solutions for Dirichlet boundary-value problems given in the following quasilinear elliptic system
\[
\begin{cases}
\begin{aligned}
-\operatorname{div} \sigma (x, u, Du) + b(x, u, Du) &= f (x, u, Du)\text{ in }\Omega,\\
u &= 0\text{ on }\partial \Omega.
\end{aligned}
\end{cases}
\]
We prove the needed result, relying on the theory of Young measures, Galerkin's approximation and weak monotonicity assumptions on \(\sigma \), in reflexive Orlicz-Sobolev spaces.Operator estimates for problems in domains with singularly curved boundary: Dirichlet and Neumann conditionshttps://zbmath.org/1541.351972024-09-27T17:47:02.548271Z"Borisov, D. I."https://zbmath.org/authors/?q=ai:borisov.denis-i"Suleimanov, R. R."https://zbmath.org/authors/?q=ai:suleimanov.r-rSummary: We consider a system of second-order semilinear elliptic equations in a multidimensional domain with an arbitrarily curved boundary contained in a narrow layer along the unperturbed boundary. The Dirichlet or Neumann condition is imposed on the curved boundary. In the case of the Neumann condition, rather natural and weak conditions are additionally imposed on the structure of the curving. Under these conditions, we show that the homogenized problem is one for the same system of equations in the unperturbed problem with a boundary condition of the same kind as on the perturbed boundary. The main result is operator \(W_2^1\)- and \(L_2\)-estimates.Coexistence states in a cross-diffusion system of a competition modelhttps://zbmath.org/1541.351982024-09-27T17:47:02.548271Z"Cui, Lu"https://zbmath.org/authors/?q=ai:cui.lu"Li, Shanbing"https://zbmath.org/authors/?q=ai:li.shanbingSummary: The main goal of this paper is to study the stationary problem for a Lotka-Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by \textit{K. Kuto} and \textit{T. Tsujikawa} [J. Differ. Equations 258, No. 5, 1801--1858 (2015; Zbl 1308.35091)] and \textit{Q. Wang} et al. [Discrete Contin. Dyn. Syst. 35, No. 3, 1239--1284 (2015; Zbl 1327.92050)], where the corresponding Neumann problem is analyzed.Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional caseshttps://zbmath.org/1541.351992024-09-27T17:47:02.548271Z"Hajaiej, Hichem"https://zbmath.org/authors/?q=ai:hajaiej.hichem"Liu, Tianhao"https://zbmath.org/authors/?q=ai:liu.tianhao"Zou, Wenming"https://zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms:
\[
\begin{cases}
-\Delta u=\lambda_1 u+ \mu_1 |u|^{2p-2}u+\beta |u|^{p-2}|v|^p u+\theta_1 u\log u^2, & x\in \Omega, \\
-\Delta v=\lambda_2 v+ \mu_2 |v|^{2p-2}v+\beta |u|^p |v|^{p-2}v+\theta_2 v\log v^2, & x\in \Omega, \\
u=v=0, & x \in \partial \Omega,
\end{cases}
\]
where \(\Omega \subset\mathbb{R}^N\) is a bounded smooth domain, \(2p=2^* =\frac{2N}{N-2}\) is the Sobolev critical exponent. When \(N\geq 5\), for different ranges of \(\beta, \lambda_i, \mu_i, \theta_i, i=1,2\), we obtain existence and nonexistence results of positive solutions via variational methods. The special case \(N=4\) was studied by the first author et al. [J. Geom. Anal. 34, No. 6, Paper No. 182, 44 p. (2024; Zbl 1537.35180)]. Note that for \(N\geq 5\), the critical exponent is given by \(2p\in (2,4)\); whereas for \(N=4\), it is \(2p=4\). In the higher-dimensional cases \(N\geq 5\) brings new difficulties, and requires new ideas. Besides, we also study the Brézis-Nirenberg problem with logarithmic perturbation
\[
-\Delta u=\lambda u+\mu |u|^{2p-2}u+\theta u \log u^2 \quad \text{ in }\Omega,
\]
where \(\mu >0\), \(\theta <0\), \(\lambda \in\mathbb{R}\), and obtain the existence of positive local minimum and least energy solution under some certain assumptions.Principal eigenvalues and eigenfunctions to Lane-Emden systems on general bounded domainshttps://zbmath.org/1541.352002024-09-27T17:47:02.548271Z"Leite, Edir Junior Ferreira"https://zbmath.org/authors/?q=ai:leite.edir-ferreira-jun"Montenegro, Marcos"https://zbmath.org/authors/?q=ai:montenegro.marcosSummary: We prove the existence of at least a curve of principal eigenvalues for two-parameter Lane-Emden systems under Dirichlet boundary conditions for general bounded domains. The nonhomogeneous counterpart is also addressed. Part of the main results (Theorems 1.1-1.3) are based on some deep ideas introduced in the seminal paper [\textit{H. Berestycki} et al., Commun. Pure Appl. Math. 47, No. 1, 47--92 (1994; Zbl 0806.35129)] and on two fundamental tools, both new and of independent interest: Aleksandrov-Bakelman-Pucci estimates (Theorem 2.1) and Harnack-Krylov-Safonov inequalities (Theorem 5.1) associated to Lane-Emden systems in smooth domains.Symmetry results of solutions for elliptic systems with linear couplingshttps://zbmath.org/1541.352012024-09-27T17:47:02.548271Z"Li, Keqiang"https://zbmath.org/authors/?q=ai:li.keqiang"Wang, Shangjiu"https://zbmath.org/authors/?q=ai:wang.shangjiu"Li, Yin"https://zbmath.org/authors/?q=ai:li.yin.1"Li, Shaoyong"https://zbmath.org/authors/?q=ai:li.shaoyongSummary: This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in \(\mathbb{R}^N\). Using the Alexandrov-Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [\textit{J. Su} et al., Appl. Math. Lett. 100, Article ID 106042, 6 p. (2020; Zbl 1428.35035)].Global integrability for solutions to quasilinear elliptic systems with degenerate coercivityhttps://zbmath.org/1541.352022024-09-27T17:47:02.548271Z"Li, Ya"https://zbmath.org/authors/?q=ai:li.ya"Liu, Gaoyang"https://zbmath.org/authors/?q=ai:liu.gaoyang"Gao, Hongya"https://zbmath.org/authors/?q=ai:gao.hongyaSummary: This paper deals with global integrability for solutions to quasilinear elliptic systems involving \(N\) equations of the form
\[
\begin{cases}
-\sum\limits_{i=1}^n D_i \left(\sum\limits_{\beta =1}^N \sum\limits_{j=1}^n a^{\alpha, \beta}_{i,j} (x, u(x)) D_j u^\beta (x) \right) = f^\alpha (x), & \text{ in } \Omega, \\
u(x)=0, & \text{ on } \partial \Omega,
\end{cases}
\]
where \(\Omega\) is an open bounded subset of \(\mathbb{R}^n\), \(n>2\), \(u=(u^1, u^2, \dots, u^N): \Omega \subset \mathbb{R}^n \to \mathbb{R}^N\), \(N \geq 2\). Under degenerate coercivity condition of the diagonal coefficients and proportional condition of the off-diagonal coefficients, we obtain some global integrability results.
{\copyright} 2024 Wiley-VCH GmbH.Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled systemhttps://zbmath.org/1541.352032024-09-27T17:47:02.548271Z"Ortegón Gallego, Francisco"https://zbmath.org/authors/?q=ai:ortegon-gallego.francisco"Rhoudaf, Mohamed"https://zbmath.org/authors/?q=ai:rhoudaf.mohamed"Talbi, Hajar"https://zbmath.org/authors/?q=ai:talbi.hajarSummary: \[
\text{In this paper, we analyze the following nonlinear elliptic problem }
\begin{cases}
A(u)=\rho (u)\vert \nabla \varphi \vert^2 \text{ in }\Omega, \\
\mathrm{div}(\rho (u)\nabla \varphi)=0\text{ in }\Omega, \\
u=0 \text{ on }\partial\Omega, \\
\varphi =\varphi_0 \text{ on }\partial\Omega.
\end{cases}
\text{ where }
\]
\(A(u) = -\mathrm{div} a(x, u, \nabla u)\) is a Leray-Lions operator of order \(p\). The second member of the first equation is only in \(L^1 (\Omega)\). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.Regularity for the steady Stokes-type flow of incompressible Newtonian fluids in some generalized function settingshttps://zbmath.org/1541.352042024-09-27T17:47:02.548271Z"Tran, Minh-Phuong"https://zbmath.org/authors/?q=ai:tran.minh-phuong"Nguyen, Thanh-Nhan"https://zbmath.org/authors/?q=ai:nguyen.thanh-nhan"Nguyen, Hong-Nhung"https://zbmath.org/authors/?q=ai:nguyen.hong-nhungSummary: A study of regularity estimate for weak solution to generalized stationary Stokes-type systems involving \(p\)-Laplacian is offered. The governing systems of equations are based on steady incompressible flow of a Newtonian fluids. This paper also provides a relatively complete picture of our main results in two regards: problems with nonlinearity is regular with respect to the gradient variable; and asymptotically regular problems, whose nonlinearity satisfies a particular structure near infinity. For such Stokes-type systems, we derive regularity estimates for both velocity gradient and its associated pressure in two special classes of function spaces: the generalized Lorentz and \(\psi \)-generalized Morrey spaces.Nodal solutions for nonlinear Schrödinger systemshttps://zbmath.org/1541.352052024-09-27T17:47:02.548271Z"Zhou, Xue"https://zbmath.org/authors/?q=ai:zhou.xue"Liu, Xiangqing"https://zbmath.org/authors/?q=ai:liu.xiangqingSummary: In this article we consider the nonlinear Schrodinger system \[ \begin{gathered} -\Delta u_j + \lambda_j u_j = \sum_{i=1}^k \beta_{ij} u_i^2 u_j, \qquad \text{in}\,\, \Omega, \\ u_j (x) = 0,\qquad \text{on}\,\, \partial \Omega,\; j=1, \ldots, k, \end{gathered}\] where \(\Omega\subset \mathbb{R}^N (N=2,3)\) is a bounded smooth domain, \(\lambda_j> 0\), \(j=1, \ldots, k, \beta_{ij}\) are constants satisfying \(\beta_{jj}>0\), \(\beta_{ij}=\beta_{ji} \leq 0\) for \(1\leq i< j\leq k\). The existence of sign-changing solutions is proved by the truncation method and the invariant sets of descending flow method.On the number of positive solutions for a higher order elliptic systemhttps://zbmath.org/1541.352062024-09-27T17:47:02.548271Z"Lu, Yichen"https://zbmath.org/authors/?q=ai:lu.yichen"Feng, Meiqiang"https://zbmath.org/authors/?q=ai:feng.meiqiangSummary: Some new criteria on existence of positive solution for a higher order elliptic problem with an eigenvalue parameter are established under some sublinear conditions, which involve the principle eigenvalues of the corresponding linear problems. New results on nonexistence and multiplicity of positive solutions are also derived.Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phasehttps://zbmath.org/1541.352072024-09-27T17:47:02.548271Z"Bao, Jiguang"https://zbmath.org/authors/?q=ai:bao.jiguang.1|bao.jiguang"Liu, Zixiao"https://zbmath.org/authors/?q=ai:liu.zixiao"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.cong.4Summary: In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.On nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearityhttps://zbmath.org/1541.352082024-09-27T17:47:02.548271Z"El Mokhtar, M. E. O."https://zbmath.org/authors/?q=ai:elmokhtar.mohammed-elmokhtar-ould"Benmansour, S."https://zbmath.org/authors/?q=ai:benmansour.safia"Matallah, A."https://zbmath.org/authors/?q=ai:matallah.atikaSummary: In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in \(\mathbb R^3\). Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.Multiple positive solutions of the quasilinear Schrödinger-Poisson system with critical exponent in \(D^{1, p}(\mathbb{R}^3)\)https://zbmath.org/1541.352092024-09-27T17:47:02.548271Z"Huang, Lanxin"https://zbmath.org/authors/?q=ai:huang.lanxin"Su, Jiabao"https://zbmath.org/authors/?q=ai:su.jiabaoSummary: This paper is concerned with the quasilinear Schrödinger-Poisson system \(-\Delta_p u - l(x)\phi|u|^{p - 2}u = |u|^{p^\ast - 2}u + \mu h(x)|u|^{q - 2}u\) in \(\mathbb{R}^3\) and \(-\Delta\phi = l(x)|u|^p\) in \(\mathbb{R}^3\), where \(\mu > 0\), \(p^\ast = \frac{3p}{3 - p}\) and \(\Delta_p u = \operatorname{div}(|\nabla u|^{p - 2}\nabla u)\). By using the Ekeland's variational principle and the mountain pass theorem, we prove that the system admits two positive solutions for \(1 \leqslant q < p\) and \(1 < p < 3\), and the system admits one positive solution for \(p \leqslant q < p^\ast\) and \(\frac{3}{2} < p < 3\).
{\copyright 2024 American Institute of Physics}The logarithmic Minkowski problem in \(\mathbb{R}^2\)https://zbmath.org/1541.352102024-09-27T17:47:02.548271Z"Liu, Yude"https://zbmath.org/authors/?q=ai:liu.yude"Lu, Xinbao"https://zbmath.org/authors/?q=ai:lu.xinbao"Sun, Qiang"https://zbmath.org/authors/?q=ai:sun.qiang"Xiong, Ge"https://zbmath.org/authors/?q=ai:xiong.geSummary: A necessary condition for the existence of solutions to the logarithmic Minkowski problem in \(\mathbb{R}^2\), which turns to be stronger than the celebrated subspace concentration condition, is given. The sufficient and necessary conditions for the existence of solutions to the logarithmic problem for quadrilaterals, as well as the number of solutions, are fully characterized.Curvature estimates for a class of Hessian quotient type curvature equationshttps://zbmath.org/1541.352112024-09-27T17:47:02.548271Z"Zhou, Jundong"https://zbmath.org/authors/?q=ai:zhou.jundongSummary: In this paper, we are concerned with the hypersurface that can be locally represented as a graph and satisfies a class of Hessian quotient type curvature equations. We establish interior curvature estimates under the condition of \(0 \leq l < k \leq C_{n-1}^{p-1}\). As an application, we prove Bernstein type theorem for this type curvature equation. We also focus on closed star shaped hypersurface satisfying this type curvature equation and obtain the global curvature estimation.Predator-prey models with different starvation-driven diffusions and resourceshttps://zbmath.org/1541.352122024-09-27T17:47:02.548271Z"Chang, Youngseok"https://zbmath.org/authors/?q=ai:chang.youngseok"Choi, Wonhyung"https://zbmath.org/authors/?q=ai:choi.wonhyung"Ahn, Inkyung"https://zbmath.org/authors/?q=ai:ahn.inkyungSummary: This paper studies a Lotka-Volterra-type predator-prey model with different starvation-driven diffusions (SDDs) and resources in spatially heterogeneous environments for two species under homogeneous Neumann boundary conditions. The stability of two semitrivial steady-state solutions to the model where one species survives and the other is absent is investigated. In addition, the results are compared with a model in which both species have uniform dispersal with constant diffusion rates. We conclude that the coexistence of predator and prey occurs in the habitat from the instability of two semitrivial solutions in which only one species survives.The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domainhttps://zbmath.org/1541.352132024-09-27T17:47:02.548271Z"Chen, Yongpeng"https://zbmath.org/authors/?q=ai:chen.yongpeng"Yang, Zhipeng"https://zbmath.org/authors/?q=ai:yang.zhipeng.2|yang.zhipeng|yang.zhipeng.1Summary: In this article, we consider the following Choquard equation with upper critical exponent:
\[
-\Delta u = \mu f(x)|u|^{p-2}u + g(x)({I_\alpha}^\ast(g|u|^{2_\alpha^\ast}))|u|^{2_\alpha^\ast} - 2u,\quad x\in \Omega,
\]
where \(\mu > 0\) is a parameter, \(N > 4\), \(0 < \alpha < N\), \(I_\alpha\) is the Riesz potential, \(\frac{N}{N-2} < p < 2\), \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary, and \(f\) and \(g\) are continuous functions. For \(\mu\) small enough, using variational methods, we establish the relationship between the number of solutions and the profile of potential \(g\).The higher regularity and decay estimates for positive solutions of fractional Choquard equationshttps://zbmath.org/1541.352142024-09-27T17:47:02.548271Z"Deng, Yinbin"https://zbmath.org/authors/?q=ai:deng.yinbin"Yang, Xian"https://zbmath.org/authors/?q=ai:yang.xianSummary: In the paper, we study the higher regularity and decay estimates for positive solutions of the following fractional Choquard equations:
\[
\begin{cases}
(-\Delta)^s u+\lambda u=(I_\alpha *|u|^p)|u|^{p-2}u\quad \text{in } \mathbb{R}^N, \\
\lim\limits_{|x|\rightarrow \infty }u(x)=0,\quad u\in H^s(\mathbb{R}^N),
\tag{0.1}
\end{cases}
\]
where \(I_\alpha =\frac{1}{|x|^{N-\alpha }}\), \(\lambda >0\) is a constant, \(N\geqslant 2\), \(s\in (0,1)\), \(\alpha \in (0,N)\), \(p\in (\frac{N+\alpha }{N},\frac{N+\alpha }{N-2s})\). Let \(Q\) be a positive solution of (0.1). We obtain an optimal decay estimate for \(Q\) and reveal the relation between \(p\) and the decay rate of \(Q\) for all \(p\in (\frac{N+\alpha }{N},\frac{N+\alpha }{N-2s})\) permitting \(p<2 \). We skillfully use the properties of Bessel kernel to deduce our main result by iteration. Moreover, by the decay estimates, the stronger regular estimates for \(Q\) are obtained.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Global higher regularity and decay estimates for positive solutions of fractional equations in \(\mathbb{R}^N\)https://zbmath.org/1541.352152024-09-27T17:47:02.548271Z"Deng, Yinbin"https://zbmath.org/authors/?q=ai:deng.yinbin"Yang, Xian"https://zbmath.org/authors/?q=ai:yang.xianSummary: In the paper, we study the global higher regularity and decay estimates of the positive solutions for the following fractional equations
\[
\begin{cases}
(-\Delta)^su+u=|u|^{p-2}u\text{ in }\mathbb{R}^N, \\
\lim\limits_{|x|\to\infty}u(x)=0,\quad u\in H^s(\mathbb{R}^N), \end{cases}\tag{0.1}
\]
where \(s\in(0,1)\), \(N>2s\), \(2<p<2_s^\ast:=\frac{2N}{N-2s}\) and \((-\Delta)^s\) is the fractional Laplacian. Let \(Q\) be a positive solution of (0.1). We prove that \(Q\in C^{k,\gamma}(\mathbb{R}^N)\cap H^k(\mathbb{R}^N)\) and obtain the decay estimates of \(D^kQ\) as \(|x|\to\infty\) for all \(k\in\mathbb{N}_+\) and \(\gamma\in(0,1)\). The argument relies on the Bessel kernel, comparison principle, Fourier analysis and iteration methods.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problemshttps://zbmath.org/1541.352162024-09-27T17:47:02.548271Z"El-Houari, H."https://zbmath.org/authors/?q=ai:el-houari.hamza"Sabiki, H."https://zbmath.org/authors/?q=ai:sabiki.hajar"Moussa, H."https://zbmath.org/authors/?q=ai:moussa.hichamSummary: In this research, we analyze the existence of infinite sequences of ordered solutions for a class of non-local elliptic problem with Dirichlet boundary condition. The primary techniques employed consist of topological degree theory for mappings of type \(S_+\) and minimization arguments in a fractional Orlicz-Sobolev space. Our main results generalize some recent findings in the literature to non-smooth cases.Asymptotic uniqueness of minimizers for Hartree type equations with fractional Laplacianhttps://zbmath.org/1541.352172024-09-27T17:47:02.548271Z"Liu, Lintao"https://zbmath.org/authors/?q=ai:liu.lintao"Teng, Kaimin"https://zbmath.org/authors/?q=ai:teng.kaimin"Yuan, Shuai"https://zbmath.org/authors/?q=ai:yuan.shuaiSummary: We study the concentration and uniqueness of standing waves associated with the constraint minimization problems for the nonlinear Hartree type equations with homogeneous potentials and fractional Laplacian. This class of equations is an effective model to describe the fractional quantum mechanics with a convolution perturbation. By making full use of the Bessel kernel and adopting the iterative process, we establish the \(L^{\infty}\)-estimates and decay properties of the solutions to the fractional Hartree equations. Based on the above basic conclusions, we establish the concentration and uniqueness of the constraint minimizers by exploring some fine energy estimates and studying some uniform regularity, while establishing local Pohozăev identity and overcoming the blow-up estimates to the nonlocal operator \((-\Delta)^s\). Compared with the classical local elliptic problems, we encounter some new difficulties because of the nonlocal nature of the fractional Laplace. One of the main difficulties is that the decay estimates of the sequences of solutions to the nonlocal problems at infinity are different from those in the case of the classical local problems. Another difficulty is that we have to consider the corresponding harmonic extension problems to construct the Pohozăev identity, which will cause us to have to estimate several kinds of integrals that never appear in the classic local problems. In addition, the presence of the potential \(|x|^2\) and the Hartree term \(|x|^{2s-3}* u^2\) will affect the blow-up frequency of the minimizers, and we have to control \(s\) to achieve the optimal blow-up speed.Normalized ground state solutions for nonautonomous Choquard equationshttps://zbmath.org/1541.352182024-09-27T17:47:02.548271Z"Luo, Huxiao"https://zbmath.org/authors/?q=ai:luo.huxiao"Wang, Lushun"https://zbmath.org/authors/?q=ai:wang.lushunSummary: In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation
\[
\begin{cases}
-\Delta u -\lambda u = \left( \frac{1}{|x|^\mu} * A|u|^p\right) A|u|^{p-2} u, \\
\int_{\mathbb{R}^N} |u|^2 \mathrm{d} x=c,\quad u\in H^1 (\mathbb{R}^N,\mathbb{R}),
\end{cases}
\]
where \(c>0\), \(0<\mu <N\), \(\lambda \in \mathbb{R}\), \(A \in C^1 (\mathbb{R}^N,\mathbb{R})\). For \(p \in (2_{*, \mu}, \bar{p})\), we prove that the Choquard equation possesses normalized ground state solutions, and the set of ground states is orbitally stable. For \(p \in (\bar p,2_\mu^*)\), we find a normalized solution, which is not a global minimizer. \(2^*_\mu\) and \(2_{*, \mu}\) are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. \(\bar{p}\) is the \(L^2\)-critical exponent. Our results generalize and extend some related results.Asymptotic behavior of semilinear parabolic equations with combined nonlinearitieshttps://zbmath.org/1541.352192024-09-27T17:47:02.548271Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li"Zhang, Kaiqiang"https://zbmath.org/authors/?q=ai:zhang.kaiqiangSummary: We study the boundedness of \(H^1\cap L^\infty\) global solutions to the semilinear parabolic equations with combined nonlinearities. We establish a sufficient condition for the initial data by the potential well method, which guarantees the solution exists globally. Then we prove the a priori estimate for all global solutions. As an application of the priori estimate, combining with the potential well theory, we show that there is a sub-convergent sequence of the global flow, which converges the positive solution to the corresponding stationary equation.Normalized solutions to the fractional Schrödinger equation with critical growthhttps://zbmath.org/1541.352202024-09-27T17:47:02.548271Z"Shen, Xinsi"https://zbmath.org/authors/?q=ai:shen.xinsi"Lv, Ying"https://zbmath.org/authors/?q=ai:lv.ying"Ou, Zengqi"https://zbmath.org/authors/?q=ai:ou.zengqiSummary: In this paper, we discuss the existence of normalized solutions to the following fractional Schrödinger equation
\[
\begin{cases}
(-\Delta)^s u = \lambda u + g(u) + |u|^{2^\ast_s - 2}u, & x\in\mathbb{R}^N,\\
\int_{\mathbb{R}^N}u^2 = a^2,
\end{cases}
\]
where \(N \geq 3\), \(s\in(0, 1)\), \(a > 0\), \(2_s^\ast = 2N/(N-2s)\), \(\lambda\in\mathbb{R}\) arises as a Lagrange multiplier, \((-\Delta)^s\) is the fractional Laplace operator and \(g: \mathbb{R}\rightarrow\mathbb{R}\) satisfies \(L^2\)-supercritical conditions. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions.Multiple solutions for critical nonlocal elliptic problems with magnetic fieldhttps://zbmath.org/1541.352212024-09-27T17:47:02.548271Z"Wen, Ruijiang"https://zbmath.org/authors/?q=ai:wen.ruijiang"Yang, Jianfu"https://zbmath.org/authors/?q=ai:yang.jianfu"Yu, Xiaohui"https://zbmath.org/authors/?q=ai:yu.xiaohuiSummary: In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field:
\[
\begin{cases}
(-i\nabla-A(x))^2u = \lambda |u|^{p-2}u+\bigg(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\bigg)|u|^{2^*_\alpha-2}u\quad\text{ in}\quad \Omega,\\
u = 0\quad \partial\Omega,
\end{cases}\tag{1}
\]
where \(i\) is imaginary unit, \( N\geq4\), \(2^*_\alpha = \frac{2N-\alpha}{N-2}\) with \(0<\alpha<4\), \(\lambda>0\) and \(2\leq p<2^* = \frac{2N}{N-2} \). Suppose the magnetic vector potential \(A(x) = (A_1(x), A_2(x)\dots, A_N(x))\) is real and local Hölder continuous. We show by the Ljusternik-Schnirelman theory that (1) has at least \(cat_\Omega(\Omega)\) nontrivial solutions for \(\lambda\) small.Boundedness of solutions to singular anisotropic elliptic equationshttps://zbmath.org/1541.352222024-09-27T17:47:02.548271Z"Brandolini, Barbara"https://zbmath.org/authors/?q=ai:brandolini.barbara"Cîrstea, Florica C."https://zbmath.org/authors/?q=ai:cirstea.florica-corina-stSummary: We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form
\[
-\Delta_{\overrightarrow{p}}u+\Phi_0(u, \nabla u) = \Psi(u, \nabla u) +f \quad \text{in } \Omega, \qquad u = 0 \quad \text{on }\partial \Omega,
\]
where \(\Omega\subset \mathbb{R}^N\) (\(N\geq 2\)) is a bounded open set, \( \Delta_{\overrightarrow{p}}u = \sum_{j = 1}^N \partial_j (|\partial_j u|^{p_j-2}\partial_j u)\) and \(\Phi_0(u, \nabla u) = \left(\mathfrak{a}_0+\sum_{j = 1}^N \mathfrak{a}_j |\partial_j u|^{p_j}\right)|u|^{m-2}u \), with \(\mathfrak{a}_0>0 \), \(m, p_j>1 \), \(\mathfrak{a}_j\geq 0\) for \(1\leq j\leq N\) and \(N/p = \sum_{k = 1}^N\) \((1/p_k)>1 \). We assume that \(f \in L^r(\Omega)\) with \(r>N/p \). The feature of this study is the inclusion of a possibly singular gradient-dependent term \(\Psi(u, \nabla u) = \sum_{j = 1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j} \), where \(\theta_j>0\) and \(0\leq q_j<p_j\) for \(1\leq j\leq N \). The existence of such weak solutions is contained in a recent paper by the authors.Multiplicative control problem for a nonlinear reaction-diffusion modelhttps://zbmath.org/1541.352232024-09-27T17:47:02.548271Z"Brizitskii, R. V."https://zbmath.org/authors/?q=ai:brizitskii.roman-viktorovich|brizitskii.r-v|brizitskii.roman-victorovich"Donchak, A. A."https://zbmath.org/authors/?q=ai:donchak.a-aSummary: The paper studies a multiplicative control problem for the reaction-diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived.Nonlinear divergence equation with potentials vanishing in some direction at infinity: the exponential critical casehttps://zbmath.org/1541.352242024-09-27T17:47:02.548271Z"Cardoso, J. A."https://zbmath.org/authors/?q=ai:cardoso.jose-anderson"Carvalho, J. L."https://zbmath.org/authors/?q=ai:carvalho.jonison-lucas"de Medeiros, E. S."https://zbmath.org/authors/?q=ai:souto-de-medeiros.everaldoSummary: The purpose of this article is to state some weighted Sobolev embedding involving functions that vanishing only in a direction. In this setting we prove a weighted Trudinger-Moser type inequality and as an application, we addressed the existence of solutions to a class of elliptic equation of the form
\[
-\operatorname{div}(a(x)\nabla u)+V(x)u=K(x)f(u)\text{ in } \mathbb{R}^2,
\]
where the nonlinearity \(f\) has exponential critical growth in sense of Trudinger-Moser.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Kirchhoff-Boussinesq-type problems with positive and zero masshttps://zbmath.org/1541.352252024-09-27T17:47:02.548271Z"Carlos, Romulo D."https://zbmath.org/authors/?q=ai:carlos.romulo-d"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher"Ruviaro, Ricardo"https://zbmath.org/authors/?q=ai:ruviaro.ricardo(no abstract)Decay estimates for quasilinear elliptic equations and a Brezis-Nirenberg result in \(D^{1, p}(\mathbb{R}^N)\)https://zbmath.org/1541.352262024-09-27T17:47:02.548271Z"Carl, Siegfried"https://zbmath.org/authors/?q=ai:carl.siegfried"Tehrani, Hossein"https://zbmath.org/authors/?q=ai:tehrani.hossein-torabiSummary: We prove decay estimates for solutions of quasilinear elliptic equations in the whole \(\mathbb{R}^N\) of the form
\[
u \in X: -\operatorname{div} A(x, \nabla u)=a(x) f(x,u),
\]
where \(X=D^{1, p}(\mathbb{R}^N)\) is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis-Nirenberg type result for the energy functional \(\Phi : X \to \mathbb{R}\) related to the p-Laplacian equation in \(\mathbb{R}^N\) in the form
\[
u \in X : -\Delta_p u=a(x) g(u),
\]
saying that for the subspace \(V\) of bounded continuous functions with weight \(1+|x|^{\frac{N-p}{p}}\), a local minimizer of \(\Phi\) in the finer \(V\) topology is also a local minimizer in the \(X\)-topology. Global \(L^\infty\)-estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.On Kirchhoff type equations with singular nonlinearities, sub-critical and critical exponenthttps://zbmath.org/1541.352272024-09-27T17:47:02.548271Z"El Mokhtar, Mohammed El Mokhtar Ould"https://zbmath.org/authors/?q=ai:el-mokhtar.mohammed-el-mokhtar-ould"Aljurbua, Saleh Fahad"https://zbmath.org/authors/?q=ai:aljurbua.saleh-fahadSummary: This paper is devoted to the existence of solutions for Kirchhoff type equations with singular nonlinearities, sub-critical and critical exponent. By using the Nehari manifold and Maximum principle theorem, the existence of at least two distinct positive solutions is obtained.Nonlinear elliptic problems in weighted variable exponent Sobolev spaces with nonlocal boundary conditionshttps://zbmath.org/1541.352282024-09-27T17:47:02.548271Z"El Omari, Soumia"https://zbmath.org/authors/?q=ai:el-omari.soumia"Melliani, Said"https://zbmath.org/authors/?q=ai:melliani.said"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakirSummary: In this paper, we study the existence and uniqueness result of weak solution for nonlinear elliptic problems with non-local boundary conditions in the Weighted variable exponent Sobolev spaces \(W^{1,p(.)}(\varOmega ,\omega)\).
For the entire collection see [Zbl 1515.35012].Regularity for solutions to a class of PDE's with Orlicz growthhttps://zbmath.org/1541.352292024-09-27T17:47:02.548271Z"Giannetti, Flavia"https://zbmath.org/authors/?q=ai:giannetti.flavia"Passarelli di Napoli, Antonia"https://zbmath.org/authors/?q=ai:passarelli-di-napoli.antoniaSummary: We consider weak solutions \(u : \Omega \rightarrow \mathbb{R}\) to partial differential equations of the form
\[
\mathrm{div} a(x, Du) = 0
\]
in \(\Omega \subset \mathbb{R}^n\), \(n>2\), where the partial map \(x \mapsto a(x,\xi)\) has a suitable Sobolev regularity and satisfies growth conditions with respect to the second variable expressed through an Orlicz function \(\phi\). We prove the second order regularity of the weak solutions.A singular nonlinear problems with natural growth in the gradi aenthttps://zbmath.org/1541.352302024-09-27T17:47:02.548271Z"Hamour, Boussad"https://zbmath.org/authors/?q=ai:hamour.boussadSummary: In this paper, we consider the equation \(-\mathrm{div} (a(x,u,Du)=H(x,u,Du)+\frac{a_0 (x)}{|u|^{\theta}}+ \chi_{\{u\neq 0\}} f(x)\) in \(\varOmega\), with boundary conditions \(u=0\) on \(\partial\varOmega\), where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\), \(1<p<N\), \(-\mathrm{div} (a(x,u,Du))\) is a Leray-Lions operator defined on \(W^{1,p}_0 (\varOmega)\), \(a_0 \in L^{N/p} (\varOmega)\), \(a_0 >0\), \(0<\theta\leq 1\), \(\chi_{\{u\neq 0\}}\) is a characteristic function, \(f\in L^{N/p}(\varOmega)\) and \(H(x,s, \xi)\) is a Carathéodory function such that \(-c_0 a(x,s,\xi)\xi \leq H(x,s,\xi) \mathrm{sign}(s)\leq \gamma a(x,s,\xi)\xi\) a.e. \(x\in\varOmega\), \(\forall s\in \mathbb{R}\), \(\forall \xi \in \mathbb{R}^N\). For \(\Vert a_0\Vert_{N/p}\) and \(\Vert f\Vert_{N/p}\) sufficiently small, we prove the existence of at least one solution \(u\) of this problem which is moreover such that the function \(\exp (\delta |u|)-1\) belongs to \(W^{1,p}_0 (\varOmega)\) for some \(\delta \geq \gamma\). This solution satisfies some a priori estimates in \(W_0^{1,p} (\varOmega)\).Existence and regularity for a degenerate problem with singular gradient lower order termhttps://zbmath.org/1541.352312024-09-27T17:47:02.548271Z"Khelifi, Hichem"https://zbmath.org/authors/?q=ai:khelifi.hichemSummary: In this paper we study the existence and regularity results for nonlinear elliptic equation with degenerate coercivity and a singular gradient lower order term.Nonlinear nonlocal elliptic problems in \(\mathbb{R}^3\): existence results and qualitative propertieshttps://zbmath.org/1541.352322024-09-27T17:47:02.548271Z"Lü, Dengfeng"https://zbmath.org/authors/?q=ai:lu.dengfeng"Dai, Shu-Wei"https://zbmath.org/authors/?q=ai:dai.shuweiSummary: We consider the following nonlinear nonlocal elliptic problem:
\[
-\left(a + b\int\limits_{\mathbb{R}^3}|\nabla\psi|^2\mathrm{d}x\right)\Delta\psi + \lambda\psi = \left(\int\limits_{\mathbb{R}^3}\frac{G(\psi(y))}{|x - y|^\alpha}\mathrm{d}y\right)G^\prime(\psi),\quad x\in\mathbb{R}^3,
\]
where \(a, b > 0\) are constants, \(\lambda > 0\) is a parameter, \(\alpha\in(0, 3)\), and \(G\in\mathcal{C}^1(\mathbb{R}, \mathbb{R})\). By using variational methods, we establish the existence of least energy solutions for the above equation under conditions on the nonlinearity \(G\) we believe to be almost necessary. Some qualitative properties of the least energy solutions are also obtained.Multiple nontrivial solutions for critical \(p\)-Kirchhoff type problems in \(\mathbb{R}^N\)https://zbmath.org/1541.352332024-09-27T17:47:02.548271Z"Sabri, Khadidja"https://zbmath.org/authors/?q=ai:sabri.khadidja"El Mokhtar, Mohammed El Mokhtar Ould"https://zbmath.org/authors/?q=ai:el-mokhtar.mohammed-el-mokhtar-ould"Matallah, Atika"https://zbmath.org/authors/?q=ai:matallah.atikaSummary: In this paper, we study the existence and multiplicity of nontrivial solutions for a \(p\)-Kirchhoff equation involving critical Sobolev-Hardy exponent by using variational methods and we need to estimate the energy levels.On the solvability of an essentially nonlinear elliptic differential equation with nonlocal boundary conditionshttps://zbmath.org/1541.352342024-09-27T17:47:02.548271Z"Solonukha, O. V."https://zbmath.org/authors/?q=ai:solonukha.olesya-vSummary: Sufficient conditions for the existence of a generalized solution to a nonlinear elliptic differential equation with nonlocal boundary conditions of Bitsadze-Samarskii type are proved. The strong ellipticity condition is used for an auxiliary differential-difference operator. Under the formulated conditions, the differential-difference operator is demicontinuous, coercive, and has a semibounded variation, so the general theory of pseudomonotone operators can be applied.Existence and upper bound results for a class of nonlinear nonhomogeneous obstacle problemshttps://zbmath.org/1541.352352024-09-27T17:47:02.548271Z"Vo Minh Tam"https://zbmath.org/authors/?q=ai:vo-minh-tam."Liao, Shanli"https://zbmath.org/authors/?q=ai:liao.shanliSummary: This paper is devoted to the study of a new class of nonlinear obstacle problems involving nonhomogeneous partial differential operators and mixed boundary conditions. We provide the existence and uniqueness of the solution for the obstacle problem via applying a surjectivity theorem for set-valued mappings formulated by the sum of a set-valued pseudomonotone operator and a maximal monotone set-valued operator. Moreover, some upper bounds to the obstacle problem are established by using a regularized gap function through different norms.The existence of arbitrary multiple nodal solutions for a class of quasilinear Schrödinger equationshttps://zbmath.org/1541.352362024-09-27T17:47:02.548271Z"Wang, Kun"https://zbmath.org/authors/?q=ai:wang.kun.1|wang.kun.6|wang.kun|wang.kun.7|wang.kun.2|wang.kun.3"Huang, Chen"https://zbmath.org/authors/?q=ai:huang.chen"Jia, Gao"https://zbmath.org/authors/?q=ai:jia.gaoSummary: This paper is concerned to studying the quasilinear Schrödinger equation:
\[
-\Delta u + V(x)u - \frac{\gamma}{2}\Delta(u^2)u = |u|^{p-2}u,\; x\in \mathbb{R}^N,
\]
where \(V(x)\) is a given potential, \(\gamma > 0\) and either \(p\in(2, 2^\ast)\), \(2^\ast = \frac{2N}{N-2}\) for \(N \geqslant 4\) or \(p\in(2, 4)\) for \(N = 3\). We establish the existence of arbitrary multiple nodal solutions for the above equations.Existence and multiplicity of normalized solutions with positive energy for the Kirchhoff equationhttps://zbmath.org/1541.352372024-09-27T17:47:02.548271Z"Xu, Lin"https://zbmath.org/authors/?q=ai:xu.lin.3|xu.lin"Li, Feng"https://zbmath.org/authors/?q=ai:li.feng.1|li.feng.3|li.feng.7|li.feng|li.feng.6"Xie, Qilin"https://zbmath.org/authors/?q=ai:xie.qilinSummary: In this paper, we investigate the existence and multiplicity of normalized solutions for the following Kirchhoff equation,
\[
\begin{cases}
-\left(a + b\int_{\mathbb{R}^3}|\nabla u|^2dx\right) \Delta u - \lambda u = f(u), &\text{in } \mathbb{R}^3,\\
\int_{\mathbb{R}^3}|u|^2 dx = c,
\end{cases}\tag{P}
\]
where \(a\), \(b\), \(c\) are positive constants and \(\lambda\in\mathbb{R}\) is an unknown parameter that appears as a Lagrange multiplier. Two normal solutions, manifesting as a local minimizer or mountain pass solution, have been obtained under the mass subcritical conditions on the nonlinearity \(f\) and some suitable mass \(c\). Additionally, we employ the Symmetric Mountain Pass Theorem to establish the multiplicity of normalized solutions for problem (P). To the best of our knowledge, we extend and complement the research success in the Kirchhoff equation for a general nonlinearity with weaker subcritical mass growth.Nodal solutions for an asymptotically linear Kirchhoff-type problem in \(\mathbb{R}^N\)https://zbmath.org/1541.352382024-09-27T17:47:02.548271Z"Zhong, Xiao-Jing"https://zbmath.org/authors/?q=ai:zhong.xiaojing(no abstract)On a class of fractional \(\Gamma (.)\)-Kirchhoff-Schrödinger system typehttps://zbmath.org/1541.352392024-09-27T17:47:02.548271Z"El-Houari, Hamza"https://zbmath.org/authors/?q=ai:el-houari.hamza"Chadli, Lalla Saádia"https://zbmath.org/authors/?q=ai:chadli.lalla-saadia"Moussa, Hicham"https://zbmath.org/authors/?q=ai:moussa.hichamSummary: This paper focuses on the investigation of a Kirchhoff-Schrödinger type elliptic system involving a fractional \(\gamma(.)\)-Laplacian operator. The primary objective is to establish the existence of weak solutions for this system within the framework of fractional Orlicz-Sobolev Spaces. To achieve this, we employ the critical point approach and direct variational principle, which allow us to demonstrate the existence of such solutions. The utilization of fractional Orlicz-Sobolev spaces is essential for handling the nonlinearity of the problem, making it a powerful tool for the analysis. The results presented herein contribute to a deeper understanding of the behavior of this type of elliptic system and provide a foundation for further research in related areas.Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operatorshttps://zbmath.org/1541.352402024-09-27T17:47:02.548271Z"Akhunov, Timur"https://zbmath.org/authors/?q=ai:akhunov.timur"Korobenko, Lyudmila"https://zbmath.org/authors/?q=ai:korobenko.lyudmilaSummary: This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form \(L_1 (x) + g(x) L_2 (y)\), where \(L_1 (x)\) is one-dimensional and \(g(x)\) may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation, and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our methods do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaceshttps://zbmath.org/1541.352412024-09-27T17:47:02.548271Z"Alimohammady, Mohsen"https://zbmath.org/authors/?q=ai:alimohammady.mohsen"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carlo"Kalleji, Morteza Koozehgar"https://zbmath.org/authors/?q=ai:kalleji.morteza-koozehgarSummary: This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term \(a\) i.e., the number of subdomains of \(\Omega \setminus a^{-1}(0)\) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on \(C^{1, \alpha} (\Omega)\).On Caffarelli-Kohn-Nirenberg type problems with a sign-changing termhttps://zbmath.org/1541.352422024-09-27T17:47:02.548271Z"Baraket, Sami"https://zbmath.org/authors/?q=ai:baraket.sami"Ben Ghorbal, Anis"https://zbmath.org/authors/?q=ai:ben-ghorbal.anis"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcherSummary: In this work we show existence and multiplicity of positive solutions using the sub-supersolution method in Caffarelli-Kohn-Nirenberg type problems with a sign-changing term. More precisely, using the sub-supersolution method, we study the following class of singular problem:
\[
\begin{cases}
-\operatorname{div} \big(|x|^{-ap} |\nabla u|^{p-2} \nabla u \big) = |x|^{-(a+1)p+c} h(x)u^{-\gamma} + |x|^{-(a+1)p+c} f(x,u) \text{ in } \Omega, \\
u>0 \text{ in } \Omega, \\
u=0 \text{ on } \partial \Omega,
\end{cases}
\]
where \(\Omega\) is a bounded smooth domain in \({\mathbb{R}}^N\) with \(N \geq 3\), \(1< p<N\), \(0 \leq a < \frac{N-p}{p}\), \(c>0\), and \(\gamma >0\). The hypotheses on the functions \(h\) and \(f\) allow to use sub-supersolutions and Mountain Pass Theorem.On the regularity and existence of weak solutions for a class of degenerate singular elliptic problemhttps://zbmath.org/1541.352432024-09-27T17:47:02.548271Z"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashantaSummary: In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of \(p\)-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary inside the domain. We provide sufficient conditions on the weight function, on the singular exponent and the source function to establish regularity and existence results.A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operatorhttps://zbmath.org/1541.352442024-09-27T17:47:02.548271Z"Baldelli, Laura"https://zbmath.org/authors/?q=ai:baldelli.laura"Ciani, Simone"https://zbmath.org/authors/?q=ai:ciani.simone"Skrypnik, Igor"https://zbmath.org/authors/?q=ai:skrypnik.igor-igorievich"Vespri, Vincenzo"https://zbmath.org/authors/?q=ai:vespri.vincenzoSummary: In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type
\[
\sum\limits_{i = 1}^s \partial_{ii} u+ \sum\limits_{i = s+1}^N \partial_i \bigg(A_i(x, u, \nabla u) \bigg) = 0, \quad x \in \Omega \subset \subset \mathbb{R}^N \quad \text{ for } \quad 1\leq s \leq N-1,
\]
where each operator \(A_i\) behaves directionally as the singular \(p \)-Laplacian, \( 1< p < 2\) and the supercritical condition \(p+(N-s)(p-2)>0\) holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.Singular orthotropic functionals with nonstandard growth conditionshttps://zbmath.org/1541.352452024-09-27T17:47:02.548271Z"Bousquet, Pierre"https://zbmath.org/authors/?q=ai:bousquet.pierre"Brasco, Lorenzo"https://zbmath.org/authors/?q=ai:brasco.lorenzo"Leone, Chiara"https://zbmath.org/authors/?q=ai:leone.chiaraSummary: We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restrictions on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers as well.Analysis of quasi-variational-hemivariational inequalities with applications to Bingham-type fluidshttps://zbmath.org/1541.352462024-09-27T17:47:02.548271Z"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Chao, Yang"https://zbmath.org/authors/?q=ai:chao.yang"He, Jiahong"https://zbmath.org/authors/?q=ai:he.jiahong"Dudek, Sylwia"https://zbmath.org/authors/?q=ai:dudek.sylwiaSummary: In this paper we study the sensitivity analysis of elliptic quasi-variational-hemivariational inequalities with constraint. The upper semicontinuity property of the solution map with respect to a parameter is established. An application to the steady-state incompressible Navier-Stokes equation with mixed boundary conditions in a model for a generalized Newtonian fluid of Bingham-type is provided. The boundary conditions represent a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier-Fujita frictional slip condition. Furthermore, a sensitivity result is proved for the weak formulation of the model when all the data are subjected to perturbations. Finally, for the Bingham-type fluids, an optimal control problem is studied.Comparison result for quasi-linear elliptic equations with general growth in the gradienthttps://zbmath.org/1541.352472024-09-27T17:47:02.548271Z"Alvino, Angelo"https://zbmath.org/authors/?q=ai:alvino.angelo"Betta, Maria Francesca"https://zbmath.org/authors/?q=ai:betta.maria-francesca"Mercaldo, Anna"https://zbmath.org/authors/?q=ai:mercaldo.anna"Volpicelli, Roberta"https://zbmath.org/authors/?q=ai:volpicelli.robertaSummary: In this paper we prove a comparison result for a class of Dirichlet boundary problems whose model is
\[
\left\{ \begin{array}{ll}
-\Delta u = {\beta}|\nabla u|^q +c u + f &\text{ in } \Omega\\
u = 0 &\text{ su } \partial \Omega,
\end{array} \right.
\] where \(\Omega\) is an open bounded subset of \({{\mathbb{R}}^N}\), \(N > 2 \). We also prove an existence and uniqueness result for weak solution to these problems.On supercritical elliptic problems: existence, multiplicity of positive and symmetry breaking solutionshttps://zbmath.org/1541.352482024-09-27T17:47:02.548271Z"Cowan, Craig"https://zbmath.org/authors/?q=ai:cowan.craig"Moameni, Abbas"https://zbmath.org/authors/?q=ai:moameni.abbasThe paper is devoted to the study of radial and nonradial classical solutions of equations of the following type
\[
-\Delta u +V(x) u = a(x) u^{p-1} \text{ in } \Omega \subset {\mathbb R}^n .
\]
\(\Omega\) is either the full space or \(\Omega\) is a bounded subset and \(u=0\) on \(\partial\Omega\). Here the exponent \(p>2\) is supercritical in the sense of Sobolev embeddings. It is assumed that \(\Omega\) satisfies certain symmetry assumptions (namely the \(m\)-revolution condition). Let \({\mathbb R}^n={\mathbb R}^k \times {\mathbb R}^l\), \(k+l=n\). A domain \(\Omega \subset {\mathbb R}^n\) is a domain of \textit{double revolution}, if it is invariant under rotations of the first \(k\) variables and also under rotations of the last \(l\) variables. \(m\)-revolution condition may be defined similarly.
Some results obtained in the paper under review:
\(\bullet\) On annual radial domains new types of positive nonradial solutions to the equation \(-\Delta u = a(u) u^{p-1}\) are obtained. \par \(\bullet\) Several types of new positive nonradial solutions to the equation \(-\Delta u = |x|^\alpha u^{p-1} \) in the unit ball \(B_1\) in \({\mathbb R}^N\) (on a range of supercritical \(p\)) are obtained.
\(\bullet\) The existaence of a positive classical solution to the equation \(-\Delta u + u = |x|^\alpha u^{p-1} \) in \({\mathbb R}^N={\mathbb R}^n \times {\mathbb R}^n\) is proved for \(\frac{2N+2\alpha-4}{N-2} < p < \frac{2N+2\alpha}{N-2}\). For large \(\alpha\) the authors obtain a nonradial solution.
\(\bullet\) The existence of a positive classical solution to the equation \(-\Delta u + \frac{u}{|x|^\alpha } = u^{p-1} \) in \(B_1\) is proved for \(2 < p < \frac{2N+2\alpha-4}{N-2}\), \(\alpha > 2\). For large \(\alpha\) the solution is nonradial.
\(\bullet\) For various problems on radial domains the authors show that ground states are nonradial.
\(\bullet\) If \(\Omega\subset {\mathbb R}^N\) is a bounded domain which is also a domain of triple revolution, the existence, and some multiplicity results (for a range of supercritical \(p\)) for the equation \(-\Delta u = a(x) u^{p-1}\) are proved.
The proofs are based on the variational approach -- a minimax principle for general semilinear elliptic problems restricted to a given convex subset.
Several new Sobolev embeddings are also established for functions having a mild monotonicity on symmetric domains.
Reviewer: Michael Perelmuter (Kyïv)Elliptic equations involving supercritical Sobolev growth with mixed Dirichlet-Neumann boundary conditionshttps://zbmath.org/1541.352492024-09-27T17:47:02.548271Z"de Assis, Heitor R."https://zbmath.org/authors/?q=ai:de-assis.heitor-r"Faria, Luiz F. O."https://zbmath.org/authors/?q=ai:faria.luiz-f-oSummary: This paper concerns elliptic problems involving supercritical Sobolev growth without the (AR) condition and with a mixed boundary Dirichlet-Neumann type condition. The conditions imposed on the nonlinearity considered here generalizes several previous papers, including that presented in the work that inspired this paper, due to \textit{E. Colorado} and \textit{I. Paral} [J. Funct. Anal. 199, No. 2, 468--507 (2003; Zbl 1034.35041)]. Beyond that, we present some complementary results, concerning the non-existence of solutions to a class of elliptic problems and a comparison result inspired by the case of Dirichlet boundary conditions, presented by the work of \textit{A. Ambrosetti} et al. [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)].Radial solutions with prescribed number of nodes to an asymptotically linear elliptic problem: a parabolic flow approachhttps://zbmath.org/1541.352502024-09-27T17:47:02.548271Z"Ishiwata, Michinori"https://zbmath.org/authors/?q=ai:ishiwata.michinori"Li, Haoyu"https://zbmath.org/authors/?q=ai:li.haoyuSummary: Based on a parabolic flow, we study the existence of radial solutions with a prescribed number of nodes to an asymptotically linear elliptic problem. Moreover, for the problem defined on the unit ball, we establish a nonexistence result concerning the solutions with more than the prescribed number of nodes, see Assertion (2) of Theorem 1.3. This is a significant difference from the problem defined on the whole space since the latter admits solution with an arbitrarily large number of nodes. This is proved in Theorem 1.1.Concentration analysis for elliptic critical equations with no boundary control: ground-state blow-uphttps://zbmath.org/1541.352512024-09-27T17:47:02.548271Z"Mesmar, Hussein"https://zbmath.org/authors/?q=ai:mesmar.hussein"Robert, Frédéric"https://zbmath.org/authors/?q=ai:robert.fredericSummary: We perform the apriori analysis of solutions to critical nonlinear elliptic equations on manifolds with boundary. The solutions are of minimizing type. The originality is that we impose no condition on the boundary, which leads us to assume \(L^2\)-concentration. We also analyze the effect of a non-homogeneous nonlinearity that results in the fast convergence of the concentration point.The Brezis-Nirenberg problem in 4Dhttps://zbmath.org/1541.352522024-09-27T17:47:02.548271Z"Pistoia, Angela"https://zbmath.org/authors/?q=ai:pistoia.angela"Rocci, Serena"https://zbmath.org/authors/?q=ai:rocci.serenaSummary: We address the existence of blowing-up solutions for the Brezis-Nirenberg problem in 4D.Unusual existence theorems for nonlocal inhomogeneous elliptic equationshttps://zbmath.org/1541.352532024-09-27T17:47:02.548271Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioSummary: In this paper, we prove two unusual existence theorems for nonlocal inhomogeneous elliptic equations. A very particular case of one of them reads as follows: Let \(k : [0, 1] \times \mathbb{R} \to \mathbb{R}\) be a continuous function such that \(k(x, \cdot)\) is nondecreasing for all \(x \in [0, 1]\) and \(k(x_0, \cdot)\) is not constant for some \(x_0 \in [0, 1]\).
Then, for every \(a > \inf_{\xi \in \mathbf{R}} \int_0^1 K(x, \xi) \, \mathrm{d} x\) (where \(K(x, \xi) = \int_0^\xi k(x, t) \, \mathrm{d}t\)) and for every convex set \(S \subseteq C^0 ([0, 1])\) dense in \(L^2 ([0, 1])\), there exists \(\tilde{\delta} \in S\) having the following property: for every continuous function \(f : [0, 1] \times \mathbb{R} \to \mathbb{R}\) and for every nonincreasing nonpositive function \(\gamma : \mathbb{R} \to \mathbb{R}\), there exists \(\epsilon > 0\) such that, for each \(\lambda \in [- \epsilon, \epsilon]\), the problem
\[
\begin{gathered}
- u'' = \lambda f(x, u) + \gamma \left(\int\limits_0^1 K (x, u (x)) \mathrm{d} x \right) k(x, u) + \tilde{\delta}(x) \text{ in } [0, 1], \\
u(0) = u(1) = 0, \\
\int\limits_0^1 K(x, u(x)) \, \mathrm{d}x < a
\end{gathered}
\]
has at least one classical solution.
Besides the presence of the convex dense set \(S\), the most important novelty is that \(\tilde{\delta}\) is fully independent of \(f\) and \(\gamma\). Moreover, we have the localization of the found solution expressed by the inequality \(\int_{\Omega} K(x, u(x)) \, \mathrm{d}x < a\).\(p\)-harmonic functions in the upper half-spacehttps://zbmath.org/1541.352542024-09-27T17:47:02.548271Z"Abreu, E."https://zbmath.org/authors/?q=ai:abreu.emerson-a-m"Clemente, R."https://zbmath.org/authors/?q=ai:clemente.rodrigo-g"do Ó, J. M."https://zbmath.org/authors/?q=ai:do-o.joao-m-bezerra"Medeiros, E."https://zbmath.org/authors/?q=ai:souto-de-medeiros.everaldoSummary: This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space \(\mathbb{R}^N_+\) (\(N\geq 3\)) satisfying nonlinear boundary conditions for \(1<p<N\). Moreover, the symmetry of positive solutions is shown by using the method of moving planes.Hölder estimate for a tug-of-war game with \(1< p< 2\) from Krylov-Safonov regularity theoryhttps://zbmath.org/1541.352552024-09-27T17:47:02.548271Z"Arroyo, Ángel"https://zbmath.org/authors/?q=ai:arroyo.angel"Parviainen, Mikko"https://zbmath.org/authors/?q=ai:parviainen.mikkoSummary: We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the \(p\)-Laplacian with \(1< p< 2\). For this version, the asymptotic Hölder continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.Existence of solutions for \(p(x)\)-Laplacian elliptic BVPs on a variable Sobolev space via fixed point theoremshttps://zbmath.org/1541.352562024-09-27T17:47:02.548271Z"Ayadi, Souad"https://zbmath.org/authors/?q=ai:ayadi.souad"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Afshari, Hojjat"https://zbmath.org/authors/?q=ai:afshari.hojjat"Sahlan, Monireh Nosrati"https://zbmath.org/authors/?q=ai:sahlan.monireh-nosratiSummary: In this paper, we prove some existence theorems for elliptic boundary value problems within the \(p(x)\)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer's and Schauder's theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings.On a new singular and degenerate extension of the \(p\)-Laplace operatorhttps://zbmath.org/1541.352572024-09-27T17:47:02.548271Z"Baravdish, George"https://zbmath.org/authors/?q=ai:baravdish.george"Cheng, Yuanji"https://zbmath.org/authors/?q=ai:cheng.yuanji"Svensson, Olof"https://zbmath.org/authors/?q=ai:svensson.olofSummary: We study a novel degenerate and singular elliptic operator \(\widetilde{\Delta}_{(\tau, \chi)}\) defined by
\[
\widetilde{\Delta}_{(\tau, \chi)} u = \tau (x, D u) (| D u | \Delta_1 u + \chi (x, Du)\Delta_\infty u),
\]
where the singular weights \(\tau (x, s) > 0\) and \(\chi (x, s) \geq 0\) are continuous functions on \(\Omega \times \mathbb{R}^n \setminus \{0\}\). The operator \(\widetilde{\Delta}_{(\tau, \chi)}\) is an extension of
\[
\Delta_{(p, q)} u = |Du|^q \Delta_1 u + (p - 1) |Du|^{p - 2} \Delta_\infty u, \quad p \geq 1, \ q \geq 0,
\]
introduced by the authors and \textit{F. Aström} in [Commun. Pure Appl. Anal. 19, No. 7, 3477--3500 (2020; Zbl 1448.35295)], which in turn is an extension of the \(p\)-Laplace operator \(\Delta_p\). We establish the well-posedness of the Neumann boundary value problem for the parabolic equation \(u_t = \widetilde{\Delta}_{(\tau, \chi)} u\) in the framework of viscosity solutions. For the solution \(u\), the weight \(\chi\) controls the evolution along the tangential and the normal directions, respectively, on the level surface of \(u\). The weight \(\tau\) controls the total speed of the evolution of \(u\).
We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator \(\widetilde{\Delta}_{(\tau, \chi)}\) gives better results than both the Perona-Malik [\textit{P. Perona} and \textit{J. Malik}, ``Scale-space and edge detection using anisotropic diffusion'', IEEE Trans. Pattern Anal. Mach. Intell. 12 (7), 629--639 (1990)] and total variation (TV) methods [\textit{T. F. Chan} and \textit{J. Shen}, Image processing and analysis. Variational, PDE, wavelet, and stochastic methods. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2005; Zbl 1095.68127)] when applied to image enhancement.The Robin problem for quasi-linear elliptic equation \(p(x)\)-Laplacian in a domain with conical boundary pointhttps://zbmath.org/1541.352582024-09-27T17:47:02.548271Z"Borsuk, Mikhail"https://zbmath.org/authors/?q=ai:borsuk.mikhail-vSummary: This paper is a survey of our last results about bounded weak solutions to the Robin boundary and the Robin transmission problems for an elliptic quasi-linear second-order equation with the variable \(p(x)\)-Laplacian in a conical bounded \(n\)-dimensional domain.
For the entire collection see [Zbl 1497.42002].Existence and multiplicity of solutions for a class of Kirchhoff-Boussinesq-type problems with logarithmic growthhttps://zbmath.org/1541.352592024-09-27T17:47:02.548271Z"Carlos, Romulo D."https://zbmath.org/authors/?q=ai:carlos.romulo-d"Mbarki, Lamine"https://zbmath.org/authors/?q=ai:mbarki.lamine"Yang, Shuang"https://zbmath.org/authors/?q=ai:yang.shuangSummary: In this paper, two problems related to the following class of elliptic Kirchhoff-Boussinesq-type models are analyzed in the subcritical (\(\beta = 0\)) and critical (\(\beta = 1\)) cases:
\[
\Delta^2 u - \Delta_p u = \tau|u|^{q - 2} u\ln|u| + \beta|u|^{2_{\ast\ast} - 2}u \text{ in }\Omega \text{ and } \Delta u = u = 0\text{ on }\partial\Omega,
\]
where \(\tau > 0\), \(2 < p < 2^\ast = \frac{2N}{N-2}\) for \(N \geq 3\) and \(2_{\ast\ast} = \infty\) for \(N = 3\), \(N = 4\), \(2_{\ast\ast} = \frac{2N}{N-4}\) for \(N \geq 5\). The first one is concerned with the existence of a nontrivial ground-state solution via variational methods. As for the second problem, we prove the multiplicity of such a solution using the Mountain Pass Theorem.Asymptotic behaviour of operators sum of \(p\)-Laplacianshttps://zbmath.org/1541.352602024-09-27T17:47:02.548271Z"Chipot, Michel"https://zbmath.org/authors/?q=ai:chipot.michelSummary: The goal of this paper is to study the asymptotic behaviour of a linear combination of operators of the \(p \)-Laplacian type in cylinder like domains having some directions going to infinity. The case of a combination of two such operators has been the topic of numerous studies in the recent years.Nehari manifold approach for superlinear double phase problems with variable exponentshttps://zbmath.org/1541.352612024-09-27T17:47:02.548271Z"Crespo-Blanco, Ángel"https://zbmath.org/authors/?q=ai:crespo-blanco.angel"Winkert, Patrick"https://zbmath.org/authors/?q=ai:winkert.patrickThe manuscript deals with the existence and multiplicity of weak solutions to the following double phase problem
\[
\begin{cases}- \Delta_{p(z)}^1 u(z) - \Delta_{q(z)}^{\mu(z)} u(z) =f(z,u(z)) \quad \mbox{in } \Omega, &\\
u=0 \quad \mbox{on }\partial \Omega, &\end{cases}
\]
where \(\Omega \subseteq \mathbb{R}^N\) is a bounded domain with a Lipschitz boundary \(\partial \Omega\), and the authors impose a Dirichlet boundary condition. Given \(r \in C(\overline{\Omega})\) with \(1<r(z)\) for all \(z \in \overline{\Omega}\), by \(\Delta_{r(z)}^{\mu(z)}\) we denote the following weighted differential operator
\[
\Delta_{r(z)}^{\mu(z)} u=\mathrm{div } [\mu(z)|\nabla u|^{r(z)-2}\nabla u] \quad \mbox{for all } u \in W^{1,r(z)}(\Omega),
\]
with \(0 \leq \mu(\cdot) \in L^\infty(\Omega)\). When \(\mu \equiv 1\) we write \(\Delta_{r(z)}^1\). In the above problem, the differential operator is the sum of two such operators (double phase problem). The authors work in the variable exponents setting, hence pose the problem in an appropriate Musielak-Orlicz Sobolev space. The exponents \(p,q\) are properly linked each other and also fulfil suitable conditions. The reaction (right hand side) has superlinear type growth at zero and infinity, and satisfies certain technical conditions.
The authors first obtain a priori bounds for weak solutions to a class of suitable general problems, then prove the existence of a positive and a negative weak solution to the above double phase problem via the mountain pass theorem, involving auxiliary functionals truncated at zero. Further, they obtain the existence of a sign-changing solution to the above double phase problem, by solving a minimization problem on a modified version of the Nehari manifold.
Reviewer: Calogero Vetro (Palermo)Existence and uniqueness for anisotropic quasilinear elliptic equations involving singular nonlinearitieshttps://zbmath.org/1541.352622024-09-27T17:47:02.548271Z"Esposito, Francesco"https://zbmath.org/authors/?q=ai:esposito.francesco"Sciunzi, Berardino"https://zbmath.org/authors/?q=ai:sciunzi.berardino"Trombetta, Alessandro"https://zbmath.org/authors/?q=ai:trombetta.alessandroSummary: We prove the existence and uniqueness of nonnegative solutions to the following singular anisotropic elliptic equation involving the Finsler \(p\)-Laplace operator:
\[
-\Delta^H_p u = \frac{f(x)}{u^{\gamma}} \quad \text{in }\Omega
\]
where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, \(p>1, \gamma >0, N \geq 2, f\geq 0\) in \(\Omega\) (not identically zero), \(H\) is a Finsler norm, and it is subject to zero Dirichlet boundary conditions. In particular, we obtain our results under very general summability assumptions on the source term \(f\).High energy solutions for \(p\)-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearityhttps://zbmath.org/1541.352632024-09-27T17:47:02.548271Z"Goel, Divya"https://zbmath.org/authors/?q=ai:goel.divya"Rawat, Sushmita"https://zbmath.org/authors/?q=ai:rawat.sushmita"Sreenadh, K."https://zbmath.org/authors/?q=ai:sreenadh.konijetiSummary: This article deals with the study of the following Kirchhoff-Choquard problem:
\[
\begin{gathered}
M\left( \int\limits_{\mathbb{R}^N} |\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left( \int\limits_{\mathbb{R}^N}\frac{F(u)(y)}{|x-y|^{\mu}}\,dy \right) f(u), \;\text{in }\mathbb{R}^N, \\
u>0, \; \text{in } \mathbb{R}^N,
\end{gathered}
\]
where \(M\) models Kirchhoff-type nonlinear term of the form \(M(t) = a + bt^{\theta -1}\), where \(a, b > 0\) are given constants; \(1<p<N, \Delta_p = \mathrm{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator; potential \(V \in C^2 (\mathbb{R}^N); f\) is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for \(\theta \in \left[ 1, \frac{2N-\mu}{N-p}\right)\) via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.Solutions of an anisotropic elliptic problem involving nonlinear termshttps://zbmath.org/1541.352642024-09-27T17:47:02.548271Z"Razani, A."https://zbmath.org/authors/?q=ai:razani.abdolrahman|razani.a"Soltani, T."https://zbmath.org/authors/?q=ai:soltani.taimazSummary: Using variational methods, the existence and multiplicity of weak solutions for the following Neumann anisotropic problem
\[
- \sum^{N}_{i = 1} \frac{\partial}{\partial x_{i}} \bigg(|\frac{\partial u}{\partial x_{i}}|^{p_{i}^{(x) - 2}} \frac{\partial u}{\partial x_{i}} \bigg) + \sum^{N}_{i = 1} a(x) |u|^{p_{i}^{(x) - 2}} u = \lambda f(x, u) + \mu g(x, u)
\]
on a bounded domain are proved.Normalized solution for \(p\)-Kirchhoff equation with a \(L^2\)-supercritical growthhttps://zbmath.org/1541.352652024-09-27T17:47:02.548271Z"Ren, Zhi-min"https://zbmath.org/authors/?q=ai:ren.zhimin"Lan, Yong-yi"https://zbmath.org/authors/?q=ai:lan.yongyiSummary: In this paper, we investigate the following \(p\)-Kirchhoff equation
\[
\begin{cases}
(a + b \int_{\mathbb{R}^N}(|\nabla u|^p + |u|^p)dx)(- \nabla_p u + |u|^{p-2} u) = |u|^{s-u}u + \mu u, x \in \mathbb{R}^N, \\
\int_{\mathbb{R}^N} |u|^2 dx = \rho,
\end{cases}
\]
where \(a > 0\), \(b \geq 0\), \(\rho > 0\) are constants, \(p^* = \frac{Np}{N - p}\) is the critical Sobolev exponent, \(\mu\) is a Lagrange multiplier, \(- \Delta_p u = - \operatorname{div}(|\nabla u|^{p - 2}\nabla u)\), \(2 < p < N < 2p\), \(\mu \in \mathbb{R}\), and \(s \in (2\frac{N + 2}{N}p - 2, p^*)\). We demonstrate that the \(p\)-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.Weighted \(p(\cdot)\)-Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applicationshttps://zbmath.org/1541.352662024-09-27T17:47:02.548271Z"Vallejos, Lucas Alejandro"https://zbmath.org/authors/?q=ai:vallejos.lucas-alejandro"Vidal, Raúl Emilio"https://zbmath.org/authors/?q=ai:vidal.raul-emilioSummary: We generalize different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups, using different techniques. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class \(A_{p(\cdot),p^{\ast}(\cdot)}\), where the exponent \(p(\cdot)\) satisfies appropriate hypotheses, and in this case we obtain first order weighted Poincaré inequalities with variable exponents. For Carnot groups we also establish higher order weighted Poincaré inequalities with variable exponents. For these results the crucial part is proving the boundedness of the fractional integral operator on Lebesgue spaces with weighted and variable exponents on spaces of homogeneous type. These results extend those obtained by \textit{X. Li} et al. [Acta Math. Sin., Engl. Ser. 31, No. 7, 1067--1085 (2015; Zbl 1339.46033)], by considering weighted inequalities. They can also be viewed as the extension of weighted Poincaré and Sobolev inequalities widely studied by many authors to the case of variable exponents.
Finally, we use these weighted Poincaré inequalities to establish the existence and uniqueness of a minimizer to the Dirichlet energy integral for a problem involving a degenerate \(p(\cdot)\)-Laplacian with zero boundary values in Carnot groups.Monge-Ampère operators and valuationshttps://zbmath.org/1541.352672024-09-27T17:47:02.548271Z"Knoerr, Jonas"https://zbmath.org/authors/?q=ai:knoerr.jonasSummary: Two classes of measure-valued valuations on convex functions related to Monge-Ampère operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on \(\mathbb{R}^n\) that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations of these valuations and a description in terms of mixed Monge-Ampère operators are established, as well as a characterization of \(\mathrm{SO}(n)\)-equivariant valuations in terms of Hessian measures.Solution of some problems for the string vibration equation in a half-Strip by quadratureshttps://zbmath.org/1541.352682024-09-27T17:47:02.548271Z"Jokhadze, O. M."https://zbmath.org/authors/?q=ai:dzhokhadze.otar-mikhajlovich"Kharibegashvili, S. S."https://zbmath.org/authors/?q=ai:kharibegashvili.s-sSummary: For the inhomogeneous string vibration equation in a half-strip, we consider a problem periodic in the spatial variable and a mixed problem. The solutions of these problems in the form of finite sums are obtained by quadratures. When solving these problems, we use the characteristic rectangle identity, Riemann invariants, and the method of characteristics.Direct and inverse initial boundary value problems for heat equation with non-classical boundary conditionhttps://zbmath.org/1541.352692024-09-27T17:47:02.548271Z"Sadybekov, M."https://zbmath.org/authors/?q=ai:sadybekov.makhmud-abdysametovich|sadybekov.makhmud-a"Derbissaly, B."https://zbmath.org/authors/?q=ai:derbissaly.bauyrzhan-o(no abstract)Control of parabolic equations with inverse square Infinite potential wellshttps://zbmath.org/1541.352702024-09-27T17:47:02.548271Z"Shao, Arick"https://zbmath.org/authors/?q=ai:shao.arickSummary: This survey summarises a presentation recently given by the author at the Ghent Methusalem Junior Analysis Seminar. The talk discussed the recent result of \textit{A. Enciso} et al. [``Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates'', Preprint, \url{arXiv:2112.04457}], joint with Alberto Enciso (ICMAT) and Bruno Vergara (Brown), as well as the main ideas of its proof.
In [loc. cit.], we consider heat operators on a bounded convex domain, with a critically singular potential diverging as the inverse square of the distance to the boundary of the domain. We address the problem of boundary null controllability -- whether one can drive the solution from any initial data to zero via suitable boundary data. We establish a null control result for such operators in all spatial dimensions, in particular providing the first result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the relevant boundary asymptotics and the appropriate energy for this problem.
For the entire collection see [Zbl 1537.35003].New contributions to a complex system of quadratic heat equations with a generalized kernels: global solutionshttps://zbmath.org/1541.352712024-09-27T17:47:02.548271Z"Otsmane, Sarah"https://zbmath.org/authors/?q=ai:otsmane.sarah"Mennouni, Abdelaziz"https://zbmath.org/authors/?q=ai:mennouni.abdelazizSummary: In this work, we propose new contributions to a complex system of quadratic heat equations with a generalized kernel of the form: \(\partial_t z=\mathfrak{L} z+ \widetilde{z}^2\), \(\partial_t \widetilde{z}=\mathfrak{L} \widetilde{z}+ z^2\), \(t>0\), with initial conditions \(z_0=u_0+v_0\), \(\widetilde{z}_0=\widetilde{u}_0+\widetilde{v}_0\), and \(\mathfrak{L}\) is a linear operator with \(e^{t\mathcal{L}}\) its semigroup having a generalized heat kernel \(G\) satisfying in particular \(G(t,x)= t^{-\frac{N}{d}} G(1,xt^{-1/d})\), \(d>0\), \(t>0\) and \(x\in \mathbb{R}^N\). Under conditions on the parameters \(\sigma_1\), \(\widetilde{\sigma}_1\), \(\rho_1\), and \(\widetilde{\rho_1}\) we show results on global-in time solution for small data \(u_0(x)\sim c|x|^{-d\sigma_1}\), \(v_0(x)\sim c|x|^{-d\rho_1}\), \(\widetilde{u}_0(x)\sim c|x|^{-d\widetilde{\sigma}_1}\) and \(\widetilde{v}_0(x)\sim c|x|^{-d\widetilde{\rho}_1}\) as \(|x|\rightarrow \infty\), (\(|c|\) is sufficiently small). We investigate the global existence of solutions to the given system.Modified \(\alpha\)-parameterized differential transform method for solving nonlinear generalized Gardner equationhttps://zbmath.org/1541.352722024-09-27T17:47:02.548271Z"Al-Rozbayani, Abdulghafor M."https://zbmath.org/authors/?q=ai:al-rozbayani.abdulghafor-m"Qasim, Ahmed Farooq"https://zbmath.org/authors/?q=ai:qasim.ahmed-farooqSummary: In this article, we present a novel enhancement to the \(\alpha\)-parameterized differential transform method (PDTM) for solving nonlinear boundary value problems. The proposed method is applied to solve the generalized Gardner equation by utilizing genetic algorithms to obtain optimal parameter values. Our proposed approach extends the general differential transformation method, allowing for the use of various values for the coefficient \(\alpha\). Our solution procedure offers a distinct advantage by allowing the original differential transformation method to be divided into multiple steps, thereby illustrating specific solution properties for nonlinear boundary value problems. Additionally, possible alternative solutions based on varying parameter values are also explored and discussed. The results with those obtained through the DTM method and exact solutions are compared to confirm the accuracy of our method and its efficiency in reaching the exact solution quickly.Plateau flow or the heat flow for half-harmonic mapshttps://zbmath.org/1541.352732024-09-27T17:47:02.548271Z"Struwe, Michael"https://zbmath.org/authors/?q=ai:struwe.michaelSummary: Using the interpretation of the half-Laplacian on \(S^1\) as the Dirichlet-to-Neumann operator for the Laplace equation on the ball \(B\), we devise a classical approach to the heat flow for half-harmonic maps from \(S^1\) to a closed target manifold \(N\subset\mathbb{R}^n\), recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When \(N\) is a smoothly embedded, oriented closed curve \(\Gamma\subset\mathbb{R}^n\), the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.Stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noiseshttps://zbmath.org/1541.352742024-09-27T17:47:02.548271Z"Bendahmane, M."https://zbmath.org/authors/?q=ai:bendahmane.mostafa"Nzeti, H."https://zbmath.org/authors/?q=ai:nzeti.herbert"Tagoudjeu, J."https://zbmath.org/authors/?q=ai:tagoudjeu.jacques"Zagour, M."https://zbmath.org/authors/?q=ai:zagour.mohamed(no abstract)Local exponential stabilization of Rogers-McCulloch and FitzHugh-Nagumo equations by the method of backsteppinghttps://zbmath.org/1541.352752024-09-27T17:47:02.548271Z"Chowdhury, Shirshendu"https://zbmath.org/authors/?q=ai:chowdhury.shirshendu"Dutta, Rajib"https://zbmath.org/authors/?q=ai:dutta.rajib"Majumdar, Subrata"https://zbmath.org/authors/?q=ai:majumdar.subrataSummary: In this article, we study the exponential stabilization of some one-dimensional nonlinear coupled parabolic-ODE systems, namely Rogers-McCulloch and FitzHugh-Nagumo systems, in the interval \((0, 1)\) by boundary feedback. Our goal is to construct an explicit linear feedback control law acting only at the right end of the Dirichlet boundary to establish the local exponential stabilizability of these two different nonlinear systems with a decay \(e^{ \omega t}\), where \(\omega \in (0, \delta ]\) for the FitzHugh-Nagumo system and \(\omega \in (0, \delta )\) for the Rogers-McCulloch system and \(\delta\) is the system parameter that presents in the ODE of both coupled systems. The feedback control law, derived by the backstepping method forces the exponential decay of solution of the closed-loop nonlinear system in both \(\mathrm{L}^2(0, 1)\) and \(\mathrm{H}^1(0, 1)\) norms, respectively, if the initial data is small enough. We also show that the linearized FitzHugh-Nagumo system is not stabilizable with exponential decay \(e^{-\omega t} \), where \(\omega > \delta \).A priori estimates for solutions of FitzHugh-Rinzel systemhttps://zbmath.org/1541.352762024-09-27T17:47:02.548271Z"De Angelis, Monica"https://zbmath.org/authors/?q=ai:de-angelis.monicaIn this paper, the FitzHugh-Rinzel model describing the complex behavior of biological systems is studied. Some properties related to the fundamental solution \(H\) of a nonlinear integro-differential equation are evaluated in order to obtain a priori estimates and asymptotic effects. The obtained estimates show that the solution of the FitzHugh-Rinzel system is bounded for all \(t\). Some relationships on convolutions which characterize the explicit solution, are presented. The obtained results show how the effects due to the initial perturbation are vanishing when \(t\) tends to infinity, and simultaneously, as time increases, the effect of the nonlinear source remains bounded.
Reviewer: Angela Slavova (Sofia)The uniform spreading speed in cooperative systems with non-uniform initial datahttps://zbmath.org/1541.352772024-09-27T17:47:02.548271Z"Hou, Ru"https://zbmath.org/authors/?q=ai:hou.ru"Wang, Zhian"https://zbmath.org/authors/?q=ai:wang.zhian"Xu, Wen-Bing"https://zbmath.org/authors/?q=ai:xu.wenbing|xu.wenbing.1"Zhang, Zhitao"https://zbmath.org/authors/?q=ai:zhang.zhitaoSummary: This paper considers the spreading speed of cooperative nonlocal dispersal systems with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which decreasingly depends only on the smallest decay rate of initial data. The decreasing property of the uniform spreading speed in the smallest decay rate further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components.Global solutions to a modified Fisher-KPP equationhttps://zbmath.org/1541.352782024-09-27T17:47:02.548271Z"Huy, Nguyen Dinh"https://zbmath.org/authors/?q=ai:huy.nguyen-dinh"Tuan, Nguyen Anh"https://zbmath.org/authors/?q=ai:tuan.nguyen-anhSummary: We study a Cauchy problem for a non-focal Fisher-KPP equation. We demonstrate in this study that as long as the habitat limit of the considered population (with the density described by the solution) is large enough relative to the growth rate, there is always a unique global solution to the problem regardless of the size of the non-negative initial data. The idea of the work can be outlined as follows. First, we prove the local existence and uniqueness of the mild solution. Second, we improve the temporal regularity of the solutions and show that the non-negativity of the initial data is preserved for this solution. Having proved these preliminary steps, we derive an energy estimate by which we can control the solution for all time.Dynamical properties of a new SIR epidemic modelhttps://zbmath.org/1541.352792024-09-27T17:47:02.548271Z"Li, Lei"https://zbmath.org/authors/?q=ai:li.lei.9"Ni, Wenjie"https://zbmath.org/authors/?q=ai:ni.wenjie"Wang, Mingxin"https://zbmath.org/authors/?q=ai:wang.mingxinSummary: Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number \(\mathcal{R}_0 \), and show that the disease-free equilibrium point is globally asymptotically stable when \(\mathcal{R}_0\le1 \), while the unique positive equilibrium point is globally asymptotically stable when \(\mathcal{R}_0>1 \). Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number \(\mathcal{R}_0\) plays a crucial role.Symmetry analysis and wave solutions of the Fisher equation using conformal fractional derivativeshttps://zbmath.org/1541.352802024-09-27T17:47:02.548271Z"Saini, Shalu"https://zbmath.org/authors/?q=ai:saini.shalu"Kumar, Rajeev"https://zbmath.org/authors/?q=ai:kumar.rajeev"Deeksha"https://zbmath.org/authors/?q=ai:deeksha."Arora, Rishu"https://zbmath.org/authors/?q=ai:arora.rishu"Kumar, Kamal"https://zbmath.org/authors/?q=ai:kumar.kamalSummary: In the present article, the time fractional Fisher equation is considered in conformal form to study the application of the Lie classical method and quantitative analysis. The Lie symmetry method has been applied to find the infinitesimal generators and symmetry reductions of the fractional Fisher equation. The obtained reduced form of the equation is solved by the method of \(G^\prime/G\), which gives different forms of solutions. The theory of bifurcation has been utilized in the reduced form to check the stability and nature of critical points by transforming the equations into an autonomous system. Some phase portraits have been drawn at different critical points by the use of maple.Life-span of solutions for a nonlinear parabolic systemhttps://zbmath.org/1541.352812024-09-27T17:47:02.548271Z"Tayachi, Slim"https://zbmath.org/authors/?q=ai:tayachi.slimSummary: In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system \(\partial_t u= \Delta u + |v|^{p-1} v\), \(\partial_t v = \Delta v + |u|^{q-1}u\), \(q \geq p \geq 1\), \(q > 1\) with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.Spreading speed of a lattice time-periodic Lotka-Volterra competition system with bistable nonlinearityhttps://zbmath.org/1541.352822024-09-27T17:47:02.548271Z"Wang, Hongyong"https://zbmath.org/authors/?q=ai:wang.hongyong"Pan, Chaohong"https://zbmath.org/authors/?q=ai:pan.chaohong(no abstract)Propagation dynamics of a three species predator-prey system with a pair of strong-weak competing predatorshttps://zbmath.org/1541.352832024-09-27T17:47:02.548271Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.234"Yang, Fei-Ying"https://zbmath.org/authors/?q=ai:yang.feiying"Li, Wan-Tong"https://zbmath.org/authors/?q=ai:li.wan-tongSummary: We are concerned with propagation phenomenon of a three-species system involving a single prey and a pair of strong-weak competing predators. More accurately, we shall mainly consider situations when the simple prey is an indigenous species, and one of the two predators is aboriginal, while the other is alien. In any case, under certain parameters, three species can finally coexist. Naturally, we have obtained that when a weak competing predator is an aboriginal species, the strong predator can successfully invade the environment and become dominant species, leading to the extinction of its weak counterpart. Particularly, we can construct traveling wave solution where the weak predator invades the environment inhabited by its strong competitor and replaces aboriginal strong counterpart, which depends on higher biomass rate of the weak predator. At last, we get the minimal wave speeds for these traveling waves in these situations.Dynamics of a competition model with intra- and interspecific interference in the unstirred chemostathttps://zbmath.org/1541.352842024-09-27T17:47:02.548271Z"Wang, Lin"https://zbmath.org/authors/?q=ai:wang.lin|wang.lin.5|wang.lin.13|wang.lin.1|wang.lin.10|wang.lin.6|wang.lin.7"Wu, Jianhua"https://zbmath.org/authors/?q=ai:wu.jianhua|wu.jianhua.1Summary: This paper deals with an unstirred chemostat system with intra- and interspecific interference. We first investigate the uniqueness and asymptotic behaviors of the positive steady state solution of the single species model. The results show that the species cannot survive when the intraspecific interference is large enough. For two species system, we find that the two species can coexist when the influence of interspecific interference is relatively small, and that the system has no positive steady-state solution when the strength of interspecific interference is large enough. Furthermore, we study the influence of diffusion rate on species coexistence. The results show that for large diffusion rate, both species will be washed out, and the competition exclusion occurs at small diffusion rate. This indicates that species coexistence only occurs at intermediate diffusion rate.Curved fronts in time-periodic reaction-diffusion equations with monostable nonlinearityhttps://zbmath.org/1541.352852024-09-27T17:47:02.548271Z"Zhang, Suobing"https://zbmath.org/authors/?q=ai:zhang.suobing"Bu, Zhen-Hui"https://zbmath.org/authors/?q=ai:bu.zhenhui"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.1Summary: Recently, it was found that there exists various curved fronts with nonplanar level sets, which has been attracting increasing attention. In this paper, we study periodic traveling curved fronts for reaction-diffusion equations with time-periodic monostable nonlinearity in \(\mathbb{R}^2 \). By using the comparison principle and constructing super-sub solutions, we first establish the existence of periodic traveling curved fronts. Furthermore, we show that when the given perturbation decays at infinity under some weighted sense, the periodic traveling curved front is the globally asymptotic stability by introducing mild subsolutions.Nonlocal Cahn-Hilliard equation with degenerate mobility: incompressible limit and convergence to stationary stateshttps://zbmath.org/1541.352862024-09-27T17:47:02.548271Z"Elbar, Charles"https://zbmath.org/authors/?q=ai:elbar.charles"Perthame, Benoît"https://zbmath.org/authors/?q=ai:perthame.benoit"Poiatti, Andrea"https://zbmath.org/authors/?q=ai:poiatti.andrea"Skrzeczkowski, Jakub"https://zbmath.org/authors/?q=ai:skrzeczkowski.jakubSummary: The link between compressible models of tissue growth and the Hele-Shaw free boundary problem of fluid mechanics has recently attracted a lot of attention. In most of these models, only repulsive forces and advection terms are taken into account. In order to take into account long range interactions, we include a surface tension effect by adding a nonlocal term which leads to the degenerate nonlocal Cahn-Hilliard equation, and study the incompressible limit of the system. The degeneracy and the source term are the main difficulties. Our approach relies on a new \(L^{\infty}\) estimate obtained by De Giorgi iterations and on a uniform control of the energy despite the source term. We also prove the long-term convergence to a single constant stationary state of any weak solution using entropy methods, even when a source term is present. Our result shows that the surface tension in the nonlocal (and even local) Cahn-Hilliard equation will not prevent the tumor from completely invading the domain.Local solvability and dilation-critical singularities of supercritical fractional heat equationshttps://zbmath.org/1541.352872024-09-27T17:47:02.548271Z"Fujishima, Yohei"https://zbmath.org/authors/?q=ai:fujishima.yohei"Hisa, Kotaro"https://zbmath.org/authors/?q=ai:hisa.kotaro"Ishige, Kazuhiro"https://zbmath.org/authors/?q=ai:ishige.kazuhiro"Laister, Robert"https://zbmath.org/authors/?q=ai:laister.robertSummary: We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a \textit{dilation-critical singularity} (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.Some qualitative properties for the Kirchhoff total variation flowhttps://zbmath.org/1541.352882024-09-27T17:47:02.548271Z"Boudjeriou, Tahir"https://zbmath.org/authors/?q=ai:boudjeriou.tahir(no abstract)On the solvability of some parabolic equations involving nonlinear boundary conditions with \(L^1\) datahttps://zbmath.org/1541.352892024-09-27T17:47:02.548271Z"Taourirte, Laila"https://zbmath.org/authors/?q=ai:taourirte.laila"Charkaoui, Abderrahim"https://zbmath.org/authors/?q=ai:charkaoui.abderrahim"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:eddine-alaa.nour|alaa.noureddineSummary: We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and \(L^1\) data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models.Existence and regularity results for degenerate parabolic problems in the presence of strongly increasing regularizing lower-order terms and \(L^m\)-data/Dirac masshttps://zbmath.org/1541.352902024-09-27T17:47:02.548271Z"Abdellaoui, Mohammed"https://zbmath.org/authors/?q=ai:abdellaoui.mohammed-amin"Redwane, Hicham"https://zbmath.org/authors/?q=ai:redwane.hichamSummary: We study the existence and regularity results for degenerate parabolic problems in the presence of strongly increasing regularizing lower-order terms and \(L^m\)-data/Dirac mass.Degenerate singular parabolic problems with natural growthhttps://zbmath.org/1541.352912024-09-27T17:47:02.548271Z"El Ouardy, Mounim"https://zbmath.org/authors/?q=ai:el-ouardy.mounim"El Hadfi, Youssef"https://zbmath.org/authors/?q=ai:el-hadfi.youssef"Sbai, Abdelaaziz"https://zbmath.org/authors/?q=ai:sbai.abdelaazizSummary: In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term
\[
\begin{cases}
\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^q)|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^p}{u^{\gamma}}=f & \text{in } Q, \\
u(x,t)=0 & \text{on } \Gamma, \\
u(x,t=0)=u_0 (x) & \text{in } \Omega,
\end{cases}
\]
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\), \(N>2\), \(Q\) is the cylinder \(\Omega \times (0,T)\), \(T>0\), \(\Gamma\) the lateral surface \(\partial \Omega \times (0,T)\), \(2\leq p<N\), \(a(x,t)\) and \(b(x,t)\) are positive measurable bounded functions, \(q\geq 0\), \(0\leq\gamma <1\), and \(f\) non-negative function belongs to the Lebesgue space \(L^m (Q)\) with \(m>1\), and \(u_0 \in L^{\infty}(\Omega)\) such that
\[
\forall\omega\subset\subset\Omega\, \exists D_{\omega}> 0:\, u_0 \geq D_{\omega}\text{ in }\omega.
\]
More precisely, we study the interaction between the term \(u^q\) (\(q>0\)) and the singular lower order term \(d(x,t)|\nabla u|^p u^{-\gamma}\) (\(0<\gamma<1\)) in order to get a solution to the above problem. The regularizing effect of the term \(u^q\) on the regularity of the solution and its gradient is also analyzed.Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostathttps://zbmath.org/1541.352922024-09-27T17:47:02.548271Z"Tchouanti, Josué"https://zbmath.org/authors/?q=ai:tchouanti.josueSummary: We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative trait whose dynamics is described by a diffusion process. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show by probabilistic arguments that the semi-group of the stochastic trait dynamics admits a density. We deduce that the diffusion-growth-fragmentation equation admits a function solution with a certain Besov regularity.A degenerate migration-consumption model in domains of arbitrary dimensionhttps://zbmath.org/1541.352932024-09-27T17:47:02.548271Z"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelSummary: In a smoothly bounded convex domain \(\Omega\subset\mathbb{R}^n\) with \(n\geq 1\), a no-flux initial-boundary value problem for
\[
\begin{cases}
u_t =\Delta\left(u\phi \left(v\right)\right), \\
v_t =\Delta v-uv,
\end{cases}
\]
is considered under the assumption that near the origin, the function \(\phi\) suitably generalizes the prototype given by
\[
\phi (\xi)=\xi^{\alpha},\qquad \xi \in [0,\xi_0].
\]
By means of separate approaches, it is shown that in both cases \(\alpha \in (0,1)\) and \(\alpha \in [1, 2]\) some global weak solutions exist which, inter alia, satisfy
\[
C(T):=\underset{t\in (0,T)}{\mathrm{ess\,sup}}\int_{\Omega}u(\cdot, t)\ln u(\cdot,t)<\infty \qquad \text{for all }T>0,
\]
with \(\sup_{T>0}C(T) < \infty\) if \(\alpha \in [1, 2]\).On regularity and asymptotic stability for semilinear nonlocal pseudo-parabolic equationshttps://zbmath.org/1541.352942024-09-27T17:47:02.548271Z"Dao Trong Quyet"https://zbmath.org/authors/?q=ai:dao-trong-quyet."Dang Thi Phuong Thanh"https://zbmath.org/authors/?q=ai:dang-thi-phuong-thanh.Summary: We deal with a class of nonlocal pseudo-parabolic equations involving strong nonlinearities. The questions on existence, regularity and stability of solutions are addressed by using local estimates, fixed point arguments, and the relation between the Hilbert scales and fractional Sobolev spaces.Dynamics of a nonlinear pseudo-parabolic equation with fading memoryhttps://zbmath.org/1541.352952024-09-27T17:47:02.548271Z"Peng, Xiaoming"https://zbmath.org/authors/?q=ai:peng.xiaoming"Shang, Yadong"https://zbmath.org/authors/?q=ai:shang.yadong"Yu, Jiali"https://zbmath.org/authors/?q=ai:yu.jialiSummary: This paper is concerned with the nonlinear pseudo-parabolic equation with fading memory. First, we prove the existence, uniqueness and continuous dependence of weak solutions when \(\rho\) and \(f\) have polynomial growth of critical order. Then, we establish the existence and optimal regularity of the global attractor. The result extends and improves some existing results.
{\copyright 2024 American Institute of Physics}Classification of initial energy in a pseudo-parabolic equation with variable exponents and singular potentialhttps://zbmath.org/1541.352962024-09-27T17:47:02.548271Z"Sun, Xizheng"https://zbmath.org/authors/?q=ai:sun.xizheng"Han, Zhiqing"https://zbmath.org/authors/?q=ai:han.zhiqing"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchenSummary: This paper deals with a pseudo-parabolic equation with singular potential and variable exponents. First, we determine the existence and uniqueness of weak solutions in Sobolev spaces with variable exponents. Second, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely using the initial energy. Third, we obtain lower and upper bounds of blow-up time for all possible initial energy. The results in this paper are compatible with the corresponding problems with constant exponents. Part results of the paper extend the recent ones in
[\textit{W. Lian} et al., J. Differ. Equations 269, No. 6, 4914--4959 (2020; Zbl 1448.35322)],
[\textit{R. Xu} and \textit{J. Su}, J. Funct. Anal. 264, No. 12, 2732--2763 (2013; Zbl 1279.35065)], and
[\textit{W. Liu} and \textit{J. Yu}, J. Funct. Anal. 274, No. 5, 1276--1283 (2018; Zbl 1383.35033)].Higher order interpolative geometries and gradient regularity in evolutionary obstacle problemshttps://zbmath.org/1541.352972024-09-27T17:47:02.548271Z"Kim, Sunghan"https://zbmath.org/authors/?q=ai:kim.sunghan"Nyström, Kaj"https://zbmath.org/authors/?q=ai:nystrom.kajThe obstacle problem for a large class of parabolic partial differential operators of the type
\[
Hu\equiv\operatorname{div}(\bar{a}(\nabla u))-\frac{\partial u}{\partial t}
\]
is considered. An important special case is the Evolutionary \(p\)-Laplace operator
\[
\operatorname{div}(|\nabla u|^{p-2}\nabla u)-\frac{\partial u}{\partial t},\quad p>2.
\]
A basic reference is \textit{E. DiBenedetto}'s book [Degenerate parabolic equations. New York, NY: Springer-Verlag (1993; Zbl 0794.35090)].
The solution \(u=u(x,t)\) is required to stay above a given obstacle \(\psi\), say \(u\geq\psi\). It is an important feature that here no assumptions about the time derivative \(\psi_t\) are made; neither does \(u_t\) appear explicitly. (This crucial point of view was present in [\textit{T. Kuusi} et al., J. Math. Pures Appl. (9) 101, No. 2, 119--151 (2014; Zbl 1322.35091)] and [\textit{P. Lindqvist} and \textit{M. Parviainen}, J. Funct. Anal. 263, No. 8, 2458--2482 (2012; Zbl 1257.35112)]). Optimal \(C^{1,d}\)-regularity is proved, when the obstacle is regular enough. It is the Hölder continuity of the space derivative \(\nabla u=\left(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)\) that is obtained.
Special attention is paid to the free boundary \(\partial(u>\psi)\) using some method from [\textit{J. Andersson} et al., J. Differ. Equations 259, No. 6, 2167--2179 (2015; Zbl 1318.35051)]. Needless to say, the present work is technically advanced, the regularity theory being used.
Reviewer: Peter Lindqvist (Trondheim)On a criterion for log-convex decay in non-selfadjoint dynamicshttps://zbmath.org/1541.352982024-09-27T17:47:02.548271Z"Johnsen, Jon"https://zbmath.org/authors/?q=ai:johnsen.jonSummary: The short-time and global behaviour are studied for autonomous linear evolution equations defined by generators of uniformly bounded holomorphic semigroups in a Hilbert space. A general criterion for log-convexity in time of the norm of the solution is treated. Strict decrease and differentiability at the initial time results, with a derivative controlled by the lower bound of the negative generator, which is proved strictly accretive with equal numerical and spectral abscissas.
For the entire collection see [Zbl 1497.42002].Short-time asymptotics for game-theoretic \(p\)-Laplacian and Pucci operatorshttps://zbmath.org/1541.352992024-09-27T17:47:02.548271Z"Berti, Diego"https://zbmath.org/authors/?q=ai:berti.diegoSummary: Let \(\Omega\) be a domain of \(\mathbb{R}^N\), \(N \geq 2\), with non empty boundary \(\Gamma\). In these notes, we deal with the solution \(u\) of \(u_t = F\left (\nabla u, \nabla^2 u\right)\) in \(\Omega \times (0, \infty)\), such that \(u\) is initially zero in \(\Omega\) and equals one on \(\Gamma\) for all positive times. Here, \(F\) is the \textit{game-theoretic \(p\)-Laplacian} \(\Delta_p^G\) or either one of the \textit{Pucci's extremal operators} \(\mathcal{M}^\pm\). In the spirit of works by Varadhan and Magnanini-Sakaguchi in the case of the same initial-boundary problem for the \textit{heat equation}, we summarize recent results regarding the connection between the behavior for small times and the geometry of \(\Omega\). In particular, we present asymptotic formulas as \(t \rightarrow 0^+\) for both the values of \(u\) and of its \(q\)-means on balls touching \(\Gamma\).
For the entire collection see [Zbl 1497.42002].Examples of equivariant Lagrangian mean curvature flowhttps://zbmath.org/1541.353002024-09-27T17:47:02.548271Z"Lotay, Jason D."https://zbmath.org/authors/?q=ai:lotay.jason-deanSummary: We describe important examples of Lagrangian mean curvature flow in \(\mathbb{C}^2\) which are invariant under a circle action. Through these examples, we see compact and non-compact situations, long-time existence, singularities forming via explicit models, and significant objects in Riemannian and symplectic geometry, including the Clifford torus, Whitney sphere and Lawlor necks.
For the entire collection see [Zbl 1497.42002].Scattering problems for the wave equation in 1D: D'Alembert-type representations and a reconstruction methodhttps://zbmath.org/1541.353012024-09-27T17:47:02.548271Z"Kalimeris, Konstantinos"https://zbmath.org/authors/?q=ai:kalimeris.konstantinos"Mindrinos, Leonidas"https://zbmath.org/authors/?q=ai:mindrinos.leonidasSummary: We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index. Analogous formulae exist already in the literature, but in the current work this is derived in a natural way for general incident field, by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is also valid for phaseless data.Wave equation for Sturm-Liouville operator with singular potentialshttps://zbmath.org/1541.353022024-09-27T17:47:02.548271Z"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-v"Shaimardan, Serikbol"https://zbmath.org/authors/?q=ai:shaimardan.serikbol"Yeskermessuly, Alibek"https://zbmath.org/authors/?q=ai:yeskermessuly.alibekSummary: The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.Global well-posedness of fourth-order Petrovsky equation with weak and strong damping termshttps://zbmath.org/1541.353032024-09-27T17:47:02.548271Z"Pang, Yue"https://zbmath.org/authors/?q=ai:pang.yue"Yin, Yufeng"https://zbmath.org/authors/?q=ai:yin.yufeng(no abstract)Potential well method for a class of fourth order wave equations with Newtonian potentialhttps://zbmath.org/1541.353042024-09-27T17:47:02.548271Z"Vu, Ngo Tran"https://zbmath.org/authors/?q=ai:vu.ngo-tran"Dung, Dao Bao"https://zbmath.org/authors/?q=ai:dung.dao-bao"Freitas, Mirelson M."https://zbmath.org/authors/?q=ai:freitas.mirelson-mSummary: In this paper, we investigate the initial boundary value problem for a fourth order wave equation with Newtonian potential. We establish firstly the local existence of solutions by Banach fixed point theorem. Using the potential well argument, we show the global existence, finite time blow-up, asymptotic behavior of solutions since the initial energy is not over the depth of the potential well. Finally, when the initial energy is supercritical, we give some explicit criterion for blow-up in finite time.Solvability of certain one-dimensional semilinear hyperbolic systems and hyperbolic Liouville equationhttps://zbmath.org/1541.353052024-09-27T17:47:02.548271Z"Huh, Hyungjin"https://zbmath.org/authors/?q=ai:huh.hyungjin(no abstract)Analytical solution of the mixed problem on a segment for one-dimensional ionization equations in the case of constant velocities of atoms and ionshttps://zbmath.org/1541.353062024-09-27T17:47:02.548271Z"Gavrikov, M. B."https://zbmath.org/authors/?q=ai:gavrikov.m-b"Taiurskii, A. A."https://zbmath.org/authors/?q=ai:tayurskii.a-a(no abstract)Existence of global bounded smooth solutions for the one-dimensional unsteady ZND combustion modelhttps://zbmath.org/1541.353072024-09-27T17:47:02.548271Z"Chen, Honghua"https://zbmath.org/authors/?q=ai:chen.honghua"Zhao, Qing"https://zbmath.org/authors/?q=ai:zhao.qingSummary: This paper studies the existence of global bounded smooth solutions to the one-dimensional ZND combustion model. We find a sufficient condition on the initial data to obtain the existence of global bounded smooth solutions to the Cauchy problem for the ZND model.Existence of a weak solution to a generalized Riemann-type hydrodynamical equationhttps://zbmath.org/1541.353082024-09-27T17:47:02.548271Z"Xia, Shuiyan"https://zbmath.org/authors/?q=ai:xia.shuiyan(no abstract)Conservation laws with nonlocal velocity: the singular limit problemhttps://zbmath.org/1541.353092024-09-27T17:47:02.548271Z"Friedrich, Jan"https://zbmath.org/authors/?q=ai:friedrich.jan"Göttlich, Simone"https://zbmath.org/authors/?q=ai:gottlich.simone"Keimer, Alexander"https://zbmath.org/authors/?q=ai:keimer.alexander"Pflug, Lukas"https://zbmath.org/authors/?q=ai:pflug.lukasThis paper deals with the nonlocal conservation law
\[
\partial_t q(t,x)= - \partial_x\bigl( q(t,x)\mathcal{W} \bigl[ \gamma ,V(q)\bigr] (t,x)\bigr),\tag{1}
\]
where \(V'\le0\), and the nonlocal operator is defined as follows
\[
\mathcal{W} \bigl[ \gamma ,V(q)\bigr] (t,x)=\frac{1}{\eta}\int_x^\infty \gamma\left(\frac{y-x}{\eta}\right)V(q(t,y)dy,
\]
\(\eta\) is a positive constant and \(\gamma \in L^\infty \cap L^1(\mathbb{R})\) is positive, decreasing, and \(\| \gamma \|_{L^1}=1\).
The authors augment (1) with the nonnegative inital condition
\[
q(0,\cdot)=q_0\in BV(\mathbb{R}),
\]
and study the well-posedness of (1) and the convergence of the solutions of (1) to the entropy ones of
\[
\partial_t q(t,x)= - \partial_x\bigl( q(t,x)V(q (t,x))\bigr),
\]
as \(\eta\to0\).
Reviewer: Giuseppe Maria Coclite (Bari)Scalar conservation law in a bounded domain with strong source at boundaryhttps://zbmath.org/1541.353102024-09-27T17:47:02.548271Z"Xu, Lu"https://zbmath.org/authors/?q=ai:xu.luSummary: We consider a scalar conservation law with source in a bounded open interval \(\Omega\subseteq\mathbb{R}\). The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function \(\varrho\) with an intensity function \(V:\Omega\rightarrow \mathbb{R}_+\) that grows to infinity at \(\partial\Omega\). We define the entropy solution \(u \in L^\infty\) and prove the uniqueness. When \(V\) is integrable, \(u\) satisfies the boundary conditions introduced by \textit{F. Otto} [C. R. Acad. Sci., Paris, Sér. I 322, No. 8, 729--734 (1996; Zbl 0852.35013)], which allows the solution to attain values at \(\partial\Omega\) different from the given boundary data. When the integral of \(V\) blows up, \(u\) satisfies an energy estimate and presents essential continuity at \(\partial\Omega\) in a weak sense.Delta-shock for a class of systems of conservation laws of the Keyfitz-Kranzer typehttps://zbmath.org/1541.353112024-09-27T17:47:02.548271Z"Li, Shiwei"https://zbmath.org/authors/?q=ai:li.shiweiAuthor's abstract: Riemann problem for a class of systems of conservation laws of Keyfitz-Kranzer type is solved. The Riemann solutions contain three kinds of interesting structures, two of which contain vacuum and the other includes delta-shock. The generalized Rankine-Hugoniot relation and improved entropy condition are proposed to solve the delta-shock. Besides, with the use of the vanishing viscosity method, all of the existence, uniqueness, and stability of the solutions involving the delta-shock are proved.
Reviewer: Lingda Xu (Hong Kong)Interactions of delta shock waves in a pressureless hydrodynamic modelhttps://zbmath.org/1541.353122024-09-27T17:47:02.548271Z"Wang, Yixuan"https://zbmath.org/authors/?q=ai:wang.yixuan"Sun, Meina"https://zbmath.org/authors/?q=ai:sun.meina(no abstract)On Gibbs measures and topological solitons of exterior equivariant wave mapshttps://zbmath.org/1541.353132024-09-27T17:47:02.548271Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjornSummary: We consider \(k\)-equivariant wave maps from the exterior spatial domain \(\mathbb{R}^3 \setminus B(0,1)\) into the target \(\mathbb{S}^3\). This model has infinitely many topological solitons \(\mathcal{Q}_{n,k}\), which are indexed by their topological degree \(n \in \mathbb{Z}\). For each \(n \in \mathbb{Z}\) and \(k \geq 1\), we prove the existence and invariance of a Gibbs measure supported on the homotopy class of \(\mathcal{Q}_{n,k}\). As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.Boundary value problems for a parabolic-hyperbolic equation with nonlinear loaded termshttps://zbmath.org/1541.353142024-09-27T17:47:02.548271Z"Abdullaev, O. Kh."https://zbmath.org/authors/?q=ai:abdullaev.obidjon-khayrullayevich(no abstract)Boundary value problems for a parabolic-hyperbolic equation with a superposition of operators of the first and second ordershttps://zbmath.org/1541.353152024-09-27T17:47:02.548271Z"Islomov, B. I."https://zbmath.org/authors/?q=ai:islomov.bozor-islomovich"Yuldashev, T. K."https://zbmath.org/authors/?q=ai:yuldashev.tursan-kamaldinovich|yuldashev.tursun-kamaldinovich"Kylyshbayeva, G. K."https://zbmath.org/authors/?q=ai:kylyshbayeva.g-k(no abstract)A problem in an unbounded domain with combined Tricomi and Frankl conditions on one boundary characteristic for one class of mixed-type equationshttps://zbmath.org/1541.353162024-09-27T17:47:02.548271Z"Mirsaburov, M."https://zbmath.org/authors/?q=ai:mirsaburov.m"Turaev, R. N."https://zbmath.org/authors/?q=ai:turaev.rasul-nSummary: In this work, in an unbounded domain, we prove the correctness of the problem with combined Tricomi and Frankl conditions on one boundary characteristic for one class of mixed-type equations.A problem with shift for mixed-type equation in domain, the elliptical part of which is a horizontal striphttps://zbmath.org/1541.353172024-09-27T17:47:02.548271Z"Zunnunov, R. T."https://zbmath.org/authors/?q=ai:zunnunov.r-t(no abstract)Symmetry results for Serrin-type problems in doubly connected domainshttps://zbmath.org/1541.353182024-09-27T17:47:02.548271Z"Borghini, Stefano"https://zbmath.org/authors/?q=ai:borghini.stefanoSummary: In this work, we employ the technique developed in [\textit{V. Agostiniani} et al., ``On the Serrin problem for ring-shaped domains'', Preprint, \url{arXiv:2109.11255}] to prove rotational symmetry for a class of Serrin-type problems for the standard Laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.Symmetry breaking solutions for a two-phase overdetermined problem of Serrin-typehttps://zbmath.org/1541.353192024-09-27T17:47:02.548271Z"Cavallina, Lorenzo"https://zbmath.org/authors/?q=ai:cavallina.lorenzo"Yachimura, Toshiaki"https://zbmath.org/authors/?q=ai:yachimura.toshiakiSummary: In this paper, we consider an overdetermined problem of Serrin-type for a two-phase elliptic operator with piecewise constant coefficients. We show the existence of infinitely many branches of nontrivial symmetry breaking solutions which bifurcate from any radially symmetric configuration satisfying some condition on the coefficients.
For the entire collection see [Zbl 1497.42002].Linear stability analysis of overdetermined problems with non-constant datahttps://zbmath.org/1541.353202024-09-27T17:47:02.548271Z"Onodera, Michiaki"https://zbmath.org/authors/?q=ai:onodera.michiakiThe author investigates the overdetermined problem
\[
\begin{cases}
-\Delta u = (n+\alpha) \, |x|^\alpha &\text{ in }\Omega,\\
u = 0 &\text{ on }\partial \Omega,\\
{}-\displaystyle\frac{\, \partial u \,}{\, \partial \nu \,} = g_0\Big(\frac{x}{|x|}\Big) \, |x|^\beta &\text{ on }\partial \Omega,
\end{cases}\tag{1}
\]
where $\Omega$ is a bounded domain in $\mathbb R^n$, $n \ge 2$, whose boundary is represented as a polar graph about the origin. The exponents $\alpha > -n$ and $\beta$ are constant, and $g_0$ is a sufficiently smooth function defined on the spherical surface $\mathbb S^{n-1} = \partial B(0,1)$. Mind that the domain $\Omega$ is an unknown of the problem. Using the implicit function theorem as in [\textit{A. Gilsbach} and \textit{M. Onodera}, Calc. Var. Partial Differ. Equ. 60, No. 6, Paper No. 241, 19 p. (2021; Zbl 1478.35153)], and assuming $\alpha + \beta - 1 \not \in \mathbb N \setminus \{0\}$, the author constructs a domain $\Omega$ where problem (1) is solvable (see also [\textit{C. Bianchini} et al., Interfaces Free Bound. 16, No. 2, 215--241 (2014; Zbl 1297.35153)]). He also shows that if $g_0$ is close to the unit constant then the constructed domain $\Omega$ approaches the unit ball centered at the origin (Theorem 1). It is not excluded, in general, that problem (1) may have multiple solutions $(u,\Omega)$: uniqueness is proved among all smooth domains containing the origin in the special case when $\alpha-\beta+1 < 0$. In such a case the author shows, in particular, that if $g_0 \equiv 1$ and problem (1) admits a solution $u$ then the domain $\Omega$ is a ball centered at the origin (Proposition 2.1). The result is achieved by comparison with convenient radial barriers (see for instance [\textit{A. Greco} and \textit{F. Pisanu}, Math. Eng. (Springfield) 4, No. 3, Paper No. 17, 14 p. (2022; Zbl 1497.35275)] and the references therein). In the case when $g_0$ is not constant, a foliation of $\mathbb R^n \setminus \{0\}$ by non-radial surfaces is used.
Reviewer: Antonio Greco (Cagliari)Quantitative magnetic isoperimetric inequalityhttps://zbmath.org/1541.353212024-09-27T17:47:02.548271Z"Ghanta, Rohan"https://zbmath.org/authors/?q=ai:ghanta.rohan"Junge, Lukas"https://zbmath.org/authors/?q=ai:junge.lukas"Morin, Léo"https://zbmath.org/authors/?q=ai:morin.leoSummary: In 1996 Erdős showed that among planar domains of fixed area, the smallest principal eigenvalue of the Dirichlet Laplacian with a constant magnetic field is uniquely achieved on the disk. We establish a quantitative version of this inequality, with an explicit remainder term depending on the field strength that measures how much the domain deviates from the disk.Neumann eigenvalues of elliptic operators in Sobolev extension domainshttps://zbmath.org/1541.353222024-09-27T17:47:02.548271Z"Gol'dshtein, Vladimir"https://zbmath.org/authors/?q=ai:goldshtein.vladimir"Pchelintsev, Valerii"https://zbmath.org/authors/?q=ai:pchelintsev.valery-anatolevich"Ukhlov, Alexander"https://zbmath.org/authors/?q=ai:ukhlov.alexanderSummary: We obtain estimates of Neumann eigenvalues of the divergence form elliptic operators in Sobolev extension domains. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. The connection between Neumann eigenvalues of elliptic operators and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857) is given.Reducibility and nonlinear stability for a quasi-periodically forced NLShttps://zbmath.org/1541.353232024-09-27T17:47:02.548271Z"Haus, E."https://zbmath.org/authors/?q=ai:haus.emanuele"Langella, B."https://zbmath.org/authors/?q=ai:langella.beatrice"Maspero, A."https://zbmath.org/authors/?q=ai:maspero.alberto"Procesi, M."https://zbmath.org/authors/?q=ai:procesi.michelaSummary: Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus \(\mathbb{T}^2:=(\mathbb{R}/2\pi\mathbb{Z})^2\), we consider a quasi-periodically forced NLS equation on \(\mathbb{T}^2\) arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.Publisher's note: ``On the first eigenvalue of the Laplacian for polygons''https://zbmath.org/1541.353242024-09-27T17:47:02.548271Z"Indrei, Emanuel"https://zbmath.org/authors/?q=ai:indrei.emanuelPublisher's note on the authors' paper [ibid. 65, No. 4, Article ID 041506, 40 p. (2024; Zbl 1537.35263)].A variational approach to the hot spots conjecturehttps://zbmath.org/1541.353252024-09-27T17:47:02.548271Z"Rohleder, Jonathan"https://zbmath.org/authors/?q=ai:rohleder.jonathanSummary: We review a recent new approach to the study of critical points of Laplacian eigenfunctions. Its core novelty is a non-standard variational principle for the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, simply connected planar domains. This principle can be used to provide simple proofs of some previously known results on the hot spots conjecture.
For the entire collection see [Zbl 1537.35003].Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean spacehttps://zbmath.org/1541.353262024-09-27T17:47:02.548271Z"Silva, Cristiano S."https://zbmath.org/authors/?q=ai:silva.cristiano-s"Miranda, Juliana F. R."https://zbmath.org/authors/?q=ai:miranda.juliana-f-r"Araújo Filho, Marcio C."https://zbmath.org/authors/?q=ai:filho.marcio-c-araujoSummary: In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace-Beltrami and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors.A logarithmic improvement in the two-point Weyl law for manifolds without conjugate pointshttps://zbmath.org/1541.353272024-09-27T17:47:02.548271Z"Keeler, Blake"https://zbmath.org/authors/?q=ai:keeler.blakeSummary: In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold \(M\) with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, \(E_\lambda (x,y)\), of the projection operator from \(L^2(M)\) onto the direct sum of eigenspaces with eigenvalue smaller than \(\lambda^2\) as \(\lambda\rightarrow\infty\). In the regime where \(x,y\) are restricted to a compact neighborhood of the diagonal in \(M\times M\), we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for \(E_\lambda\) and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case \(E_\lambda (x,x)\). When \(x,y\) avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for \(E_\lambda \). Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the \(C^\infty\)-topology to a universal scaling limit at an inverse logarithmic rate.Semiregular non-commutative harmonic oscillators: some spectral asymptotic propertieshttps://zbmath.org/1541.353282024-09-27T17:47:02.548271Z"Malagutti, Marcello"https://zbmath.org/authors/?q=ai:malagutti.marcello"Parmeggiani, Alberto"https://zbmath.org/authors/?q=ai:parmeggiani.albertoSummary: The study is devoted to spectral analysis of systems of PDEs, namely, a class of systems containing certain quantum optics models such as the Jaynes-Cummings model. More in detail, the research deals with spectral Weyl asymptotics for a semiregular system, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch.
For the entire collection see [Zbl 1537.35003].Scattering of Maxwell potentials on curved spacetimeshttps://zbmath.org/1541.353292024-09-27T17:47:02.548271Z"Taujanskas, Grigalius"https://zbmath.org/authors/?q=ai:taujanskas.grigaliusSummary: We report on the recent construction of a scattering theory for Maxwell potentials on curved spacetimes [\textit{J.-P. Nicolas} and \textit{G. Taujanskas}, ``Conformal scattering of Maxwell potentials'', Preprint, \url{arXiv:2211.14579}].
For the entire collection see [Zbl 1537.35003].Spectral gap for obstacle scattering in dimension 2https://zbmath.org/1541.353302024-09-27T17:47:02.548271Z"Vacossin, Lucas"https://zbmath.org/authors/?q=ai:vacossin.lucasSummary: We study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a noneclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an \textit{open hyperbolic quantum map}, achieved by \textit{S. Nonnenmacher} et al. [Ann. Math. (2) 179, No. 1, 179--251 (2014; Zbl 1293.81022)]. In fact, we obtain a spectral gap for this type of object, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of \textit{S. Dyatlov} et al. [J. Am. Math. Soc. 35, No. 2, 361--465 (2022; Zbl 1487.58032)] to apply this fractal uncertainty principle in our context.Solvability of some Fredholm integro-differential equations with mixed diffusion in a squarehttps://zbmath.org/1541.353312024-09-27T17:47:02.548271Z"Efendiev, Messoud"https://zbmath.org/authors/?q=ai:efendiev.messoud"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitaliSummary: We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems in a square in two dimensions with periodic boundary conditions. They contain the normal diffusion in one direction and the superdiffusion in the other direction. We work in a constrained subspace of \(H^2\) using the fixed point technique. The elliptic equation involves a second order differential operator satisfying the Fredholm property. It is established that, under reasonable technical assumptions, the convergence in the appropriate function spaces of the integral kernels yields the existence and convergence in \(H_0^2\) of the solutions. We generalize the results obtained in our preceding work [J. Differ. Equations 284, 83--101 (2021; Zbl 1510.45009)] for the analogous equation considered in the whole \(\mathbb{R}^2\) which contained a non-Fredholm operator.Hydrodynamic traffic flow models including random accidents: a kinetic derivationhttps://zbmath.org/1541.353322024-09-27T17:47:02.548271Z"Chiarello, Felisia Angela"https://zbmath.org/authors/?q=ai:chiarello.felisia-angela"Göttlich, Simone"https://zbmath.org/authors/?q=ai:gottlich.simone"Schillinger, Thomas"https://zbmath.org/authors/?q=ai:schillinger.thomas"Tosin, Andrea"https://zbmath.org/authors/?q=ai:tosin.andreaSummary: We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentialshttps://zbmath.org/1541.353332024-09-27T17:47:02.548271Z"Kepka, Bernhard"https://zbmath.org/authors/?q=ai:kepka.bernhardSummary: We consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. More precisely, we study the case in which the shear is dominant compared with the dilation and the collision operator has homogeneity \(\gamma >0\). We prove that solutions with initially high temperature remain close and converge to a Maxwellian distribution with temperature going to infinity. Furthermore, we give precise asymptotic formulas for the temperature. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both noncutoff and cutoff kernels.Incompressible Navier-Stokes-Fourier limit of 3D stationary Boltzmann equationhttps://zbmath.org/1541.353342024-09-27T17:47:02.548271Z"Wu, Lei"https://zbmath.org/authors/?q=ai:wu.lei.1"Ouyang, Zhimeng"https://zbmath.org/authors/?q=ai:ouyang.zhimengSummary: We consider the 3D stationary Boltzmann equation in convex domains with diffuse-reflection boundary condition. We rigorously derive the steady incompressible Navier-Stokes-Fourier system and justify the asymptotic convergence as the Knudsen number \({\varepsilon}\) shrinks to zero. The proof is based on an intricate analysis of boundary layers with geometric correction and focuses on technical difficulties caused by the singularity in collision kernel \(k(v,v')\) and the perturbed remainder estimates.The global strong solutions of the 3D incompressible Hall-MHD system with variable densityhttps://zbmath.org/1541.353352024-09-27T17:47:02.548271Z"An, Shu"https://zbmath.org/authors/?q=ai:an.shu"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.8|chen.jing.10|chen.jing.7|chen.jing.11|chen.jing.3|chen.jing.16|chen.jing.9|chen.jing.5|chen.jing|chen.jing.12|chen.jing.4|chen.jing.2"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin.1Summary: In this paper, we focus on the well-posedness problem of the three-dimensional incompressible viscous and resistive Hall-magnetohydrodynamics system (Hall-MHD) with variable density. We mainly prove the existence and uniqueness issues of the density-dependent incompressible Hall-magnetohydrodynamic system in critical spaces on \(\mathbb{R}^3\).The 3-D time-dependent Navier-Stokes equations on multi-connected domains with inhomogeneous boundary conditions and spectral hyperviscosityhttps://zbmath.org/1541.353362024-09-27T17:47:02.548271Z"Avrin, Joel"https://zbmath.org/authors/?q=ai:avrin.joel-dSummary: We consider the time-dependent 3-D Navier-Stokes equations (NSE) on a multi-connected bounded domain \(\Omega \subset \mathbb{R}^3\) with inhomogeneous boundary data \(\beta \in H^{1/2}(\Gamma)\) on \(\partial \Omega =\Gamma\), where \(\Gamma\) is a union of Lipschitz continuous surfaces \(\Gamma_0, \Gamma_1, \dots, \Gamma_l\). This assumption includes the particular case when the \(\Gamma_i\) are disjoint, the stationary version of which is classically known as Leray's problem. Existence results for Leray's problem have either assumed flux conditions beyond the general flux condition necessitated by compatibility constraints, or required size restrictions on the data. Here we incorporate a spectral hyperviscosity term in the time-dependent case and obtain existence and foundational results, assuming only the general flux condition and without imposing size restrictions on the boundary data \(\beta\). For any such \(\beta \in H^{1/2}(\Gamma)\) we establish global existence and uniqueness of mild solutions. Then on any interval \([0, T]\) on which these solutions and the corresponding NSE solution share a common \(H^1\)-bound (as is present on local intervals of existence of strong solutions, in certain special cases, or as is commonly assumed in achieving strong convergence results in numerical studies) we show for slightly more regular \(\beta\) that the spectrally-hyperviscous solutions converge strongly and uniformly in \(H^1 (\Omega)\) to the NSE solution as the spectral hyperviscosity term vanishes in the limit of key parameters. To achieve this robust sense of approximation of the NSE system, an involved setup and specially-adapted semigroup techniques assume essential roles, and the exposition is new for the case \(\beta =0\) as well. Our final results adapt the NSE reformulation in [\textit{J.-G. Liu} et al., J. Comput. Phys. 229, No. 9, 3428--3453 (2010; Zbl 1307.76029)] to recast our approximating system in a form potentially more adaptable to computation.Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equationshttps://zbmath.org/1541.353372024-09-27T17:47:02.548271Z"Barker, Blake"https://zbmath.org/authors/?q=ai:barker.blake"Melinand, Benjamin"https://zbmath.org/authors/?q=ai:melinand.benjamin"Zumbrun, Kevin"https://zbmath.org/authors/?q=ai:zumbrun.kevin-rThe 1D compressible nonsteady Navier-Stokes equations are considered, with noncharacteristic inflow-outflow boundary conditions. The existence, uniqueness, and stability of steady solutions of the full (nonisentropic) above system on a bounded interval are studied, for large-amplitude data. The spectral stability is studied by using numerical Evans function investigations -- see [\textit{B. Barker} et al., SIAM J. Appl. Dyn. Syst. 17, No. 2, 1766--1785 (2018; Zbl 1395.65131)]. Important tools are the Morse index of the linearized operator about the wave, the Brouwer degree and a ``Cauchy-to-boundary value'' map \(\Psi\). Roughly speaking, \(\Psi\) realizes the correspondence between the conditions of the problem and the solution -- see (1.7), (1.8). The properties of \(\Psi\) and the existence theorem are given in Sections 2--3. The uniqueness is proved in Proposition 4.1: if \(\Psi\) is bijective for some particular data, then we have uniqueness; otherwise ``there is at least one choice of data possessing multiple solutions''. I would say that, to some extent, the conclusion seems to be contained in the hypothesis (which is very strong). The spectral stability and the Evans function are explained, by using the stability index. Numerical results for simple gases are given in Section 6. The computation of the Evans function is obtained with the package STABLAB. Very interesting plots illustrate the obtained results. Some very useful numerical and analytical examples concerning the non uniqueness cases are given in the last section, where an abstract bifurcation result is also obtained. Details regarding the MATLAB-based package STABLAB (used here) are given in the appendix -- see [\textit{B. Barker} et al., Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376, No. 2117, Article ID 20170184, 25 p. (2018; Zbl 1402.65052)].
Reviewer: Gelu Paşa (Bucureşti)Low Mach number limit on perforated domains for the evolutionary Navier-Stokes-Fourier systemhttps://zbmath.org/1541.353382024-09-27T17:47:02.548271Z"Basarić, Danica"https://zbmath.org/authors/?q=ai:basaric.danica"Chaudhuri, Nilasis"https://zbmath.org/authors/?q=ai:chaudhuri.nilasisSummary: We consider the Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat-conducting fluid on a domain perforated by tiny holes. First, we identify a class of dissipative solutions to the Oberbeck-Boussinesq approximation as a low Mach number limit of the primitive system. Secondly, by proving the weak-strong uniqueness principle, we obtain strong convergence to the target system on the lifespan of the strong solution.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Onset of nonlinear instabilities in monotonic viscous boundary layershttps://zbmath.org/1541.353392024-09-27T17:47:02.548271Z"Bian, D."https://zbmath.org/authors/?q=ai:bian.dongfen|bian.dongping|bian.da|bian.desong"Grenier, E."https://zbmath.org/authors/?q=ai:grenier.edouard|grenier.emmanuelSummary: In this paper, we study the nonlinear stability of a shear layer profile for Navier-Stokes equations near a boundary. More precisely, we investigate the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile, we obtain that the nonlinearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude \(O(\nu^{1/4})\) only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.Enhanced and unenhanced dampings of the Kolmogorov flowhttps://zbmath.org/1541.353402024-09-27T17:47:02.548271Z"Chen, Zhi-Min"https://zbmath.org/authors/?q=ai:chen.zhiminSummary: The Kolmogorov flow represents the stationary sinusoidal solution \((\sin y, 0)\) to a two-dimensional spatially periodic Navier-Stokes system, driven by an external force. This system admits the additional non-stationary solution \((\sin y, 0) + e^{- \nu t}(\sin y, 0)\), which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity \(\nu\). Enhanced damping or enhanced dissipation of the problem is obtained by presenting higher decay rate for the difference between a solution and the non-stationary basic solution. Moreover, for the understanding of the metastability problem in an explicit manner, a variety of exact solutions are presented to show enhanced and unenhanced dampings.Some new properties of a suitable weak solution to the Navier-Stokes equationshttps://zbmath.org/1541.353412024-09-27T17:47:02.548271Z"Crispo, Francesca"https://zbmath.org/authors/?q=ai:crispo.francesca"Maremonti, Paolo"https://zbmath.org/authors/?q=ai:maremonti.paolo"Grisanti, Carlo Romano"https://zbmath.org/authors/?q=ai:grisanti.carlo-romanoThe authors consider the problem of energy conservation for suitable weak solutions of the Navier-Stokes equation. The existence of weak solutions satisfying an energy inequality goes back to the work of \textit{J. Leray} [Acta Math. 63, 193--248 (1934; JFM 60.0726.05)] in 1934, and it is well known that smooth solutions satisfy an energy equality. The gap between the energy inequality and the energy equality remains a very persistent problem, and has seen a number of recent negative results in the convex integration framework.
In this paper, the authors estimate the deficit in the energy inequality -- that is how far a suitable weak solution is from satisfying the energy -- by giving a specific quantity \(H(s,t)\) that is defined in terms of the limit of solutions of the mollified Navier-Stokes equation. They do this by making use of a new strong convergence result for solutions of the mollified Navier-Stokes equation to suitable weak solutions. This represents an interesting new approach, because it allows the deficit in the energy inequality to be expressed as a limit of quantities coming from a globally well posed problem.
For the entire collection see [Zbl 1467.76003].
Reviewer: Evan Miller (Hamilton)Full compressible Navier-Stokes equations with the Robin boundary condition on temperaturehttps://zbmath.org/1541.353422024-09-27T17:47:02.548271Z"Dong, Wenchao"https://zbmath.org/authors/?q=ai:dong.wenchao(no abstract)Global solutions to the rotating Navier-Stokes equations with large data in the critical Fourier-Besov spaceshttps://zbmath.org/1541.353432024-09-27T17:47:02.548271Z"Fujii, Mikihiro"https://zbmath.org/authors/?q=ai:fujii.mikihiroSummary: We consider the initial value problem for the 3D incompressible Navier-Stokes equations with the Coriolis force. The aim of this paper is to prove the existence of a unique global solution with \textit{arbitrarily large} initial data in the scaling critical Fourier-Besov spaces \(\widehat{\dot{B}}_{p, \sigma}^{\frac{3}{p}-1} (\mathbb{R}^3)^3\) (\(2 \leqslant p < 4\), \(1 \leqslant \sigma < \infty\)), provided that the size of the Coriolis parameter is sufficiently large. Moreover, if the initial data additionally belong to the scaling sub-critical spaces, we obtain an explicit relationship between the initial data and the Coriolis force, which ensures the existence of a unique global solution.
{\copyright} 2023 Wiley-VCH GmbH.Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficienthttps://zbmath.org/1541.353442024-09-27T17:47:02.548271Z"He, Zihui"https://zbmath.org/authors/?q=ai:he.zihui"Liao, Xian"https://zbmath.org/authors/?q=ai:liao.xianSummary: We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function.
For some stationary symmetric flows, we reformulate the Navier-Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.Instantaneous unboundedness of the entropy and uniform positivity of the temperature for the compressible Navier-Stokes equations with fast decay densityhttps://zbmath.org/1541.353452024-09-27T17:47:02.548271Z"Li, Jinkai"https://zbmath.org/authors/?q=ai:li.jinkai"Xin, Zhouping"https://zbmath.org/authors/?q=ai:xin.zhoupingSummary: This paper concerns the physical behaviors of any solutions to the one-dimensional compressible Navier-Stokes equations for viscous and heat conductive gases with constant viscosities and heat conductivity for fast decaying density at far fields only. First, it is shown that the specific entropy becomes not uniformly bounded immediately after the initial time, as long as the initial density decays to vacuum at the far field at the rate not slower than \(O(\frac 1{|x|^{\ell_\rho}})\) with \(\ell_\rho > 2\). Furthermore, for faster decaying initial density, i.e., \(\ell_\rho \geq 4\), a sharper result is discovered that the absolute temperature becomes uniformly positive at each positive time, no matter whether it is uniformly positive or not initially, and consequently the corresponding entropy behaves as \(O(-\log (\varrho_0(x)))\) at each positive time, independent of the boundedness of the initial entropy. Such phenomena are in sharp contrast to the case with slowly decaying initial density of the rate no faster than \(O(\frac 1{x^2})\), for which our previous works
[Adv. Math. 361, Article ID 106923, 50 p. (2020; Zbl 1433.35236);
Commun. Pure Appl. Math. 75, No. 11, 2393--2445 (2022; Zbl 1510.35211);
Sci. China, Math. 66, No. 10, 2219--2242 (2023; Zbl 1525.35193)]
show that the uniform boundedness of the entropy can be propagated for all positive time and thus the temperature decays to zero at the far field. These give a complete answer to the problem concerning the propagation of uniform boundedness of the entropy for the heat conductive ideal gases and, in particular, show that the algebraic decay rate 2 of the initial density at the far field is sharp for the uniform boundedness of the entropy. The tools to prove our main results are based on some scaling transforms, including the Kelvin transform, and a Hopf type lemma for a class of degenerate equations with possible unbounded coefficients.Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinderhttps://zbmath.org/1541.353462024-09-27T17:47:02.548271Z"Liu, Jitao"https://zbmath.org/authors/?q=ai:liu.jitaoSummary: In this paper, we study the initial boundary value problem and vanishing viscosity limit for incompressible axisymmetric Navier-Stokes equations \textit{with swirls} in the exterior of a cylinder under Navier-slip boundary condition. In the first part, we prove the existence of a unique global solution with the axisymmetric initial data \(\boldsymbol{u}_0^\nu\in L_\sigma^2(\Omega)\) and axisymmetric force \(\boldsymbol{f}\in L^2([0,T];L^2(\Omega))\). This result improves the initial regularity condition on the global well-posedness result obtained by
\textit{K. Abe} and \textit{G. Seregin} [Proc. R. Soc. Edinb., Sect. A, Math. 150, No. 4, 1671--1698 (2020; Zbl 1446.35092)]
and extends their boundary condition. In the second part, we make the first attempt to investigate the inviscid limit of unforced viscous axisymmetric flows \textit{with swirls} and prove that the viscous axisymmetric flows with swirls converge to inviscid axisymmetric flows without swirls under the condition \(\|ru_{0 \theta}^\nu\|_{L^2(\Omega)}=\mathcal{O}(\sqrt{\nu})\). Some new uniform estimates, independent of the viscosity, are obtained here. The second result can be thought as a follow-up work to the previous work by
\textit{K. Abe} [J. Math. Pures Appl. (9) 137, 1--32 (2020; Zbl 1437.35562)],
where the inviscid limit for the same equations \textit{without swirls} in an infinite cylinder was studied.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On the hydrostatic approximation of Navier-Stokes-Maxwell system with Gevrey datahttps://zbmath.org/1541.353472024-09-27T17:47:02.548271Z"Liu, Ning"https://zbmath.org/authors/?q=ai:liu.ning"Paicu, Marius"https://zbmath.org/authors/?q=ai:paicu.marius"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.3Summary: In this paper, we prove the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a 2-D striped domain with initial data around some nonzero background magnetic field in Gevrey-2 class. Then we rigorously justify the limit from the scaled anisotropic equations to the associated hydrostatic system and provide with the precise convergence rate. Finally, with small initial data in Gevrey-\(\frac{3}{2}\) class, we also extend the lifespan of thus obtained solutions to a longer time interval.The optimal \(L^2\) decay rate of the velocity for the general FENE dumbbell modelhttps://zbmath.org/1541.353482024-09-27T17:47:02.548271Z"Luo, Zhaonan"https://zbmath.org/authors/?q=ai:luo.zhaonan"Luo, Wei"https://zbmath.org/authors/?q=ai:luo.wei.2"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: In this paper we mainly study large time behavior for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. The sharp \(L^2\) decay rate was obtained on the co-rotational case. We prove that the optimal \(L^2\) decay rate of the velocity of the general FENE dumbbell model is \((1+t)^{-\frac{d}{4}}\) with \(d \geq 2\). Our obtained result is sharp and improves considerably the previous result in [\textit{W. Luo} and \textit{Z. Yin}, Arch. Ration. Mech. Anal. 224, No. 1, 209--231 (2017; Zbl 1366.35134)].On unique solvability of the time-periodic problem for the Navier-Stokes equationhttps://zbmath.org/1541.353492024-09-27T17:47:02.548271Z"Nakatsuka, Tomoyuki"https://zbmath.org/authors/?q=ai:nakatsuka.tomoyukiSummary: We consider the existence and uniqueness of time-periodic solutions to the Navier-Stokes equation in the whole space. We decompose periodic solutions into steady and purely periodic parts, and we analyze the equations they should satisfy. Based on the analysis of the purely periodic solutions represented by the Fourier transform to the Stokes equation, their additional regularity in time can be obtained and we use it to construct a time-periodic solution of the Navier-Stokes equation. Furthermore, we show that if the time-periodic solution is sufficiently small in an appropriate sense, then the Navier-Stokes equation admits no other solution in the same class.Numerical study on how advection delays and removes singularity formation in the Navier-Stokes equationshttps://zbmath.org/1541.353502024-09-27T17:47:02.548271Z"Ohkitani, Koji"https://zbmath.org/authors/?q=ai:ohkitani.kojiSummary: We numerically study a distorted version of the Euler and Navier-Stokes equations, which are obtained by depleting the advection term systematically. It is known that in the inviscid case some solutions blow up in finite time when advection is totally discarded, [\textit{P. Constantin}, Commun. Math. Phys. 104, 311--326 (1986; Zbl 0655.76041)]. Taking a pair of orthogonally offset vortex tubes and the Taylor-Green vortex as initial data, we show the following. (1) Blowup persists even with viscosity when advection is discarded, and (2) for small viscosity, the time of blowup increases logarithmically as we reinstate advection using a continuous parameter, which would be consistent with the regularity of the Navier-Stokes equations. A tiny mismatch in the coefficient of the advection term, as minute as parts per trillion, throws the system out of compactness and leads to blowup.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Asymptotic behavior of the 3D incompressible Navier-Stokes equations with dampinghttps://zbmath.org/1541.353512024-09-27T17:47:02.548271Z"Peng, Fuxian"https://zbmath.org/authors/?q=ai:peng.fuxian"Jin, Xueting"https://zbmath.org/authors/?q=ai:jin.xueting"Yu, Huan"https://zbmath.org/authors/?q=ai:yu.huanSummary: In this paper, we consider the 3D incompressible Navier-Stokes equations with damping term \(| u |^{\beta - 1} u\) (\(\beta \geq 1\)). First, by using a different and simple method from
\textit{X. Cai} and \textit{L. Lei} [Acta Math. Sci., Ser. B, Engl. Ed. 30, No. 4, 1235--1248 (2010; Zbl 1240.35379)],
\textit{Y. Jia} et al. [Nonlinear Anal., Real World Appl. 12, No. 3, 1736--1747 (2011; Zbl 1216.35088)],
\textit{Z. Jiang} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 13, 5002--5009 (2012; Zbl 1391.76093)] and
\textit{H. Yu} and \textit{X. Zheng} [J. Math. Anal. Appl. 477, No. 2, 1009--1018 (2019; Zbl 1421.35257)],
for any \(\beta \geq 1\), we prove that the weak solutions decay to zero in \(L^2\) as time tends to infinity; for any \(\beta \geq 3\), we derive optimal decay rates of the \(L^2\)-norm of the solutions. Second, we obtain the decay rate with some appropriate space weighted estimates, which is the first result on the 3D damped Navier-Stokes equations to our knowledge.On uniqueness of mild \(L^{3, \infty}\)-solutions on the whole time axis to the Navier-Stokes equations in unbounded domainshttps://zbmath.org/1541.353522024-09-27T17:47:02.548271Z"Taniuchi, Yasushi"https://zbmath.org/authors/?q=ai:taniuchi.yasushiSummary: This paper is concerned with the uniqueness of bounded continuous \(L^{3,\infty}\)-solutions on the whole time axis \(\mathbb{R}\) or the half-line \((-\infty, T)\) to the Navier-Stokes equations in 3-dimensional unbounded domains. When \(\Omega\) is an unbounded domain, it is known that a small solution in \(BC(\mathbb{R}; L^{3,\infty})\) is unique within the class of solutions which have sufficiently small \(L^{\infty}(\mathbb{R}; L^{3,\infty})\)-norm; i.e., if two solutions \(u\) and \(v\) exist for the same force \(f\), \textit{both} \(u\) and \(v\) are small, then the two solutions coincide. There is another type of uniqueness theorem. \textit{R. Farwig} et al. [Commun. Partial Differ. Equations 40, No. 10, 1884--1904 (2015; Zbl 1334.35203)] showed that if two solutions \(u\) and \(v\) exist for the same force \(f\), \(u\) is small and if \(v\) has a precompact range \(\mathscr{R}(v) := \{v(t); -\infty<t<T\}\) in \(L^{3, \infty}\), then the two solutions coincide. However, there exist many solutions which do not have precompact range. In this paper, instead of the precompact range condition, by assuming some decay property of \(v(x, t)\) with respect to the spatial variable \(x\) near \(t=-\infty\), we show a modified version of the above-mentioned uniqueness theorem. As a by-product, in the \textit{half-space} \(\mathbb{R}^3_+\), we obtain a non-existence result of backward self-similar \(L^{3, \infty}\)-solutions sufficiently close to some homogeneous function \(Q(x/|x|)/|x|\) in a certain sense.Error estimates for finite element discretizations of the instationary Navier-Stokes equationshttps://zbmath.org/1541.353532024-09-27T17:47:02.548271Z"Vexler, Boris"https://zbmath.org/authors/?q=ai:vexler.boris"Wagner, Jakob"https://zbmath.org/authors/?q=ai:wagner.jakobSummary: In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the \(L^{\infty} (I; L^2 (\Omega)), L^2 (I; H^1 (\Omega))\) and \(L^2 (I; L^2 (\Omega))\) norms have been shown. The main result of the present work extends the error estimate in the \(L^{\infty} (I; L^2 (\Omega))\) norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the \(L^{\infty} (I; L^2 (\Omega))\) error estimate, also allow us to show best approximation type error estimates in the \(L^2 (I; H^1 (\Omega))\) and \(L^2 (I; L^2 (\Omega))\) norms, which complement this work.Non-uniqueness in law of three-dimensional Navier-Stokes equations diffused via a fractional Laplacian with power less than one halfhttps://zbmath.org/1541.353542024-09-27T17:47:02.548271Z"Yamazaki, Kazuo"https://zbmath.org/authors/?q=ai:yamazaki.kazuoSummary: Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has Hölder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hopf regularity. One of the novelties of this work is that in order to handle a transport error, we consider phase functions convected by not only a mollified velocity field but a sum of that with a mollified Ornstein-Uhlenbeck process if the noise is additive and a product of that with a mollified exponential Brownian motion if the noise is linearly multiplicative.Liouville-type theorems for the 3D stationary MHD equationshttps://zbmath.org/1541.353552024-09-27T17:47:02.548271Z"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.11|zhang.hui.40|zhang.hui.32|zhang.hui.9|zhang.hui.6|zhang.hui.14|zhang.hui.20|zhang.hui.4|zhang.hui.2|zhang.hui.15|zhang.hui.13|zhang.hui.12|zhang.hui.5|zhang.hui.8|zhang.hui.3|zhang.hui.21|zhang.hui|zhang.hui.30|zhang.hui.31"Zu, Qian"https://zbmath.org/authors/?q=ai:zu.qianSummary: In this paper, we consider the Liouville-type theorems for the 3D stationary incompressible MHD equations. Using the Caccioppoli type estimate, we proved the smooth solutions \((u, b)\) are identically equal to zero when \((u, b)\in L^p(\mathbb{R}^3)\), \(p\in(\frac{3}{2}, 3)\). In addition, under an additional assumption in the setting of the Sobolev space of negative order \(\dot{H}^{-1}(\mathbb{R}^3)\), we can extend the index \(p\in(3, +\infty)\). In fact, our results combine with the result of \textit{B. Yuan} and \textit{Y. Xiao} [J. Math. Anal. Appl. 491, No. 2, Article ID 124343, 9 p. (2020; Zbl 1450.35091)] that \(p\in[2, \frac{9}{2}]\), which implies a very intriguing and novel result for the 3D stationary MHD equations with \(p\in(\frac{3}{2}, +\infty)\).The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domainhttps://zbmath.org/1541.353562024-09-27T17:47:02.548271Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.69|zhang.jie.50|zhang.jie.45|zhang.jie.9|zhang.jie.1|zhang.jie.12|zhang.jie.37|zhang.jie.3|zhang.jie.6|zhang.jie.56|zhang.jie|zhang.jie.7|zhang.jie.4|zhang.jie.21|zhang.jie.13|zhang.jie.43|zhang.jie.53|zhang.jie.16|zhang.jie.8|zhang.jie.51|zhang.jie.14"Liu, Wenjun"https://zbmath.org/authors/?q=ai:liu.wenjun|liu.wenjun-jSummary: In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter \(\varepsilon\in(0,\infty)\), we consider the case that if the orders of the horizontal and vertical viscous coefficients \(\mu\) and \(\nu\) are \(\mu=O(1)\) and \(\nu=O(\varepsilon^\alpha)\), and the orders of magnetic diffusion coefficients \(\kappa\) and \(\sigma\) are \(\kappa=O(1)\) and \(\sigma=O(\varepsilon^\alpha)\), with \(\alpha>2\), then the limiting system is the PEHM as \(\varepsilon\) goes to zero. For \(H^1\)-initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as \(\varepsilon\) tends to zero. For \(H^1\)-initial data with additional regularity \((\partial_z \tilde{A}_0,\partial_z \tilde{B}_0)\in L^p(\Omega)(2<p<\infty)\), we slightly improve the well-posed result in
[\textit{C. Cao} et al., J. Funct. Anal. 272, No. 11, 4606--4641 (2017; Zbl 1366.35123)]
to extend the local-in-time strong convergences to the global-in-time one. For \(H^2\)-initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as \(\varepsilon\) goes to zero. Moreover, the rate of convergence is of the order \(O(\varepsilon^{\gamma/2})\), where \(\gamma=\min\{2,\alpha-2\}\) with \(\alpha\in(2,\infty)\). It should be noted that in contrast to the case \(\alpha>2\), the case \(\alpha=2\) has been investigated by
\textit{L. Du} and \textit{D. Li} [``The primitive equations with magnetic field approximation of the 3D MHD equations'', Preprint, \url{arXiv:2208.01985}],
in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order \(O(\varepsilon)\).
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Stability of contact lines in 2D stationary Bénard convectionhttps://zbmath.org/1541.353572024-09-27T17:47:02.548271Z"Zheng, Yunrui"https://zbmath.org/authors/?q=ai:zheng.yunruiSummary: We consider the evolution of contact lines for thermal convection of viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Here we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the elliptic analysis near corners and \textit{a priori} estimates of a geometric formulation of the Boussinesq equations.The incompressible \(\alpha\)-Euler equations in the exterior of a vanishing diskhttps://zbmath.org/1541.353582024-09-27T17:47:02.548271Z"Busuioc, Adriana Valentina"https://zbmath.org/authors/?q=ai:busuioc.adriana-valentina"Iftimie, Dragos"https://zbmath.org/authors/?q=ai:iftimie.dragos"Filho, Lopes Milton C."https://zbmath.org/authors/?q=ai:filho.lopes-milton-c"Nussenzveig Lopes, Helena Judith"https://zbmath.org/authors/?q=ai:nussenzveig-lopes.helena-jSummary: In this article we consider the \(\alpha \)-Euler equations in the exterior of a small fixed disk of radius \(\varepsilon \). We assume that the initial potential vorticity is compactly supported and independent of \(\varepsilon \), and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on \(\varepsilon \). We prove that the solution of this problem converges, as \(\varepsilon\to 0\), to the solution of a modified \(\alpha \)-Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity.On stability and instability of \(C^{1, \alpha}\) singular solutions to the 3D Euler and 2D Boussinesq equationshttps://zbmath.org/1541.353592024-09-27T17:47:02.548271Z"Chen, Jiajie"https://zbmath.org/authors/?q=ai:chen.jiajie"Hou, Thomas Y."https://zbmath.org/authors/?q=ai:hou.thomas-yizhaoSummary: Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging
[\textit{A. J. Majda} and \textit{A. L. Bertozzi}, Vorticity and incompressible flow. Cambridge: Cambridge University Press (2002; Zbl 0983.76001);
\textit{J. D. Gibbon}, Physica D 237, No. 14--17, 1894--1904 (2008; Zbl 1143.76389);
\textit{A. Kiselev}, in: Proceedings of the international congress of mathematicians 2018, ICM 2018. Volume III. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 2363--2390 (2018; Zbl 1448.35398);
\textit{T. D. Drivas} and \textit{T. M. Elgindi}, EMS Surv. Math. Sci. 10, No. 1, 1--100 (2023; Zbl 1532.35359);
\textit{P. Constantin}, Bull. Am. Math. Soc., New Ser. 44, No. 4, 603--621 (2007; Zbl 1132.76009)].
In
[Ann. Math. (2) 194, No. 3, 647--727 (2021; Zbl 1492.35199)]
(see also [\textit{T. M. Elgindi} et al., ``On the stability of self-similar blow-up for \(C^{1,\alpha}\) solutions to the incompressible Euler equations on \(\mathbb{R}^3\)'', Preprint, \url{arXiv:1910.14071}]),
\textit{T. Elgindi} proved that the 3D axisymmetric Euler equations with no swirl and \(C^{1, \alpha}\) initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with \(C^{1, \alpha}\) initial velocity and boundary develop a stable asymptotically self-similar (or approximately self-similar) finite time singularity
[\textit{J. Chen} and \textit{T. Y. Hou}, Commun. Math. Phys. 383, No. 3, 1559--1667 (2021; Zbl 1485.35071)]
in the same setting as the Hou-Luo blowup scenario
[\textit{G. Luo} and \textit{T. Y. Hou}, Proc. Natl. Acad. Sci. USA 111, No. 36, 12968--12973 (2014; Zbl 1431.35115);
Multiscale Model. Simul. 12, No. 4, 1722--1776 (2014; Zbl 1316.35235)].
On the other hand, the authors of
[\textit{A. F. Vasseur} and \textit{M. Vishik}, Commun. Math. Phys. 378, No. 1, 557--568 (2020; Zbl 1446.35114)] and
[\textit{L. Lafleche} et al., J. Math. Pures Appl. (9) 155, 140--154 (2021; Zbl 1484.76016)]
recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in
[Vasseur and Vishik, loc. cit.] and
[Lafleche et al., loc. cit.]
require some strong regularity assumption on the initial data, which is not satisfied by the \(C^{1, \alpha}\) velocity field. In this paper, we generalize the analysis of
Elgindi [loc. cit.],
Chen and Hou [loc. cit.],
Vasseur and Vishik [loc. cit.] and
Lafleche et al. [loc. cit.] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with \(C^{1, \alpha}\) velocity are unstable under the notion of stability introduced in
Vasseur and Vishik [loc. cit.] and
Lafleche et al. [loc. cit.].
These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions.On the local well-posedness for the relativistic Euler equations for a liquid bodyhttps://zbmath.org/1541.353602024-09-27T17:47:02.548271Z"Ginsberg, Daniel"https://zbmath.org/authors/?q=ai:ginsberg.daniel"Lindblad, Hans"https://zbmath.org/authors/?q=ai:lindblad.hansThe authors consider the motion of a perfect fluid in the 4D globally hyperbolic spacetime \((M,g)\), described by Einstein's equations: \(R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R=T_{\mu \nu }\), assuming units such that the speed of light is equal to one, where \(R_{\mu \nu }\) is the Ricci curvature of the Lorentzian metric \(g\), \(R=g_{\mu \nu }R_{\mu \nu }\) the scalar curvature, \(T\) the energy-momentum tensor of a perfect fluid: \(T_{\mu \nu }=(\rho +p)u_{\mu }u_{\nu }+pg_{\mu \nu }\), \(u=u^{\mu }\partial _{\mu }\) being the fluid velocity, \(g(u,u)=-1\) and \(g(u,\tau )<0\), where \(\tau \) is the future-directed timelike vector defining the time axis in \((M,g)\), \(\rho \geq 0\) being the energy density of matter and \(p\geq 0\) the pressure. Introducing the rescaled fluid velocity \(v_{\mu }=\sqrt{\sigma }u_{\mu }\), the authors obtain the system \(v^{\nu }\nabla _{\nu }v+\frac{1}{2}\nabla _{\mu }\sigma =0\), \(v^{\nu }\nabla _{\nu }e(\sigma )+\nabla _{\mu }v^{\mu }=0 \), in the domain \(\mathcal{D}_{t}\) occupied by the fluid at time \(t\), with \( e(\sigma )=log(n(\sigma )/\sqrt{\sigma })\), where \(\sigma =\frac{p+\rho }{n}\) is the enthalpy, \(n(\sigma )\) being obtained by inverting the previous relation after expressing \(p=P(n)\), \(\rho =\rho (n)\). The initial conditions \(p=0\), on \(\partial \mathcal{D}_{t}\), \(g(\mathcal{N},v)=0\), on \(\Lambda =\cup _{0\leq t\leq T}\partial \mathcal{D}_{t}\), are imposed, where \( \mathcal{N}\) is the outward-pointing unit normal vector field to \(\Lambda \) . The first main result of the paper proves that the preceding problem, with initial data \(u\mid _{t=0}=\overset{o}{u}\), \(\rho \mid _{t=0}=\overset{o}{ \rho }\) has a unique solution \(u^{\mu }(t)\in H^{r}(\mathcal{D}_{t})\), \( 0\leq \mu \leq 3\), \(\rho (t)\in H^{r}(\mathcal{D}_{t})\) with \(\rho =E(n)\), \( p=P(n)\) for \(t\leq T_{0}\) for some \(0<T_{0}\leq T\). Here the authors chose \( r\geq 10\), a globally hyperbolic spacetime \((M\times \lbrack 0,T],g)\), a subset \(\mathcal{D}_{0}\subset M\times \{t=0\}\) which is diffeomorphic to the Euclidean unit ball, invertible functions \(P,E\in C^{\infty }(\mathbb{R} _{\geq 0};\mathbb{R}_{\geq 0})\) so that the sound speed \(\eta \) defined through \(\eta ^{2}=\frac{d}{d\rho }P(\rho )\) satisfies \(\eta ^{2}\leq 1\) and \(\eta ^{2}\geq 1-\delta \) for \(\delta \) sufficiently small, and initial data \((\overset{o}{u},\overset{o}{\rho })\) with \(\sum_{\mu =}^{3}\left\Vert \overset{o}{u}^{\mu }\right\Vert _{H^{r}(\mathcal{D}_{0})}+\left\Vert \overset{o}{\rho }\right\Vert _{H^{r}(\mathcal{D}_{0})}<\infty \), \(S\in \mathcal{S}\), \(\overset{o}{\rho }\geq \rho _{1}>0\), and which satisfies compatibility conditions. For the proof, the authors first introduce the energy \(\mathcal{E}_{0}(t)=\mathcal{E}_{0}(t)+\int_{0}^{t}\int_{\mathcal{D} _{t}}T_{\mu \nu }\mathcal{L}_{\tau }g^{\mu \nu }\) where \(\mathcal{E} _{0}(0)=\int_{\mathcal{D}_{t}}\rho u_{\tau }^{2}+pg(\overline{u},\overline{u} )\), where \(\mathcal{L}_{\tau }g\) is the Lie derivative of \(g\) with respect to \(\tau \). They prove a higher-order version of the preceding energy identity, introducing Lagrangian coordinates which fix the boundary. They use curl and divergence estimates. They introduce a neighboring problem in the Newtonian case \((\partial _{t}+v^{k}\partial _{k})v_{i}+\delta ^{ij}\partial _{j}h=f^{i}\), \((\partial _{t}+v^{k}\partial _{k})e(h)+\partial _{i}v^{i}=g\), in \(\mathcal{D}_{t}\), where \(i=1,2,3\), \(f,g\) are given functions, \(h\) denotes the Newtonian enthalpy, defined through the equation of state \(p=p(\rho )\), by \(\rho h^{\prime }(\rho )=p(\rho )\), \(\rho \) now being the mass density, and \(e(h)=log\rho (h)\). They prove a well-posedness result for this neighboring problem, considering the equations of motion of a compressible barotropic fluid, \(\rho (\partial _{t}+v^{j}\partial _{j})v_{i}+\partial _{i}p=0\), \((\partial _{t}+v^{j}\partial _{j})\rho +\rho divv=0\), in \(\mathcal{D}_{t}\), where \(p=P(\rho )\) for a given function \(P\) with \(P(0)>0\), subject to the boundary conditions \(p=0\), \( n_{t}+v^{j}n_{j}=0\), on \(\partial \mathcal{D}_{t}\), that they rewrite in terms of the enthalpy defined by \(h(\rho )=P(\rho )/\rho \), and they prove an existence result for this final system in Lagrangian coordinates. They prove uniform energy estimates for a smoothed problem. Coming back to the original problem, the authors introduce a regularization and define the tangentially regularized velocity and they write the associated problem. They build the corresponding framework, they propose higher-order equations and energies for the velocity vector field and they prove uniform estimates for the enthalpy and divergence and curl estimates for the relativistic velocity and coordinates. Gathering all these results, they can conclude the proof of the main results. The proofs of the different results are given with complete details and the paper ends with six appendices.
Reviewer: Alain Brillard (Riedisheim)The time-asymptotic expansion for the compressible Euler equations with dampinghttps://zbmath.org/1541.353612024-09-27T17:47:02.548271Z"Huang, Feimin"https://zbmath.org/authors/?q=ai:huang.feimin"Wu, Xiaochun"https://zbmath.org/authors/?q=ai:wu.xiaochunSummary: In 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave \((\bar{v}, \bar{u})\) of the porous media equation. Geng et al. proposed a time-asymptotic expansion around the diffusion wave \((\bar{v}, \bar{u})\), which is a better asymptotic profile than \((\bar{v}, \bar{u})\). In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time-asymptotic expansion.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Singularity formation for the cylindrically symmetric rotating relativistic Euler equations of Chaplygin gaseshttps://zbmath.org/1541.353622024-09-27T17:47:02.548271Z"Hu, Yanbo"https://zbmath.org/authors/?q=ai:hu.yanbo"Guo, Houbin"https://zbmath.org/authors/?q=ai:guo.houbinSummary: This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}A new high-order Camassa-Holm-type equation for shallow water waves moving over a shear flowhttps://zbmath.org/1541.353632024-09-27T17:47:02.548271Z"Wang, Yunbo"https://zbmath.org/authors/?q=ai:wang.yunbo"Kang, Jing"https://zbmath.org/authors/?q=ai:kang.jing"Fan, Ying"https://zbmath.org/authors/?q=ai:fan.yingSummary: We derive a new quasilinear nonlocal shallow water equation of Camassa-Holm type with higher-order nonlinearities from the governing equations for large-amplitude gravity water waves, which incorporates an ambient underlying linear shear flow and can be recognized as the higher-order generalization of the compressible hyper-elastic rod model in the material science. The mathematical modeling is implemented for the rotational water flow in the general form compared with the known models. Moreover, the local well-posedness in Besov spaces of the corresponding Cauchy problem is studied.Qualitative analysis of smooth solution for the Euler equations of Chaplygin gashttps://zbmath.org/1541.353642024-09-27T17:47:02.548271Z"Wu, Xinglong"https://zbmath.org/authors/?q=ai:wu.xinglong"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.bolingSummary: This manuscript is devoted to studying the blow-up phenomena and instability of the smooth solution for the isentropic Chaplygin gas equations in \(\mathbb{R}^N\) for any dimension \(N \geq 1\). We first give two blow-up phenomena of the Chaplygin gas equations, if the initial data satisfy some conditions (compact support or spherical symmetry). Next, the dynamics and instability of a family of solutions for the Cauchy problem of equation (1.5) is investigated, if the velocity \(u = c (t) r\).Formation of singularities in plasma ion dynamicshttps://zbmath.org/1541.353652024-09-27T17:47:02.548271Z"Bae, Junsik"https://zbmath.org/authors/?q=ai:bae.junsik"Choi, Junho"https://zbmath.org/authors/?q=ai:choi.junho"Kwon, Bongsuk"https://zbmath.org/authors/?q=ai:kwon.bongsukSummary: We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for \(C^1\) blow-up of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the pressureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave \(C^1\) class in a finite time when the gradients of the Riemann functions are initially large.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Equilibrium configurations of a symmetric body immersed in a stationary Navier-Stokes flow in a planar channelhttps://zbmath.org/1541.353662024-09-27T17:47:02.548271Z"Berchio, Elvise"https://zbmath.org/authors/?q=ai:berchio.elvise"Bonheure, Denis"https://zbmath.org/authors/?q=ai:bonheure.denis"Galdi, Giovanni P."https://zbmath.org/authors/?q=ai:galdi.giovanni-paolo"Gazzola, Filippo"https://zbmath.org/authors/?q=ai:gazzola.filippo"Perotto, Simona"https://zbmath.org/authors/?q=ai:perotto.simonaSummary: We study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier-Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time.Analysis of a two phase flow model of biofilm spreadhttps://zbmath.org/1541.353672024-09-27T17:47:02.548271Z"Carpio, Ana"https://zbmath.org/authors/?q=ai:carpio.ana"Duro, Gema"https://zbmath.org/authors/?q=ai:duro.gemaSummary: We consider a quasi-stationary problem describing the status of velocities, pressures and chemicals affecting cell behavior within a biofilm. The model couples stationary transport equations and compressible Stokes systems with convection-reaction-diffusion equations. We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the quasi-stationary system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.Remarks on the smoothness of the \(C^{1,\alpha}\) asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equationshttps://zbmath.org/1541.353682024-09-27T17:47:02.548271Z"Chen, Jiajie"https://zbmath.org/authors/?q=ai:chen.jiajie|chen.jiajie.1|chen.jiajie.2Summary: We show that the constructions of \(C^{1,\alpha}\) asymptotically self-similar singularities for the three-dimensional (3D) Euler equations by \textit{T. Elgindi} [Ann. Math. (2) 194, No. 3, 647--727 (2021; Zbl 1492.35199)], and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by \textit{J. Chen} and \textit{T. Y. Hou} [Commun. Math. Phys. 383, No. 3, 1559--1667 (2021; Zbl 1485.35071)] can be extended to construct singularity with velocity \(\mathbf{u}\in C^{1,\alpha}\) that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted Hölder estimates with weights vanishing near the singular point. Our results are inspired by the recent work of \textit{D. Córdoba} et al. [``Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2$'', Preprint, \url{arXiv:2308.12197}] that it is possible to construct a \(C^{1,\alpha}\) singularity for the 3D axisymmetric Euler equations without swirl and with velocity \(\mathbf{u}\in C^{\infty}(\mathbb{R}^3\backslash\{0\})\).
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Abundant variant wave patterns by coupled Boussinesq-Whitham-Broer-Kaup equationshttps://zbmath.org/1541.353692024-09-27T17:47:02.548271Z"Chen, Shuangqing"https://zbmath.org/authors/?q=ai:chen.shuangqing"Li, Minghao"https://zbmath.org/authors/?q=ai:li.minghao"Guan, Bing"https://zbmath.org/authors/?q=ai:guan.bing"Li, Yuchun"https://zbmath.org/authors/?q=ai:li.yuchun"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.14|wang.yong.8|wang.yong.16|wang.yong.25|wang.yong.13|wang.yong.11|wang.yong.2|wang.yong.55|wang.yong.1|wang.yong.7|wang.yong.17|wang.yong.18|wang.yong.35|wang.yong.3|wang.yong.27|wang.yong.5|wang.yong.10|wang.yong.41|wang.yong.15|wang.yong.32|wang.yong.6"Lin, Xiaoqiang"https://zbmath.org/authors/?q=ai:lin.xiaoqiang"Liu, Tianqing"https://zbmath.org/authors/?q=ai:liu.tianqingSummary: This paper focus on searching for abundant wave patterns to time-fractional coupled Boussinesq-Whitham-Broer-Kaup equations, which plays a crucial role in obtaining the novel insights for propagational dynamics of shallow water. Through the innovative application of complete discrimination system for polynomial method (CDSPM) and conformable fractional transformation, the whole types of solutions for the fractional equations presented in the existing literatures are obtained. In particular, we obtain the solitary wave type solutions and Jacobian elliptic functions type solutions, and the later are initially founded. The existence and applicability of the obtained solutions are proved by concrete examples. Finally, the nature of wave propagation and the fractional characteristics are intuitively revealed by graphical simulations.Optimal time decay of the compressible micropolar fluids with discontinuous initial data in 3Dhttps://zbmath.org/1541.353702024-09-27T17:47:02.548271Z"Chen, Songzhuang"https://zbmath.org/authors/?q=ai:chen.songzhuangSummary: The global existence of weak solutions to Cauchy problem for compressible micropolar fluids with discontinuous initial data has been established in [\textit{M. Chen} et al., Commun. Math. Sci. 13, No. 1, 225--247 (2015; Zbl 1309.35071)]. However, the optimal decay rates of these weak solutions remain an open problem. The current work is to show the optimal decay rates of these weak solutions in \(L^r\)-norm with \(2 < r \leq \infty\) and the optimal decay rates of the first order derivative of the velocity and micro-rotational velocity in \(L^2\)-norm which are the same as the rates of classical solutions in [\textit{Q. Liu} and \textit{P. Zhang}, J. Differ. Equations 260, No. 10, 7634--7661 (2016; Zbl 1341.35121)]. In this process, we combine electromagnetic and fluid decomposition, frequency domain decomposition, and construct a series of temporal energy functional to solve complex nonlinear problems with low regularity.Existence and degenerate regularity of trajectory statistical solution for the 3D incompressible non-Newtonian micropolar fluidshttps://zbmath.org/1541.353712024-09-27T17:47:02.548271Z"Chen, Xiaolin"https://zbmath.org/authors/?q=ai:chen.xiaolin"Yang, Hujun"https://zbmath.org/authors/?q=ai:yang.hujun"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaoling"Zhao, Caidi"https://zbmath.org/authors/?q=ai:zhao.caidiSummary: In this article, the authors investigate the existence and degenerate regularity of trajectory statistical solutions for the 3D incompressible non-Newtonian micropolar fluids in bounded domain. They first prove the existence of the trajectory attractor, and construct the trajectory statistical solution via the trajectory attractor and generalized Banach limit. Then they establish that the trajectory statistical solutions possess degenerate regularity of Lusin's type when the associated Grashof number is small enough.Quantitative derivation of a two-phase porous media system from the one-velocity Baer-Nunziato and Kapila systemshttps://zbmath.org/1541.353722024-09-27T17:47:02.548271Z"Crin-Barat, Timothée"https://zbmath.org/authors/?q=ai:crin-barat.timothee"Shou, Ling-Yun"https://zbmath.org/authors/?q=ai:shou.lingyun"Tan, Jin"https://zbmath.org/authors/?q=ai:tan.jinSummary: We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer-Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer-Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood-Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the \textit{overdamping phenomenon}, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several \textit{auxiliary unknowns} allowing us to recover crucial \(\mathcal{O}(\varepsilon)\) bounds.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd and the London Mathematical Society}Norm inflation for the viscous nonlinear wave equationhttps://zbmath.org/1541.353732024-09-27T17:47:02.548271Z"de Roubin, Pierre"https://zbmath.org/authors/?q=ai:de-roubin.pierre"Okamoto, Mamoru"https://zbmath.org/authors/?q=ai:okamoto.mamoruThe authors study the viscous wave equation exhibiting polynomial-type nonlinearity and on specific Sobolev spaces with negative index. Here, the index is defined from the degree of scaling invariance property which is satisfied by the equation. The main question is whether the problem is ill-posed (well-posed), as a function of the dimension of the space and the degree of the nonlinearity. Here, under conditions, it is proved that the existence of solutions that having a norm inflation with respect to the time. This reflecting the fact of ``ill-posedness'' of the problem.
The paper is well written and accessible to all mathematicians, even for PhD-students.
Reviewer: Philippe Briet (Toulon)Stability and exponential decay for the compressible viscous non-resistive MHD systemhttps://zbmath.org/1541.353742024-09-27T17:47:02.548271Z"Dong, Boqing"https://zbmath.org/authors/?q=ai:dong.boqing"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahong"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaopingSummary: How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in \(\mathbb{R}^3\) or \(\mathbb{T}^3\) is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in \(\mathbb{T}^3\) when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Propagation and energy of the dressed solitons in the Thomas-Fermi magnetoplasmahttps://zbmath.org/1541.353752024-09-27T17:47:02.548271Z"El-Monier, S. Y."https://zbmath.org/authors/?q=ai:el-monier.s-y"Atteya, A."https://zbmath.org/authors/?q=ai:atteya.aSummary: A theoretical investigation is presented for dust-acoustic (DA) waves in a collisionless Thomas-Fermi magnetoplasma. The plasma system consists of electrons, ions, and negatively charged dust grains, all existing in a quantizing magnetic field. The Korteweg-de Vries (KdV) and KdV type equations are derived by using the reductive perturbation method. The solutions of these evolved equations are obtained. The contribution of higher-order corrections to the DA is investigated. The electric field and the soliton energy were also derived. The K-dV and dressed soliton energies are depleted as the dust temperature and magnetic field increase. But they magnify as obliqueness increases. The present results are beneficial in understanding the waves propagating in Thomas-Fermi magnetoplasma that are applicable for high-intensity laser-solid matter interaction experiments and astrophysical compact objects such as white dwarfs.Long time well-posedness and full justification of a Whitham-Green-Naghdi systemhttps://zbmath.org/1541.353762024-09-27T17:47:02.548271Z"Emerald, Louis"https://zbmath.org/authors/?q=ai:emerald.louis"Paulsen, Martin Oen"https://zbmath.org/authors/?q=ai:paulsen.martin-oenSummary: We establish the full justification of a ``Whitham-Green-Naghdi'' system modeling the propagation of surface gravity waves with bathymetry in the shallow water regime. It is an asymptotic model of the water waves equations with the same dispersion relation. The model under study is a nonlocal quasilinear symmetrizable hyperbolic system without surface tension. We prove the consistency of the general water waves equations with our system at the order of precision \(O( \mu^2(\varepsilon + \beta))\), where \(\mu\) is the shallow water parameter, \( \varepsilon\) the nonlinearity parameter, and \(\beta\) the topography parameter. Then we prove the long time well-posedness on a time scale \(O(\frac{1}{\max \{ \varepsilon , \beta \}})\). Lastly, we show the convergence of the solutions of the Whitham-Green-Naghdi system to the ones of the water waves equations on the later time scale.Lump and hybrid solutions for a (3+1)-dimensional Boussinesq-type equation for the gravity waves over a water surfacehttps://zbmath.org/1541.353772024-09-27T17:47:02.548271Z"Feng, Chun-Hui"https://zbmath.org/authors/?q=ai:feng.chunhui"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Yang, Dan-Yu"https://zbmath.org/authors/?q=ai:yang.danyu"Gao, Xiao-Tian"https://zbmath.org/authors/?q=ai:gao.xiao-tianSummary: In this paper, we investigate a (3+1)-dimensional Boussinesq-type equation, which can elucidate the gravity waves over a water surface: (1) Via the long-wave limit method, the \(M\) th-order lump solutions are constructed, where \(M\) is a positive integer. Hybrid solutions are also worked out, which are composed of the lumps and solitons/breathers. (2) Spatial structures of the first-, second- and third-order lumps are graphically discussed. (3) Interactions between/among the first-order lump and one soliton/two solitons/the first-order breather are graphically presented.Global weak solutions of the Serre-Green-Naghdi equations with surface tensionhttps://zbmath.org/1541.353782024-09-27T17:47:02.548271Z"Guelmame, Billel"https://zbmath.org/authors/?q=ai:guelmame.billelSummary: In this paper we consider the Serre-Green-Naghdi equations with surface tension. Smooth solutions of this system conserve an \(H^1\)-equivalent energy. We prove the existence of global weak dissipative solutions for any relatively small-energy initial data. We also prove that the Riemann invariants of the solutions satisfy a one-sided Oleinik inequality.Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the Benney-Luke equation using optimal system of Lie subalgebrashttps://zbmath.org/1541.353792024-09-27T17:47:02.548271Z"Hussain, A."https://zbmath.org/authors/?q=ai:hussain.akhtar"Usman, M."https://zbmath.org/authors/?q=ai:usman.muhammad.1|usman.mustofa|usman.murat|usman.muhammad-rashid|usman.muhammad|usman.mohammad|usman.mahamood"Zaman, F. D."https://zbmath.org/authors/?q=ai:zaman.fiazud-din"Ibrahim, T. F."https://zbmath.org/authors/?q=ai:ibrahim.tarek-fawzi"Dawood, A. A."https://zbmath.org/authors/?q=ai:dawood.a-aSummary: Nonlinear evolution equations (NLE-Es) arise in a variety of domains, including fluid dynamics, biological sciences, the study of solid states, optical fibres, coastal engineering, ocean physics, and nonlinear complex physical systems. This work examines the Lie symmetry study of the Benney-Luke (B-L) equation relying on two nonzero real parameters. The Lie infinitesimal generators, the one-dimensional optimal system, and the geometric vector fields are all obtained using the Lie symmetry technique. To start, we find close-form invariant solutions by employing symmetry reduction of Lie subalgebras. In some reduction cases, we transform the B-L equation into a variety of non-linear ordinary differential equations (NL-ODEs), which have the benefit of providing a significant number of closed-form solitary wave solutions. The traveling wave solutions contain special functional parameter solutions, trigonometric function solutions, rational function solutions, and hyperbolic trigonometric function solutions. The dynamical profile of closed-form wave solutions exhibits periodic solitons, dark peakon solitons, bright solitons, and bright solitons (bell shape) which we reveal in our study for the first time. By using Noether's method, we also determine the conservation laws. Finally, three-dimensional diagrams are used to reveal the dynamical investigation of some known solutions.On the existence, regularity and uniqueness of \(L^p\)-solutions to the steady-state 3D Boussinesq system in the whole space and with gravity accelerationhttps://zbmath.org/1541.353802024-09-27T17:47:02.548271Z"Jarrín, Oscar"https://zbmath.org/authors/?q=ai:jarrin.oscarSummary: We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for \(3<p\le +\infty\) we prove the existence of weak \(L^p\)-solutions. Moreover, within the framework of a slightly modified system, we discuss the possibly non-existence of \(L^p\)-solutions for \(1\le p \le 3\). Then, we use the more general setting of the \(L^{p,\infty}\)-spaces to show that weak solutions and their derivatives are Hölder continuous functions, where the maximum gain of regularity is determined by the initial regularity of the external forces and the gravitational acceleration. As a bi-product, we get a new regularity criterion for the steady-state Navier-Stokes equations. Furthermore, in the particular homogeneous case when the external forces are equal to zero; and for a range of values of the parameter \(p\), we show that weak solutions are not only smooth enough, but also they are identical to the trivial (zero) solution. This result is of independent interest, and it is also known as the Liouville-type problem for the steady-state Boussinesq system.The Batchelor-Howells-Townsend spectrum: large velocity casehttps://zbmath.org/1541.353812024-09-27T17:47:02.548271Z"Jolly, M. S."https://zbmath.org/authors/?q=ai:jolly.michael-summerfield"Wirosoetisno, D."https://zbmath.org/authors/?q=ai:wirosoetisno.djokoSummary: We consider the behaviour of a passive tracer \(\theta\) governed by \(\partial_t\theta+u\cdot\nabla\theta=\Delta\theta+g\) in two space dimensions with prescribed smooth random incompressible velocity \(u(x,t)\) and source \(g(x)\). In [J. Fluid Mech. 5, 113--134 (1959; Zbl 0085.39701); ibid. 5, 134--139 (1959; Zbl 0085.39702)], \textit{G. K. Batchelor} and \textit{G. K. Batchelor} et al., resp.,
predicted that the tracer (power) spectrum should scale as \(|\theta_k|^2\propto|k|^{-4}|u_k|^2\) for \(|k|\) above some \(\overline{\kappa}(u)\), with different behaviour for \(|k|\lesssim\overline{\kappa}(u)\) predicted earlier by
\textit{A. M. Obukhov} [``Structure of the temperature field in turbulent flows'', Izv. Akad. Nauk SSSR Geogr. Geofiz. 13, 58--69 (1949)] and
\textit{S. Corrsin} [J. Appl. Phys. 22, 469--473 (1951; Zbl 0044.40601)].
In this paper, we prove that the BHT scaling does indeed hold probabilistically for sufficiently large \(|k|\), asymptotically up to controlled remainders, using only bounds on the smaller \(|k|\) component.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd and the London Mathematical Society}A priori estimates for the motion of charged liquid drop: a dynamic approach via free boundary Euler equationshttps://zbmath.org/1541.353822024-09-27T17:47:02.548271Z"Julin, Vesa"https://zbmath.org/authors/?q=ai:julin.vesa"La Manna, Domenico Angelo"https://zbmath.org/authors/?q=ai:la-manna.domenico-angeloSummary: We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains \(C^{1, \alpha}\)-regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., \(C^\infty\) in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the \(C^{1, \alpha}\)-regularity assumption to be optimal. We also quantify the \(C^\infty\)-regularity via high order energy estimates which, in particular, implies the well-posedness of the problem.Strong solutions to the 3D full compressible magnetohydrodynamic flowshttps://zbmath.org/1541.353832024-09-27T17:47:02.548271Z"Liu, Junchen"https://zbmath.org/authors/?q=ai:liu.junchen"Wang, Xiuqing"https://zbmath.org/authors/?q=ai:wang.xiuqingSummary: In this paper, we deal with the strong solutions to the full compressible magnetohydrodynamic flows in a domain \(\Omega \subseteq \mathbb{R}^3\). We prove the local existence and unique of strong solution under three different types of boundary conditions provided that the initial data satisfies a natural compatibility condition. The initial density of such a strong solution is allowed to contain vacuum states. Compared to the classical compressible MHD, the key issue in the limit passage is to obtain the a priori estimates of the absolute temperature, some new estimate ideas are introduced to estimate these results.Stability and long time asymptotic for the 2D anisotropic micropolar Rayleigh-Bénard problem with horizontal dissipationhttps://zbmath.org/1541.353842024-09-27T17:47:02.548271Z"Luo, Zehua"https://zbmath.org/authors/?q=ai:luo.zehua"Li, Dan"https://zbmath.org/authors/?q=ai:li.dan"Wang, Yuzhu"https://zbmath.org/authors/?q=ai:wang.yuzhu|wang.yuzhu.1Summary: In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh-Bénard problem with horizontal dissipation. Based on the energy method and the bootstrapping argument, global classical solutions are proved under the assumptions of small initial data. Moreover, time-decay rates of the oscillation part of global classical solutions are obtained by combining the energy method and Poincaré type inequality.
{\copyright 2024 American Institute of Physics}Bent-half space model problem for Lamé equation with surface tensionhttps://zbmath.org/1541.353852024-09-27T17:47:02.548271Z"Maryani, Sri"https://zbmath.org/authors/?q=ai:maryani.sri"Wardayani, Ari"https://zbmath.org/authors/?q=ai:wardayani.ari"Renny"https://zbmath.org/authors/?q=ai:renny.Summary: The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the \(\mathcal{R}\)-boundedness of the solution operator families of the Lamé equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of \(N\)-dimensional Euclidean space \((N \geq 2)\). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [the first author et al., Math. Stat. 10, No. 3, 498--514 (2022; \url{doi:10.13189/ms.2022.100305})]. They investigated the \(\mathcal{R}\)-boundedness of the solution operator families in the half-space case for the model problem of the Lamé equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that \(\mathcal{R}\)-boundedness implies the maximal \(L_p\)-\(L_q\) regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.Interaction of two soliton waves in plasma including electrons with Kappa-Cairns distribution functionhttps://zbmath.org/1541.353862024-09-27T17:47:02.548271Z"Mirzaei, M."https://zbmath.org/authors/?q=ai:mirzaei.mozhgan|mirzaei.mostafa|mirzaei.mahbube|mirzaei.mohammad-javad|mirzaei.mehdi|mirzaei.mahmood.1|mirzaei.masoud|mirzaei.majid|mirzaei.maryam|mirzaei.mohsen|mirzaei.masaud"Motevalli, S. M."https://zbmath.org/authors/?q=ai:motevalli.s-mSummary: The interaction of positron acoustic soliton waves (PASWs) with the arbitrary collision angle in plasma including cold fluid positrons, stationary ions and electrons with Kappa-Cairns (K-C) distribution function have been studied. The equations of Korteweg-de Vries (KdV) and the phase shifts are obtained by employing the extended Poincaré-Lighthill-Kuo (PLK) method for the two colliding waves. The influences of parameters of the K-C distribution function (\(\kappa\) and \(\alpha\)), the collision angle \(\theta\) and the proportion of the ion (electron) and positron unperturbed densities (\(\beta_i(\beta_e)\)) on the phase shifts are investigated.Stagnation point flow over a stretching/shrinking cylinder with prescribed surface heat fluxhttps://zbmath.org/1541.353872024-09-27T17:47:02.548271Z"Najib, Najwa"https://zbmath.org/authors/?q=ai:najib.najwa"Bachok, Norfifah"https://zbmath.org/authors/?q=ai:bachok.norfifah"Arifin, Norihan Md."https://zbmath.org/authors/?q=ai:arifin.norihan-mdSummary: The steady stagnation-point flow towards a horizontal linearly stretching/shrinking cylinder immersed in an incompressible viscous fluid with prescribed surface heat flux is investigated. The governing partial differential equations in cylindrical form are transformed into ordinary differential equations by similarity transformations. The transformed equations are solved numerically by using the shooting method. Results for the skin friction coefficient, local Nusselt number, velocity profiles and temperature profiles are presented for different values of the governing parameters. Effects of the curvature parameter, stretching/shrinking parameter and Prandtl number on the flow and heat transfer characteristics are discussed. The study indicate that the solutions for a shrinking cylinder are non-unique. It is observed that the surface shear stress and heat transfer rate at the surface increase as the curvature parameter increases.
For the entire collection see [Zbl 1388.00025].On the propagation of equatorial waves interacting with a non-uniform currenthttps://zbmath.org/1541.353882024-09-27T17:47:02.548271Z"Novruzov, Emil"https://zbmath.org/authors/?q=ai:novruzov.emil-bSummary: We consider the propagation of equatorial waves of small amplitude, in a flow with an underlying non-uniform current. Without making the too restrictive rigid-lid approximation, by exploiting the available Hamiltonian structure of the problem, we derive the dispersion relation for the propagation of coupled long-waves: a surface wave and an internal wave. Also, we investigate the above-mentioned model of wave-current interactions in the general case with arbitrary vorticities.Hydrostatic approximation and optimal convergence rate for the second-grade fluid systemhttps://zbmath.org/1541.353892024-09-27T17:47:02.548271Z"Paicu, Marius"https://zbmath.org/authors/?q=ai:paicu.marius"Yu, Tianyuan"https://zbmath.org/authors/?q=ai:yu.tianyuan"Zhu, Ning"https://zbmath.org/authors/?q=ai:zhu.ningSummary: This paper is devoted to the study of the Second-Grade fluid system from three aspects. Firstly, we establish the well-posedness of the scaled anisotropic Second-Grade fluid system and the hydrostatic Second-Grade fluid system in a thin strip \(\mathbb{R}\times\mathbb{T}\) with convex Gevrey-2 initial data. Secondly, we justify the limit from the anisotropic Second-Grade fluid system to the hydrostatic Second-Grade fluid system. Finally, we prove that the hydrostatic Second-Grade fluid system will converge to the hydrostatic Navier-Stokes system as material parameter goes to zero.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Eigenvalue regularity criteria of the three-dimensional micropolar fluid equationshttps://zbmath.org/1541.353902024-09-27T17:47:02.548271Z"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandra"Wu, Fan"https://zbmath.org/authors/?q=ai:wu.fanThe work addresses some particular questions of the weak solution smoothness for the incompressible micropolar fluid equations considered in a 3D case under specified criteria for the intermediate eigenvalue of the strain matrix. These criteria are given in terms of equalities and inequalities for indices \(p\) and \(q\) of the Besov space.
Reviewer: Eugene Postnikov (Kursk)Multistability for nematic liquid crystals in cuboids with degenerate planar boundary conditionshttps://zbmath.org/1541.353912024-09-27T17:47:02.548271Z"Shi, Baoming"https://zbmath.org/authors/?q=ai:shi.baoming"Han, Yucen"https://zbmath.org/authors/?q=ai:han.yucen"Majumdar, Apala"https://zbmath.org/authors/?q=ai:majumdar.apala"Zhang, Lei"https://zbmath.org/authors/?q=ai:zhang.lei.4Summary: We study nematic configurations within three-dimensional (3D) cuboids, with planar degenerate boundary conditions on the cuboid faces, in the Landau-de Gennes framework. There are two geometry-dependent variables: the edge length of the square cross-section, \(\lambda\), and the parameter \(h\), which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimizer with a small enough cuboid size. We develop a new numerical scheme for the high-index saddle dynamics to deal with the surface energies. We report on a plethora of (meta)stable states, and their dependence on \(h\) and \(\lambda\), and in particular how the 3D states are connected with their two-dimensional counterparts on squares and rectangles. Notably, we find families of almost uniaxial stable states constructed from the topological classification of tangent unit-vector fields and study transition pathways between them. We also provide a phase diagram of competing (meta)stable states as a function of \(\lambda\) and \(h\).Stability of a class of solutions of the barotropic vorticity equation on a sphere equation on a spherehttps://zbmath.org/1541.353922024-09-27T17:47:02.548271Z"Skiba, Yuri N."https://zbmath.org/authors/?q=ai:skiba.yuri-nSummary: The linear and nonlinear stability of modons and Wu-Verkley waves, which are weak solutions of the barotropic vorticity equation on a rotating sphere, are analyzed. Necessary conditions for normal mode instability are obtained, the growth rate of unstable modes is estimated, and the orthogonality of unstable modes to the basic flow is shown. The Liapunov instability of dipole modons in the norm associated with enstrophy is proven.Quasi-periodic breathers and rogue waves to the focusing Davey-Stewartson equationhttps://zbmath.org/1541.353932024-09-27T17:47:02.548271Z"Sun, Jianqing"https://zbmath.org/authors/?q=ai:sun.jianqing"Hu, Xingbiao"https://zbmath.org/authors/?q=ai:hu.xingbiao"Zhang, Yingnan"https://zbmath.org/authors/?q=ai:zhang.yingnan.1Summary: The Davey-Stewartson equation has been instrumental in describing various physical phenomena, especially (2+1)-dimensional breathers and rogue waves. In this paper, we present a direct approach to studying the quasi-periodic breathers of the Davey-Stewartson equation. By employing Hirota's bilinear method and leveraging certain identities of theta functions, the problem is transformed into an over-determined nonlinear algebraic system, which can be formulated as a nonlinear least square problem and solved by classical numerical iterative algorithms. Through asymptotic analysis and numerical experiments, we categorize these solutions into three cases: quasi-periodic breathers, quasi-periodic stationary breathers, and quasi-periodic homoclinic orbits. The latter exhibit behavior reminiscent of quasi-periodic rogue waves, which are often observed in oceanic rogue wave phenomena. This work advances our understanding of the dynamics and properties of (2+1)-dimensional quasi-periodic breathers and rouge waves.Bifurcations, exact peakon, periodic peakons and solitary wave solutions of generalized Camassa-Holm-Degasperis-Procosi type equationhttps://zbmath.org/1541.353942024-09-27T17:47:02.548271Z"Sun, Xianbo"https://zbmath.org/authors/?q=ai:sun.xianbo|sun.xianbo.1"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin"Chen, Guanrong"https://zbmath.org/authors/?q=ai:chen.guanrong(no abstract)On an initial-boundary value problem which arises in the dynamics of a compressible ideal stratified fluidhttps://zbmath.org/1541.353952024-09-27T17:47:02.548271Z"Tsvetkov, Denis Olegovich"https://zbmath.org/authors/?q=ai:tsvetkov.denis-olegovichSummary: In this paper, we investigate the problem on small motions of a compressible ideal stratified fluid in a bounded domain. The problem is studied on the base of approach connected with application of so-called operator matrices theory, as well as abstract differential operator equations. For this purpose, Hilbert spaces and some of their subspaces are introduced. The original initial-boundary value problem reduces to the Cauchy problem for a second-order differential operator equation in the orthogonal sum of some Hilbert spaces. Further, an equation with a closed operator is associated with the resulting equation. On this basis, sufficient conditions for the existence of a solution to the corresponding problem are found.Solvability of an initial-boundary value problem for the modified Kelvin-Voigt model with memory along fluid motion trajectorieshttps://zbmath.org/1541.353962024-09-27T17:47:02.548271Z"Turbin, M. V."https://zbmath.org/authors/?q=ai:turbin.mikhail-v"Ustiuzhaninova, A. S."https://zbmath.org/authors/?q=ai:ustiuzhaninova.a-sSummary: The paper deals with proving the weak solvability of an initial-boundary value problem for the modified Kelvin-Voigt model taking into account memory along the trajectories of motion of fluid particles. To this end, we consider an approximation problem whose solvability is established with the use of the Leray-Schauder fixed point theorem. Then, based on a priori estimates, we show that the sequence of solutions of the approximation problem has a subsequence that weakly converges to the solution of the original problem as the approximation parameter tends to zero.Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaceshttps://zbmath.org/1541.353972024-09-27T17:47:02.548271Z"Vitillaro, Enzo"https://zbmath.org/authors/?q=ai:vitillaro.enzoSummary: The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.Novel soliton molecules, periodic wave and other diverse wave solutions to the new (2 + 1)-dimensional shallow water wave equationhttps://zbmath.org/1541.353982024-09-27T17:47:02.548271Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Li, Shuai"https://zbmath.org/authors/?q=ai:li.shuai|li.shuai.1"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.feng"Xu, Peng"https://zbmath.org/authors/?q=ai:xu.peng.5Summary: In this research, we focus on some novel exact solutions of the new (2 + 1)-dimensional shallow water wave equation (SWWE). First, the soliton molecules on the (x,y)-, (x,t)- and (y,t)-planes are constructed via assigning the velocity resonance conditions to the multiple soliton solutions (MSSs) that can be derived via the Hirota method. Second, the periodic wave solutions are explored by means of the new homoclinic approach. Finally, the other diverse wave solutions including the kink wave, singular wave and the singular periodic wave solutions are also plumbed by the sub-equation approach (SEA). The dynamic performances of the extracted solutions are presented graphically to unveil the nonlinear physical characteristics. As we know, the extracted solutions in this paper are all new and have not been investigated in other literature, which can help us make sense of the nonlinear dynamics of the new (2 + 1)-dimensional SWWE better.Stagnation-point flow over a nonlinearly stretching/shrinking sheet in a micropolar fluidhttps://zbmath.org/1541.353992024-09-27T17:47:02.548271Z"Yacob, Nor Azizah"https://zbmath.org/authors/?q=ai:yacob.nor-azizah"Ishak, Anuar"https://zbmath.org/authors/?q=ai:ishak.anuarSummary: The problem of a steady stagnation-point flow towards a nonlinearly stretching/shrinking sheet immersed in an incompressible micropolar fluid is studied. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation, before being solved numerically by a shooting method. The results show that increasing the material parameter \(K\) is to increase the skin friction coefficient (in absolute sense). Moreover, dual solutions are found to exist for the shrinking sheet, whereas the solution is unique for the stretching case.
For the entire collection see [Zbl 1388.00025].On an inhomogeneous slip-inflow boundary value problem for a steady viscous compressible channel flowhttps://zbmath.org/1541.354002024-09-27T17:47:02.548271Z"Yang, Wengang"https://zbmath.org/authors/?q=ai:yang.wengang(no abstract)Global well-posedness and large time behavior for the inviscid Oldroyd-B modelhttps://zbmath.org/1541.354012024-09-27T17:47:02.548271Z"Ye, Weikui"https://zbmath.org/authors/?q=ai:ye.weikui"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin.2|zhao.bin.3Summary: In this paper, we consider the 3-dimensional incompressible inviscid Oldroyd-B model. Firstly, we establish the global existence of the solutions for the inviscid Oldroyd-B model with different coupling coefficient \(k > 0\). Then, we show the connection between the solution with the parameter \(k\) that is the reciprocal of Weissenberg number. On one hand, we prove that the solutions \((u, \tau)\) depend continuously on the parameter \(k > 0\), though \((u, \tau)\) corresponds to different decays rate for different \(k\). On the other hand, when \(k \to 0\) in large time, we prove that there is a gap between the \(L^2\) norm of solution \(u^k(t, x)\) to the model of parameter \(k > 0\) with the \(L^2\) norm of solution \(u^0(t, x)\) to the model of parameter \(k = 0\). In a word, we prove that the larger positive \(k\) induces the faster decay rate of the solution, but \(k\) cannot go to zero, or the dissipation will vanish instantly.Global strong solutions to the incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum in 3D exterior domainshttps://zbmath.org/1541.354022024-09-27T17:47:02.548271Z"Yuan, Bing"https://zbmath.org/authors/?q=ai:yuan.bing"Zhang, Rong"https://zbmath.org/authors/?q=ai:zhang.rong.2"Zhou, Peng"https://zbmath.org/authors/?q=ai:zhou.peng|zhou.peng.1|zhou.peng.2Summary: The nonhomogeneous incompressible Magnetohydrodynamic Equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with slip boundary conditions. The key is the constraint of an additional initial value condition \(B_0 \in L^p\) \((1 \leqslant p < 12 / 7)\), which increase decay-in-time rates of the solutions, thus we obtain the global existence and uniqueness of strong solutions provided the gradient of the initial velocity and initial magnetic field is suitably small. In particular, the initial density is allowed to contain vacuum states and large oscillations. Moreover, the large-time behavior of the solution is also shown.Singularity for the drift-flux system of two-phase flow with the generalized Chaplygin gashttps://zbmath.org/1541.354032024-09-27T17:47:02.548271Z"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.26|zhang.jun.57|zhang.jun.10|zhang.jun.8|zhang.jun.11|zhang.jun.42|zhang.jun.31|zhang.jun.36|zhang.jun.12|zhang.jun.34|zhang.jun.1|zhang.jun|zhang.jun.16|zhang.jun.17|zhang.jun.37|zhang.jun.9|zhang.jun.6|zhang.jun.43|zhang.jun.7|zhang.jun.27|zhang.jun.5|zhang.jun.15|zhang.jun.29|zhang.jun.23|zhang.jun.2"Guo, Lihui"https://zbmath.org/authors/?q=ai:guo.lihuiSummary: This article considers the formation of singularity in the one-dimensional isentropic drift-flux system of two-phase flow with the generalized Chaplygin gas. We give an appropriate initial condition that results in the formation of singularity in finite time. Notably, the formation of singularity is accompanied by the concentration of mass. Furthermore, we verify the theoretical results.Forward dynamics of 3D double time-delayed MHD-Voight equationshttps://zbmath.org/1541.354042024-09-27T17:47:02.548271Z"Zhang, Qiangheng"https://zbmath.org/authors/?q=ai:zhang.qiangheng"Zhang, Qunli"https://zbmath.org/authors/?q=ai:zhang.qunli(no abstract)Lie symmetry analysis, optimal system, symmetry reductions and analytic solutions for a \((2+1)\)-dimensional generalized nonlinear evolution system in a fluid or a plasmahttps://zbmath.org/1541.354052024-09-27T17:47:02.548271Z"Zhou, Tian-Yu"https://zbmath.org/authors/?q=ai:zhou.tian-yu"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Shen, Yuan"https://zbmath.org/authors/?q=ai:shen.yuan.5"Cheng, Chong-Dong"https://zbmath.org/authors/?q=ai:cheng.chong-dongSummary: Nonlinear evolution equations are used to describe such nonlinear phenomena as the solitons, travelling waves and breathers in fluid mechanics, plasma physics and optics. In this paper, we investigate a \((2+1)\)-dimensional generalized nonlinear evolution system in a fluid or a plasma. Via the Lie symmetry analysis, we acquire the Lie point symmetry generators and Lie symmetry groups of that system. Via the optimal system method, we derive the optimal system of the 1-dimensional subalgebras. Based on the symmetry generators in that optimal system, we give some symmetry reductions for the \((2+1)\)-dimensional generalized nonlinear evolution system. Finally, via those symmetry reductions, we acquire some soliton, rational-type and power-series solutions.Various solutions of the (2+1)-dimensional Hirota-Satsuma-Ito equation using the bilinear neural network methodhttps://zbmath.org/1541.354062024-09-27T17:47:02.548271Z"Zhu, Guangzheng"https://zbmath.org/authors/?q=ai:zhu.guangzheng"Wang, Hailing"https://zbmath.org/authors/?q=ai:wang.hailing"Mou, Zhen-ao"https://zbmath.org/authors/?q=ai:mou.zhen-ao"Lin, Yezhi"https://zbmath.org/authors/?q=ai:lin.yezhiSummary: The Hirota-Satsuma-Ito equation is a well-known nonlinear partial differential equation in fluid mechanics. This paper deals with a (2+1)-dimensional Hirota-Satsuma-Ito equation through the bilinear neural network method. In the bilinear neural network method, a variety of neural network structures, including the single hidden layer and multi hidden layers neural network, are used to obtain the analytical solutions which are summarized to be of the following types: breathers, interaction of opposite waves, interaction of rogue wave and soliton, traveling waves and rogue waves. The feasibility and advantage of the proposed structures are illustrated by seeking these new solutions. Wave characteristics are exhibited by some plots of these obtained solutions.Resonances at the threshold for Pauli operators in dimension twohttps://zbmath.org/1541.354072024-09-27T17:47:02.548271Z"Breuer, Jonathan"https://zbmath.org/authors/?q=ai:breuer.jonathan"Kovařík, Hynek"https://zbmath.org/authors/?q=ai:kovarik.hynekSummary: It is well-known that, due to the interaction between the spin and the magnetic field, the two-dimensional Pauli operator has an eigenvalue 0 at the threshold of its essential spectrum. We show that when perturbed by an effectively positive perturbation, \(V\), coupled with a small parameter \(\varepsilon\), these eigenvalues become resonances. Moreover, we derive explicit expressions for the leading terms of their imaginary parts in the limit \(\varepsilon \searrow 0\). These show, in particular, that the dependence of the imaginary part of the resonances on \(\varepsilon\) is determined by the flux of the magnetic field. The cases of non-degenerate and degenerate zero eigenvalue are treated separately. We also discuss applications of our main results to particles with anomalous magnetic moments.Application of geometric algebra to Koga's work on quantum mechanicshttps://zbmath.org/1541.354082024-09-27T17:47:02.548271Z"Didimos, K. V."https://zbmath.org/authors/?q=ai:didimos.k-vSummary: One of the people who has offered alternatives to the Copenhagen interpretation of quantum mechanics is Toyoki Koga. Some of the important equations in quantum mechanics are the Schrödinger equation, Dirac equation, and Pauli equation; however, Koga only focused on the solutions of the first two. This article briefly introduces geometric algebra to study Koga's works, especially on the Dirac equation and how to get a translation of Koga's solution of the Dirac equation using geometric algebra.
For the entire collection see [Zbl 1531.20003].Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an intervalhttps://zbmath.org/1541.354092024-09-27T17:47:02.548271Z"Li, Junfeng"https://zbmath.org/authors/?q=ai:li.junfeng|li.junfeng.2|li.junfeng.1"Zheng, Chuang"https://zbmath.org/authors/?q=ai:zheng.chuangSummary: In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schrödinger equation posed on a bounded interval \((0, L)\) with non-homogeneous Navier or Dirichlet boundary conditions, respectively. For Navier boundary IBVP, we set up its local well-posedness if the initial data lies in \(H^s (0, L)\) with \(s \geq 0\) and \(s \neq n + 1/2\), \(n \in \mathcal{N}\), and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the \(j\)-th order data are chosen in \(H^{(s+3-j)/4}_{loc} (\mathcal{R}^+)\), for \(j = 0, 2\). For Dirichlet boundary IBVP the corresponding local well-posedness is obtained when \(s > 10/7\) and \(s \neq n+1/2\), \(n \in \mathcal{N}\), and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the \(j\)-th order data are chosen in \(H^{(s+3-j)/4}_{loc} (\mathcal{R}^+)\), for \(j = 0, 1\).Nonlinear PDE models in semi-relativistic quantum physicshttps://zbmath.org/1541.354102024-09-27T17:47:02.548271Z"Möller, Jakob"https://zbmath.org/authors/?q=ai:moller.jakob-riishede"Mauser, Norbert J."https://zbmath.org/authors/?q=ai:mauser.norbert-juliusSummary: We present the self-consistent Pauli equation, a semi-relativistic model for charged spin-1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the \(O(1/c)\) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac-Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger-Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger-Poisson equation also arises as the mean field limit of the \(N \)-body Schrödinger equation with Coulomb interaction. We propose that the Pauli-Poisson equation arises as the mean field limit \(N\to\infty\) of the linear \(N \)-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli-Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in \(1/c\) of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions \& Paul and Markowich \& Mauser, where we need methods like magnetic Lieb-Thirring estimates.Limit behaviors of pseudo-relativistic Hartree equation with power-type perturbationshttps://zbmath.org/1541.354112024-09-27T17:47:02.548271Z"Wang, Qingxuan"https://zbmath.org/authors/?q=ai:wang.qingxuan"Xu, Zefeng"https://zbmath.org/authors/?q=ai:xu.zefengSummary: We consider the following pseudo-relativistic Hartree equations with power-type perturbation,
\[
i \partial_t \psi = \sqrt{- \Delta + m^2} \psi - (\frac{1}{|x|} \ast |\psi|^2) \psi + \varepsilon |\psi|^{p - 2} \psi, \quad \text{with } (t, x) \in \mathbb{R} \times \mathbb{R}^3
\]
where \(2 < p < 3\), \(\varepsilon > 0\) and \(m > 0\), \(p = \frac{8}{3}\) can be viewed as a Slater modification. We mainly focus on the normalized ground state solitary waves \(\varphi_\varepsilon\), where \(\|\varphi_\varepsilon\|_2^2 = N\). Firstly, we prove the existence and nonexistence of normalized ground states under \(L^2\)-subcritical, \(L^2\)-critical \((p = \frac{8}{3})\) and \(L^2\)-supercritical perturbations. Secondly, we classify perturbation limit behaviors of ground states when \(\varepsilon \to 0^+\), and obtain two different blow-up profiles for \(N = N_c\) and \(N > N_c\), where \(N_c\) be regard as ``Chandrasekhar limiting mass''. We prove that \(\langle \varphi_\varepsilon, \sqrt{- \Delta} \varphi_\varepsilon \rangle \sim \varepsilon^{- \frac{2}{3p - 4}}\) for \(N = N_c\) and \(2 < p < 3\), while \(\langle \varphi_\varepsilon, \sqrt{- \Delta} \varphi_\varepsilon \rangle \sim \varepsilon^{- \frac{2}{3p - 8}}\) for \(N > N_c\) and \(\frac{8}{3} < p < 3\). Finally, we study the asymptotic behavior for \(\varepsilon \to + \infty\), and obtain an energy limit \(\lim_{\varepsilon \to + \infty} e_\varepsilon(N) = \frac{1}{2} mN\) and a vanishing rate \(\int_{\mathbb{R}^3} |\varphi_\varepsilon|^p dx \lesssim \varepsilon^{- 1}\) when \(N > N_c\) and \(\frac{8}{3} < p < 3\).Strichartz inequality for orthonormal functions associated with special Hermite operatorhttps://zbmath.org/1541.354122024-09-27T17:47:02.548271Z"Ghosh, Sunit"https://zbmath.org/authors/?q=ai:ghosh.sunit"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarup"Swain, Jitendriya"https://zbmath.org/authors/?q=ai:swain.jitendriyaIn this paper, the authors establish the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, they study the optimal behavior of the constant as a limit of a large number of functions.
Reviewer: Changxing Miao (Beijing)Soliton solutions for a quantum particle in one-dimensional boxeshttps://zbmath.org/1541.354132024-09-27T17:47:02.548271Z"Jangid, Anjali"https://zbmath.org/authors/?q=ai:jangid.anjali"Devi, Pooja"https://zbmath.org/authors/?q=ai:devi.pooja"Soni, Harsh"https://zbmath.org/authors/?q=ai:soni.harsh"Chakraborty, Aniruddha"https://zbmath.org/authors/?q=ai:chakraborty.aniruddhaSummary: In this study, we present a new analytical solution for the time-dependent Schrödinger equation for a free particle in one-dimensional case. The solution is derived by doing a non-linear transform to the linear Schrödinger equation and converting it into a Burger-like equation. We obtained an interesting non-stationary wave function where our soliton solution moves in time while maintaining its shape. The new solution is then analysed for three different cases: a periodic box, a box with hard wall boundary conditions and a periodic array of Dirac delta potentials. The resulting analytical solutions exhibit several interesting features including quantized soliton velocity and velocity bands. The analytical soliton solution that has been proposed, in our opinion, makes an important contribution to the study of quantum mechanics and we believe it will contribute significantly to our understanding of how particles behave in one-dimensional box potentials.Propagation of regularity for transport equations: a Littlewood-Paley approachhttps://zbmath.org/1541.354142024-09-27T17:47:02.548271Z"Meyer, David"https://zbmath.org/authors/?q=ai:meyer.david-a|meyer.david-c|meyer.david-g|meyer.david-j"Seis, Christian"https://zbmath.org/authors/?q=ai:seis.christianSummary: It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We propose a new approach to the study of regularity that is based on Littlewood-Paley theory, thus measuring regularity in terms of Besov norms. We recover the results that are available in the literature and extend these optimally to the diffusive setting. As a consequence, we derive sharp bounds on rates of convergence in the zero-diffusivity limit.Initial-boundary value problem for transport equations driven by rough pathshttps://zbmath.org/1541.354152024-09-27T17:47:02.548271Z"Noboriguchi, Dai"https://zbmath.org/authors/?q=ai:noboriguchi.daiSummary: In this paper, we are interested in the initial Dirichlet boundary value problem for a transport equation driven by weak geometric Hölder \(p\)-rough paths. We introduce a notion of solutions to rough partial differential equations with boundary conditions. Consequently, we will establish a well-posedness for such a solution under some assumptions stated below. Moreover, the solution is given explicitly.Solitary wave effects of Woods-Saxon potential in Schrödinger equation with 3d cubic nonlinearityhttps://zbmath.org/1541.354162024-09-27T17:47:02.548271Z"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafa"Iqbal, Muhammad Sajid"https://zbmath.org/authors/?q=ai:iqbal.muhammad-sajid"Ali, Ali Hasan"https://zbmath.org/authors/?q=ai:ali.ali-hasan"Manzoor, Zuha"https://zbmath.org/authors/?q=ai:manzoor.zuha"Ashraf, Farrah"https://zbmath.org/authors/?q=ai:ashraf.farrahSummary: In this research article, we apply the generalized projective Riccati equation method to construct traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods-Saxon potential. The generalized projective Riccati equation method is a powerful and effective mathematical tool for obtaining exact solutions of nonlinear partial differential equations, and it allows us to derive a variety of traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods-Saxon potential. These solutions contain periodic wave solutions, bright and dark soliton solutions. The study of many physical systems, such as Bose-Einstein condensates and nonlinear optics, that give rise to the nonlinear Schrödinger equation. We provide a detailed description of the generalized projective Riccati equation method in the paper, and demonstrate its usefulness in solving the nonlinear Schrödinger equation with Woods-Saxon potential. We present various graphical representations of the obtained solutions using MATLAB software, and analyze their characteristics. Our results provide new insights into the behavior of the 3d cubic focusing nonlinear Schrödinger equation with Woods-Saxon potential, and have potential applications in numerous fields of physics, as well as nonlinear optics and condensed matter physics.M-lump solutions and interactions phenomena for the (2+1)-dimensional KdV equation with constant and time-dependent coefficientshttps://zbmath.org/1541.354172024-09-27T17:47:02.548271Z"Ali, Karmina K."https://zbmath.org/authors/?q=ai:ali.karmina-k"Yilmazer, Resat"https://zbmath.org/authors/?q=ai:yilmazer.resatSummary: In this paper, Hirota's simple method and long-wave method are used to study the (2+1)-dimensional KdV equation including constant and time-dependent coefficients. One-, two-, and three-M-lump solutions are constructed for both cases. Also, the interaction phenomena of M-lump solution with one-soliton, and two-soliton waves are derived for these equations. Moreover, to investigate the physical characteristics of the obtained results, the 3-dimensional figures and contour plots are graphed. All gained solutions verify the proposed equations.Overtaking collisions of \(m\) shock waves and interactions of \(n(n\to\infty)\)-lump, \(m(m\to\infty)\)-solitons, \(\tau(\tau\to\infty)\)-periodic waves solutions to a generalized (2+1)-dimensional new KdV modelhttps://zbmath.org/1541.354182024-09-27T17:47:02.548271Z"Alshammari, F. S."https://zbmath.org/authors/?q=ai:alshammari.fahad-sameer"Albilasi, R. S."https://zbmath.org/authors/?q=ai:albilasi.r-s"Hoque, M. F."https://zbmath.org/authors/?q=ai:hoque.md-fazlul"Rohsid, H. O."https://zbmath.org/authors/?q=ai:rohsid.h-oSummary: We consider a new generalized (2+1)-dimensional KdV model to investigate \(m\) (\(m\to\infty\)) shock and \(n\) (\(n\to\infty\)) breather wave solutions via two integral schemes. For the treatment of the model in an auxiliary equation approach, we first convert a nonlinear Burger equation to an ordinary differential equation (ODE) through a certain transformation. This ODE is used as an auxiliary equation of the method to obtain \(m\) (\(m\to\infty\)) shock wave solutions of the model. For different values of the parameters, we present head on and overtaking collisions with scattering ways of particle of the \(m\) (\(m\to\infty\)) shock wave solutions. We construct \(n\) soliton solutions of the model by using Hirota-bilinear approach. We obtain one lump type breather waves, interactions of one breather wave with a kink wave, interactions of two lump type breather waves by choosing complex conjugate values of free parameters in the \(n\)-soliton solutions of the model. Finally, we introduce two lemmas, a theorem and few corollaries on the hybrid interaction (\(n\to\infty\) lumps, \(m\to\infty\) solitons and \(\tau\to\infty\) periodic waves) solutions of the model. The theories and results are illustrated with adequate examples and suitable graphs.Analytical approximate solutions of Caputo fractional KdV-Burgers equations using Laplace residual power series techniquehttps://zbmath.org/1541.354192024-09-27T17:47:02.548271Z"Burqan, Aliaa"https://zbmath.org/authors/?q=ai:burqan.aliaa-abed-al-jawwad"Khandaqji, Mona"https://zbmath.org/authors/?q=ai:khandaqji.mona"Al-Zhour, Zeyad"https://zbmath.org/authors/?q=ai:zhour.zeyad-al"El-Ajou, Ahmad"https://zbmath.org/authors/?q=ai:el-ajou.ahmad"Alrahamneh, Tasneem"https://zbmath.org/authors/?q=ai:alrahamneh.tasneemSummary: The KdV-Burgers equation is one of the most important partial differential equations, established by Korteweg and de Vries to describe the behavior of nonlinear waves and many physical phenomena. In this paper, we reformulate this problem in the sense of Caputo fractional derivative, whose physical meanings, in this case, are very evident by describing the whole time domain of physical processing. The main aim of this work is to present the analytical approximate series for the nonlinear Caputo fractional KdV-Burgers equation by applying the Laplace residual power series method. The main tools of this method are the Laplace transform, Laurent series, and residual function. Moreover, four attractive and satisfying applications are given and solved to elucidate the mechanism of our proposed method. The analytical approximate series solution via this sweet technique shows excellent agreement with the solution obtained from other methods in simple and understandable steps. Finally, graphical and numerical comparison results at different values of \(\alpha\) are provided with residual and relative errors to illustrate the behaviors of the approximate results and the effectiveness of the proposed method.Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equationshttps://zbmath.org/1541.354202024-09-27T17:47:02.548271Z"Claeys, Tom"https://zbmath.org/authors/?q=ai:claeys.tom"Glesner, Gabriel"https://zbmath.org/authors/?q=ai:glesner.gabriel"Ruzza, Giulio"https://zbmath.org/authors/?q=ai:ruzza.giulio"Tarricone, Sofia"https://zbmath.org/authors/?q=ai:tarricone.sofiaSummary: We study Jánossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg-de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir-Corwin-Quastel's integro-differential Painlevé II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg-de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions.Correction to: ``Perturbation at blow-up time of self-similar solutions for the modified Korteweg-de Vries equation''https://zbmath.org/1541.354212024-09-27T17:47:02.548271Z"Correia, Simão"https://zbmath.org/authors/?q=ai:correia.simao"Côte, Raphaël"https://zbmath.org/authors/?q=ai:cote.raphaelCorrection to the authors' paper [ibid. 248, No. 2, Paper No. 25, 44 p. (2024; Zbl 1536.35296)].
From the text: The publication of this article unfortunately contained mistakes. The name of one of the authors was not correct. The corrected name is given below.
Raphaël Côte
The original article has been corrected.On the Kadomtsev-Petviashvili equation with double-power nonlinearitieshttps://zbmath.org/1541.354222024-09-27T17:47:02.548271Z"Esfahani, Amin"https://zbmath.org/authors/?q=ai:esfahani.amin"Levandosky, Steven"https://zbmath.org/authors/?q=ai:levandosky.steven-paul"Muslu, Gulcin M."https://zbmath.org/authors/?q=ai:muslu.gulcin-mSummary: In this paper, we study the generalized KP equation with double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by \(\mu_1|u|^{p_1 - 1}u + \mu_2|u|^{p_2 - 1}u\), with \(p_1 > p_2\), encompassing cases where \(\mu_1 > 0\) and \(\mu_1 < 0 < \mu_2\). One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equationshttps://zbmath.org/1541.354232024-09-27T17:47:02.548271Z"Figueira, Renata O."https://zbmath.org/authors/?q=ai:figueira.renata-o"Panthee, Mahendra"https://zbmath.org/authors/?q=ai:panthee.mahendraSummary: This paper is devoted to obtaining new lower bounds to the radius of spatial analyticity for the solutions of modified Korteweg-de Vries (mKdV) equation and a coupled system of mKdV-type equations, starting with real analytic initial data with a fixed radius of analyticity \(\sigma_0\). Specifically, we derive almost conserved quantities to prove that the local solution can be extended to a time interval \([0, T]\) for any large \(T>0\) in such a way that the radius of analyticity \(\sigma (T)\) decays no faster than \(cT^{-1}\) for both the equations, where \(c\) is a positive constant. The results of this paper improve the ones obtained in
[\textit{R. O. Figueira} and \textit{M. Panthee}, NoDEA, Nonlinear Differ. Equ. Appl. 31, No. 4, Paper No. 68, 23 p. (2024; Zbl 07874564)] and
[\textit{R. O. Figueira} and \textit{A. A. Himonas}, J. Math. Anal. Appl. 497, No. 2, Article ID 124917, 17 p. (2021; Zbl 1462.35333)],
respectively, for the mKdV equation and a mKdV-type system.Singular manifold, auto-Bäcklund transformations and symbolic-computation steps with solitons for an extended three-coupled Korteweg-de Vries systemhttps://zbmath.org/1541.354242024-09-27T17:47:02.548271Z"Gao, Xin-Yi"https://zbmath.org/authors/?q=ai:gao.xinyi"Guo, Yong-Jiang"https://zbmath.org/authors/?q=ai:guo.yongjiang"Shan, Wen-Rui"https://zbmath.org/authors/?q=ai:shan.wenrui"Zhou, Tian-Yu"https://zbmath.org/authors/?q=ai:zhou.tian-yuSummary: Korteweg-de Vries (KdV)-type models are frequently seen during the investigations on the optical fibers, cosmic plasmas, planetary oceans and atmospheres. In this paper, for an extended three-coupled KdV system, noncharacteristic movable singular manifold and symbolic computation help us bring about four sets of the auto-Bäcklund transformations with some solitons. All of our results rely on the coefficients in that system.On the persistence of spatial analyticity for generalized KdV equation with higher order dispersionhttps://zbmath.org/1541.354252024-09-27T17:47:02.548271Z"Getachew, Tegegne"https://zbmath.org/authors/?q=ai:getachew.tegegne"Tesfahun, Achenef"https://zbmath.org/authors/?q=ai:tesfahun.achenef"Belayneh, Birilew"https://zbmath.org/authors/?q=ai:belayneh.birilewSummary: Persistence of spatial analyticity is studied for solutions of the generalized Korteweg-de Vries (KdV) equation with higher order dispersion
\[
\partial_t u+(-1)^{j+1} \partial_x^{2j+1} u = \partial_x \left(u^{2k+1} \right),
\]
where \(j \geq 2\), \(k \geq 1\) are integers. For a class of analytic initial data with a fixed radius of analyticity \(\sigma_0\), we show that the uniform radius of spatial analyticity \(\sigma (t)\) of solutions at time \(t\) cannot decay faster than \(\frac{1}{\sqrt t}\) as \(t \to \infty\). In particular, this improves a recent result due to \textit{G. Petronilho} and \textit{P. L. da Silva} [Math. Nachr. 292, No. 9, 2032--2047 (2019; Zbl 1427.35220)] for the modified Kawahara equation \((j=2, k=1)\), where they obtained a decay rate of order \(t^{-4 +}\). Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.
{\copyright} 2023 Wiley-VCH GmbH.Lump and interaction dynamics of the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equationhttps://zbmath.org/1541.354262024-09-27T17:47:02.548271Z"He, Lingchao"https://zbmath.org/authors/?q=ai:he.lingchao"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwen"Zhao, Zhonglong"https://zbmath.org/authors/?q=ai:zhao.zhonglongSummary: This paper mainly considers the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation describing propagation dynamics of nonlinear waves appearing in physics, biology and electrical networks. A simple method to derive the multiple lumps and lump molecules is proposed by means of the solutions of Grammian form and polynomial functions. The interaction dynamical behaviors of the lumps and lump molecules are analyzed with the aid of numerical simulation. The interaction solutions including lumps, lump molecules and kink solitons are obtained, where the lumps can be emitted and absorbed by solitons. The phenomenon that one lump wave exchanges between two bound solitons is presented. A special Grammian \(\tau\)-function is employed to construct the lump chain. Based on an analytical method related to dominant domain, we systematically analyze the interactions between lump chain and solitons. The lump chain can be emitted by a soliton and then absorbed by other solitons when the parameter is in a certain range. A large number of new solutions obtained in this paper are helpful to study the wave phenomena existing in the theory of water wave, soliton, optical fiber communication and other relevant scientific fields.Comparative analysis of classical and stochastic Maccari system of nonlinear equationshttps://zbmath.org/1541.354272024-09-27T17:47:02.548271Z"Iqbal, Muhammad Sajid"https://zbmath.org/authors/?q=ai:iqbal.muhammad-sajid"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafa"Sohail, Saba"https://zbmath.org/authors/?q=ai:sohail.saba"Raheem, Adil"https://zbmath.org/authors/?q=ai:raheem.adil"Hussain, Shabbir"https://zbmath.org/authors/?q=ai:hussain.shabbir"Mahmoud, Emad E."https://zbmath.org/authors/?q=ai:mahmoud.emad-eSummary: In this paper, the exact solutions of classical and stochastic Maccari system is constructed. The exact comparative solutions are examined and plotted. Interesting results in the case of multiplicative noise are formulated and graphically elaborated. The applications of the stochastic Maccari system are added for the physical purpose. The existence of results for the real part of underlying system are discussed first time for a priori estimates. The perturbations, which disturbed the formation of Langmuir waves, are geometrically expressed in this article. Due to the presence of multiplicative noise term, our system brings a real flavor to the dynamics of the problem.Cancellation properties and unconditional well-posedness for the fifth order KdV type equations with periodic boundary conditionhttps://zbmath.org/1541.354282024-09-27T17:47:02.548271Z"Kato, Takamori"https://zbmath.org/authors/?q=ai:kato.takamori"Tsugawa, Kotaro"https://zbmath.org/authors/?q=ai:tsugawa.kotaroSummary: We consider the fifth order KdV type equations and prove the unconditional well-posedness in \(H^s(\mathbb{T})\) for \(s \geq 1\). It is optimal in the sense that the nonlinear terms can not be defined in the space-time distribution framework for \(s < 1\). The main idea is to employ the normal form reduction and a kinds of cancellation properties to deal with the derivative losses.Dynamics of nonlinear time fractional equations in shallow water waveshttps://zbmath.org/1541.354292024-09-27T17:47:02.548271Z"Khater, Mostafa M. A."https://zbmath.org/authors/?q=ai:khater.mostafa-m-aSummary: This study investigates the modified nonlinear time fractional Harry Dym equation, incorporating the conformable fractional derivative. Functioning as a mathematical framework for examining nonlinear phenomena in shallow water waves, particularly solitons, this model elucidates the intricate effects of dispersion and nonlinear steepening on wave dynamics. Employing a blend of analytical and numerical methodologies, the research aims to decipher the physical implications of the equation and its interconnectedness with other nonlinear evolution equations. The model delineates the evolution of a nonlinear wave in \(1+1\) dimensions (one spatial dimension \(x\) and time \(t)\). The proposed methodology encompasses the \(\left(\frac{G'}{G},\, \frac{1}{G}\right)\) expansion method, an analytical technique, alongside three numerical schemes utilizing B-spline methods. These methodologies facilitate the exploration of the equation's behavior and enable precise computations of its solutions. The principal findings underscore the effective application of the proposed methodologies in resolving the modified nonlinear time fractional Harry Dym equation, furnishing valuable insights into its dynamics and significantly contributing to its physical interpretation. The significance of these discoveries lies in their contribution to the broader comprehension of nonlinear evolution equations and their pertinence across various scientific and engineering domains. This study provides novel insights into the modified nonlinear time fractional Harry Dym equation through a combined analytical and numerical approach. It advances the field of nonlinear dynamics and carries implications for analyzing analogous nonlinear evolution equations. These findings deepen our understanding of the equation's physical interpretation and lay the groundwork for future explorations in related domains.Solving the regularized Schamel equation by the singular planar dynamical system method and the deep learning methodhttps://zbmath.org/1541.354302024-09-27T17:47:02.548271Z"Li, Kebing"https://zbmath.org/authors/?q=ai:li.kebing"Zhou, Yuqian"https://zbmath.org/authors/?q=ai:zhou.yuqian"Liu, Qian"https://zbmath.org/authors/?q=ai:liu.qian.2"Zhang, Shengning"https://zbmath.org/authors/?q=ai:zhang.shengning"Yi, Xueqiong"https://zbmath.org/authors/?q=ai:yi.xueqiong(no abstract)Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear systemhttps://zbmath.org/1541.354312024-09-27T17:47:02.548271Z"Li, Lingfei"https://zbmath.org/authors/?q=ai:li.lingfei"Wu, Jingyu"https://zbmath.org/authors/?q=ai:wu.jingyu"Xie, Yingying"https://zbmath.org/authors/?q=ai:xie.yingyingSummary: In this paper, we propose a new variable separation method that does not need the Hirota's bilinear form and directly gives the analytic form of the solution \(u\) instead of its potential \(u_y\). This new method covers the N-soliton solution obtained by the Hirota's direct method and the multi-valued solution of the multi-linear variable separation approach (MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the ``direct separation approach'' (DSA). Taking the extended (3+1)-dimensional Kadomtsev-Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.Bilinear forms, bilinear Bäcklund transformation, soliton and breather interactions of a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the surface waves in a strait or large channelhttps://zbmath.org/1541.354322024-09-27T17:47:02.548271Z"Li, Liu-Qing"https://zbmath.org/authors/?q=ai:li.liu-qing"Gao, Yi-Tian"https://zbmath.org/authors/?q=ai:gao.yitian"Yu, Xin"https://zbmath.org/authors/?q=ai:yu.xin.1"Jia, Ting-Ting"https://zbmath.org/authors/?q=ai:jia.tingting"Hu, Lei"https://zbmath.org/authors/?q=ai:hu.lei.1"Zhang, Cai-Yin"https://zbmath.org/authors/?q=ai:zhang.cai-yinSummary: In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth and width and non-vanishing vorticity, in which \(\alpha_1(t)\), \(\beta(t)\) and \(\gamma(t)\) are the dispersive, dissipative and line-damping coefficients, respectively, where \(t\) is the temporal variable. Bilinear forms, bilinear Bäcklund transformation and multi-soliton solutions are constructed via the Hirota bilinear method under some variable-coefficient constraints. Based on those multi-soliton solutions, multi-pole, breather and hybrid solutions are derived. Effect of \(\alpha_1(t)\), \(\beta(t)\) and \(\gamma(t)\) on the solutions is discussed analytically and graphically. For the solitons, we find that \(\alpha_1(t)\) and \(\beta(t)\) are related to the velocities and characteristic lines, and the amplitudes depend on \(\gamma(t)\). For the multi-pole and breather solutions, \(\alpha_1(t)\) and \(\beta(t)\) influence the center trajectories of the solutions, while \(\gamma(t)\) influences the amplitudes. Hybrid solutions composed of the breathers and solitons are worked out and discussed graphically.Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli systemhttps://zbmath.org/1541.354332024-09-27T17:47:02.548271Z"Maltseva, Diana S."https://zbmath.org/authors/?q=ai:maltseva.diana-s"Popovych, Roman O."https://zbmath.org/authors/?q=ai:popovych.roman-oSummary: We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which corrects, enhances and generalizes many results existing in the literature. The point-symmetry pseudogroup of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system using differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended by Laplace and Darboux transformations.The focusing coupled modified Korteweg-de Vries equation with nonzero boundary conditions: the Riemann-Hilbert problem and soliton classificationhttps://zbmath.org/1541.354342024-09-27T17:47:02.548271Z"Ma, Xinxin"https://zbmath.org/authors/?q=ai:ma.xinxinSummary: The focusing coupled modified Korteweg-de Vries equation with nonzero boundary conditions is investigated by the Riemann-Hilbert approach. Three symmetries are formulated to derive compact exact solutions. The solutions include six different types of soliton solutions and breathers, such as dark-dark, bright-bright, kink-dark-dark, kink-bright-bright solitons, and a breather-breather solution.Long time asymptotics of large data in the Kadomtsev-Petviashvili modelshttps://zbmath.org/1541.354352024-09-27T17:47:02.548271Z"Mendez, Argenis J."https://zbmath.org/authors/?q=ai:mendez.argenis-j"Muñoz, Claudio"https://zbmath.org/authors/?q=ai:munoz.claudio"Poblete, Felipe"https://zbmath.org/authors/?q=ai:poblete.felipe"Pozo, Juan C."https://zbmath.org/authors/?q=ai:pozo.juan-carlosSummary: We consider the Kadomtsev-Petviashvili (KP) equations posed on \(\mathbb{R}^2\). For both models, we provide sequential in time asymptotic descriptions of solutions obtained from arbitrarily large initial data, inside regions of the plane not containing lumps or line solitons, and under minimal regularity assumptions. The proof involves the introduction of two new virial identities adapted to the KP dynamics. This new approach is particularly important in the KP-I case, where no monotonicity property was previously known. The core of our results do not require the use of the integrability of KP and are adaptable to well-posed perturbations.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On the Cauchy problem for the new shallow-water models with cubic nonlinearityhttps://zbmath.org/1541.354362024-09-27T17:47:02.548271Z"Mi, Yongsheng"https://zbmath.org/authors/?q=ai:mi.yongsheng"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.bolingSummary: This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. On the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map for the system. On the other hand, we prove the persistence properties in weighted spaces of the solution, provided that the initial potential satisfies a certain sign condition.Pseudosolitons on the generalized conformable Korteweg-de Vries equationhttps://zbmath.org/1541.354372024-09-27T17:47:02.548271Z"Noyola-Rodríguez, J."https://zbmath.org/authors/?q=ai:rodriguez.j-noyola|noyola-rodriguez.jesus"Gómez, J. C. Hernandez"https://zbmath.org/authors/?q=ai:gomez.j-c-hernandez"Ramos, S. Gatica"https://zbmath.org/authors/?q=ai:ramos.s-gatica"Pineda-Pineda, Jair J."https://zbmath.org/authors/?q=ai:pineda-pineda.jair-jSummary: We consider the Korteweg-de Vries (KdV) model with classical derivative for outflows on water surfaces where the medium is assumed to be shallow and replace the classical first order derivative in time by the generalized conformable fractional derivative of order \(\alpha\) with \(0<\alpha\le 1\) and different kernels \(T(t,\alpha)\) defining fractional derivative mentioned above. We prove the existence of new soliton-like traveling waves which we call pseudodolitons propagating in a not necessarily linear wavefront. We study the dynamical system associated to the equation with generalized conformable fractional derivative and calculate some balance laws for the energy. In general it is concluded that the pseudosolitons have two main characteristics depending on the kernel of the fractional derivative. If the fractional derivative is conformable then the pseudosolitons propagate over large distances preserving their shape and amplitude and converge to the soliton-like solutions of the classical KdV when \(\alpha\to 1\). On the other hand, if the derivative is nonconformable then the pseudosolitons only preserve the properties of the soliton-like traveling waves.Highest waves for fractional Korteweg-de Vries and Degasperis-Procesi equationshttps://zbmath.org/1541.354382024-09-27T17:47:02.548271Z"Ørke, Magnus C."https://zbmath.org/authors/?q=ai:orke.magnus-cSummary: We study traveling waves for a class of fractional Korteweg-De Vries and fractional Degasperis-Procesi equations with a parametrized Fourier multiplier operator of order \(s \in (-1, 0)\). For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal \(s\)-Hölder regularity, attained in the cusp.Local well-posedness for the Zakharov-Kuznetsov equation on the background of a bounded functionhttps://zbmath.org/1541.354392024-09-27T17:47:02.548271Z"Palacios, José Manuel"https://zbmath.org/authors/?q=ai:palacios.jose-manuelSummary: We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in \(H^s (\mathbb{R}^2)\), for \(s\in [1,2]\), on the background of an \(L^{\infty}(\mathbb{R}^3)\)-function \(\Psi (t,x,y)\), with \(\Psi (t,x,y)\) satisfying some natural extra conditions. This result not only gives us a framework to solve the ZK equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of periodic solutions. Additionally, we show the global well-posedness in the energy space \(H^1 (\mathbb{R}^2)\).Simulation study of dust magnetosonic excitations in a magnetized dusty plasmahttps://zbmath.org/1541.354402024-09-27T17:47:02.548271Z"Singla, Sunidhi"https://zbmath.org/authors/?q=ai:singla.sunidhi"Chandra, S."https://zbmath.org/authors/?q=ai:chandra.soumen|chandra.suresh.1|chandra.shiva|chandra.shalini|chandra.sarthak|chandra.satish|chandra.subodh|chandra.sanjeev|chandra.susheel|chandra.sujan|chandra.shekhar-s|chandra.suryansh|chandra.swarniv|chandra.samarth|chandra.sushil|chandra.sudip-ratan|chandra.sumir|chandra.subha|chandra.subhash-ajay|chandra.sharat|chandra.sarvesh|chandra.souvik|chandra.saroj-kumar|chandra.sanjay|chandra.saket|chandra.saurabh"Saini, N. S."https://zbmath.org/authors/?q=ai:saini.nareshpal-singhSummary: A theoretical investigation is made to study the properties of dust magnetosonic (DMS) solitons in a magnetized electron-ion-dust plasma that contains negative polarity warm dust grains, and inertialess ions as well as electrons. By using reductive perturbation technique (RPT), the Korteweg-de Vries (KdV) equation is derived. There is the formation of only positive potential DMS solitons in the high plasma-\(\beta\) limit. The effects of plasma parameters, viz., plasma-\(\beta\), electron to ion temperature ratio and dust to electron density ratio on the characteristics of DMS solitons are also studied numerically. Furthermore, we have analysed the head-on collision of DMS solitary waves and subsequent evolution of stationary structures using INSAT FORK code. Results of this investigation might be useful for understanding the nonlinear disturbances in space plasmas especially in Earth's magnetosphere region and are also useful in understanding the energy transport phenomena of nonlinear structures.Long-timescale soliton dynamics in the Korteweg-de Vries equation with multiplicative translation-invariant noisehttps://zbmath.org/1541.354412024-09-27T17:47:02.548271Z"Westdorp, R. W. S."https://zbmath.org/authors/?q=ai:westdorp.rik-w-s"Hupkes, H. J."https://zbmath.org/authors/?q=ai:hupkes.hermen-janSummary: This paper studies the behavior of solitons in the Korteweg-de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.Semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg-de Vries equationshttps://zbmath.org/1541.354422024-09-27T17:47:02.548271Z"Zhao, Qiulan"https://zbmath.org/authors/?q=ai:zhao.qiulan"Cheng, Hongbiao"https://zbmath.org/authors/?q=ai:cheng.hongbiao"Li, Xinyue"https://zbmath.org/authors/?q=ai:li.xinyueSummary: The ubiquitous Korteweg-de Vries equation is widely used in oceanic, atmospheric dynamics, and fluid mechanics and can explain almost all kinds of physical phenomena of waves, while explicit solutions are studied in nonlinear mechanics and optics. We investigate explicit solutions for the semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg-de Vries (cmKdV) equations, which describe sound waves in an inharmonic lattice. A semi-discrete four-component Frobenius-coupled cmKdV equation is presented by utilizing a zero-curvature equation. We use two different reduction methods to get a semi-discrete local Frobenius-coupled cmKdV equation and a semi-discrete nonlocal Frobenius-coupled cmKdV equation. In the paper, we study the above two new types of semi-discrete Frobenius-coupled cmKdV equations. Then the bi-Hamiltonian structures of the above two types of equations are constructed, which show that the two types of equations possess Liouville integrability. The covariant properties of the both new equations are shown by constructing their corresponding Darboux transformation (DT), respectively. Soliton solutions, semi-rational soliton solutions, and rogue wave solutions of the above two types of equations are constructed by \(N\)-fold DT and generalized \((n, N - n)\)-fold DT. Three-dimensional plots and density profiles of these explicit solutions are presented. The characteristics of these figures demonstrate the relationship between the above two types of equations.Bifurcations, exact peakon, periodic peakons and solitary wave solutions of the cubic Camassa-Holm type equationhttps://zbmath.org/1541.354432024-09-27T17:47:02.548271Z"Zhou, Yuqian"https://zbmath.org/authors/?q=ai:zhou.yuqian"Chen, Guanrong"https://zbmath.org/authors/?q=ai:chen.guanrong"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin(no abstract)Modulation instability and modulated wave patterns in a nonlinear electrical transmission line with the next-nearest-neighbor couplinghttps://zbmath.org/1541.354442024-09-27T17:47:02.548271Z"Abbagari, Souleymanou"https://zbmath.org/authors/?q=ai:abbagari.souleymanou"Houwe, Alphonse"https://zbmath.org/authors/?q=ai:houwe.alphonse"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre"Doka, Serge Yamigno"https://zbmath.org/authors/?q=ai:doka.serge-yamignoSummary: In this paper, we investigate modulation instability and solitary waves in a one-dimensional nonlinear electrical transmission line with second-neighbor interactions. Through the quasi-discrete multiple scale method, we derive coupled nonlinear Schrödinger equations. To investigate the modulation instability in the structure, we use the linear stability analysis, and an expression for the modulation instability spectrum is derived. From the analytical investigation, we show that the second neighbor coupling affects both the modulation instability growth rate and modulation instability bands. Furthermore, we show via single and coupled mode excitation that bright and dark solitary waves can propagate at the lower and upper cutoff frequencies. To evaluate the robustness of the analytical investigation, we use a direct numerical simulation of the continuous wave. We conclude that the second-neighbor couplings generate rogue waves in the structure during the long-time evolution of the plane wave. We also demonstrate that with a variation of the excitation wave number, the nonlinear electrical transmission line can support new objects as wave molecules for high values of the wave number. This feature is not yet observed in nonlinear electrical transmission lines and will be useful in many fields of physics.Investigation of solitons in magneto-optic waveguides with Kudryashov's law nonlinear refractive index for coupled system of generalized nonlinear Schrödinger's equations using modified extended mapping methodhttps://zbmath.org/1541.354452024-09-27T17:47:02.548271Z"Ahmed, Karim K."https://zbmath.org/authors/?q=ai:ahmed.karim-k"Badra, Niveen M."https://zbmath.org/authors/?q=ai:badra.niveen-m"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-m"Rabie, Wafaa B."https://zbmath.org/authors/?q=ai:rabie.wafaa-b"Mirzazadeh, Mohammad"https://zbmath.org/authors/?q=ai:mirzazadeh.mohammad"Eslami, Mostafa"https://zbmath.org/authors/?q=ai:eslami.mostafa"Hashemi, Mir Sajjad"https://zbmath.org/authors/?q=ai:hashemi.mir-sajjadSummary: In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov's law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions.Parametric effects on paraxial nonlinear Schrödinger equation in Kerr mediahttps://zbmath.org/1541.354462024-09-27T17:47:02.548271Z"Arafat, S. M. Yiasir"https://zbmath.org/authors/?q=ai:arafat.s-m-yiasir"Khan, Kamruzzaman"https://zbmath.org/authors/?q=ai:khan.kamruzzaman"Islam, S. M. Rayhanul"https://zbmath.org/authors/?q=ai:islam.s-m-rayhanul"Rahman, M. M."https://zbmath.org/authors/?q=ai:rahman.md-mizanur.1|rahman.mohammad-mamunur|rahman.md-m|rahman.m-m|rahman.md-motlubar|rahman.md-mahmudur|rahman.m-muhammad-mahboob-ur|rahman.md-mashiar|rahman.m-mahbubur|rahman.mohammad-mahabubur|rahman.m-mahboob-ur|rahman.mohammad-mafizur|rahman.m-m-hafizur|rahman.mohammad-mansur|rahman.md-mijanurSummary: In this study, we have considered the (2+1)-dimensional paraxial nonlinear Schrödinger (NLS) equation in Kerr media and used the \((w/g)\)-expansion method. The \(g^\prime\) and \((g^\prime/g^2)\)-expansion techniques have been customized from the \((w/g)\)-expansion method. We applied these two techniques to the paraxial NLS equation and found the optical soliton solutions. The optical soliton solutions are attained as the flat kink, kink, singular kink, peakon, anti-parabolic, W-shape, M-shape, bell, and periodic wave solitons in terms of free parameters. We have presented three-dimensional (3D), two-dimensional (2D) and contour plots of the obtained results and discussed the effect of the free parameters and nonlinearity of the equation by determining different parametric values, which have not been discussed in the previous literature. We have studied the impact of the Kerr nonlinearity and wavenumber on the travelling wave solutions. Moreover, we also analyze the streamlines pattern and instantaneous local directions of the wave profile. All wave phenomena are applied to signal transmission, magneto-acoustic waves in plasma, optical fiber art, coastal engineering, quantum mechanics, hydro-magnetic waves, nonlinear optics and so on. The achieved solutions prove that the proposed methods are very powerful and effective for modern science and engineering for scrutinizing nonlinear evolutionary equations.\(H^s\) bounds for the derivative nonlinear Schrödinger equationhttps://zbmath.org/1541.354472024-09-27T17:47:02.548271Z"Bahouri, Hajer"https://zbmath.org/authors/?q=ai:bahouri.hajer"Leslie, Trevor M."https://zbmath.org/authors/?q=ai:leslie.trevor-m"Perelman, Galina"https://zbmath.org/authors/?q=ai:perelman.galinaSummary: We study the derivative nonlinear Schrödinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domainshttps://zbmath.org/1541.354482024-09-27T17:47:02.548271Z"Bégout, Pascal"https://zbmath.org/authors/?q=ai:begout.pascal"Díaz, Jesús Ildefonso"https://zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonsoSummary: We consider the damped nonlinear Schrödinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term \(V(x) u\), a nonlinear term of the form \(\operatorname{i}\mu u(t, x) / | u(t, x) |\) for a given parameter \(\mu > 0\) (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term \(f(t, x)\). The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikodým property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the \(L^\infty\)-norm of the forcing term \(f(t, x)\) becomes less than \(\mu\) after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} \(v\) (here \(v = - \operatorname{i}\mu u / | u | + f\)).Averaging for the \(2d\) dispersion-managed NLShttps://zbmath.org/1541.354492024-09-27T17:47:02.548271Z"Campos, Luccas"https://zbmath.org/authors/?q=ai:campos.luccas"Murphy, Jason"https://zbmath.org/authors/?q=ai:murphy.jason"Van Hoose, Tim"https://zbmath.org/authors/?q=ai:van-hoose.timSummary: We establish global-in-time averaging for the \(L^2\)-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in \(L^2\).The cubic nonlinear fractional Schrödinger equation on the half-linehttps://zbmath.org/1541.354502024-09-27T17:47:02.548271Z"Cavalcante, Márcio"https://zbmath.org/authors/?q=ai:cavalcante.marcio-andre-araujo"Huaroto, Gerardo"https://zbmath.org/authors/?q=ai:huaroto.gerardoSummary: We study the cubic nonlinear fractional Schrödinger equation with Lévy indices \(\frac{4}{3} < \alpha < 2\) posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line \(\mathbb{R}\). Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann-Liouville fractional operator combined with Fourier restriction method of \textit{J. Bourgain} [Geom. Funct. Anal. 3, No. 3, 209--262 (1993; Zbl 0787.35098)] and some ideas of the recent work of \textit{M. B. Erdoğan} et al. [Indiana Univ. Math. J. 68, No. 2, 369--3952 (2019; Zbl 1416.35291)]. The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line \(\mathbb{R}\) for the case of the low regularity assumption, which was point out at the recent work [Erdoğan et al., loc. cit.].A weakly turbulent solution to the cubic nonlinear harmonic oscillator on \(\mathbb{R}^2\) perturbed by a real smooth potential decaying to zero at infinityhttps://zbmath.org/1541.354512024-09-27T17:47:02.548271Z"Chabert, Ambre"https://zbmath.org/authors/?q=ai:chabert.ambreSummary: We build a smooth real potential \(V(t, x)\) on \(( t_0,+\infty)\times \mathbb{R}^2\) decaying to zero as \(t\to\infty\) and a smooth solution to the associated perturbed cubic noninear harmonic oscillator whose Sobolev norms blow up logarithmically as \(t\to\infty \). Adapting the method of Faou and Raphael for the linear case, we modulate the Solitons associated to the nonlinear harmonic oscillator by time-dependent parameters solving a quasi-Hamiltonian dynamical system whose action grows up logarithmically, thus yielding logarithmic growth for the Sobolev norm of the solution.On a class of quasilinear Schrödinger equations with superlinear termshttps://zbmath.org/1541.354522024-09-27T17:47:02.548271Z"Cheng, Yongkuan"https://zbmath.org/authors/?q=ai:cheng.yongkuan"Shen, Yaotian"https://zbmath.org/authors/?q=ai:shen.yaotianSummary: In this paper, we consider a class of quasilinear Schrödinger equations arising from a model of a self-trapped electrons in quadratic or hexagonal lattices. By variational methods, we prove that this problem admits a positive solution for any positive parameter.
{\copyright 2024 American Institute of Physics}Well-posedness for the Schrödinger-KdV system on the half-linehttps://zbmath.org/1541.354532024-09-27T17:47:02.548271Z"Compaan, E."https://zbmath.org/authors/?q=ai:compaan.erin"Shin, W."https://zbmath.org/authors/?q=ai:shin.wonmin|shin.w-m|shin.won-yong|shin.woongjae|shin.wongyu|shin.wangseok|shin.wooyoung|shin.wan-seon|shin.woochang|shin.wonseok|shin.wiroy|shin.wonjae|shin.weon|shin.wook"Tzirakis, N."https://zbmath.org/authors/?q=ai:tzirakis.nikolaosSummary: In this paper we obtain improved local well-posedness results for the Schrödinger-KdV system on the half-line. We employ the Laplace-Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order to accommodate the bounded operators of the half-line problem. Our result extends the previous local results in
[\textit{M. Cavalcante} and \textit{C. Kwak}, NoDEA, Nonlinear Differ. Equ. Appl. 27, No. 5, Paper No. 45, 50 p. (2020; Zbl 1448.35440)],
[\textit{M. Cavalcante} and \textit{A. J. Corcho}, J. Evol. Equ. 20, No. 4, 1563--1596 (2020; Zbl 1466.35312)]
and
[\textit{A. A. Himonas} and \textit{F. Yan}, Appl. Numer. Math. 199, 32--58 (2024; Zbl 07856321)]
matching the results that \textit{Y. Wu} [Differ. Integral Equ. 23, No. 5--6, 569--600 (2010; Zbl 1240.35528)] obtained for the real line system. We also demonstrate the uniqueness for the full range of locally well-posed solutions. In addition we obtain global well-posedness on the half-line for the energy solutions with zero boundary data, along with polynomial-in-time bounds for higher order Sobolev norms for the Schrödinger part.Scattering, random phase and wave turbulencehttps://zbmath.org/1541.354542024-09-27T17:47:02.548271Z"Faou, Erwan"https://zbmath.org/authors/?q=ai:faou.erwan"Mouzard, Antoine"https://zbmath.org/authors/?q=ai:mouzard.antoineSummary: We start from the remark that in wave turbulence theory, exemplified by the cubic two-dimensional Schrödinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.Standing waves and global well-posedness for the 2d Hartree equation with a point interactionhttps://zbmath.org/1541.354552024-09-27T17:47:02.548271Z"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Michelangeli, Alessandro"https://zbmath.org/authors/?q=ai:michelangeli.alessandro"Scandone, Raffaele"https://zbmath.org/authors/?q=ai:scandone.raffaeleSummary: We study a class of two-dimensional nonlinear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a nontrivial adaptation of the standard properties of Schwartz symmetrization for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.Global dynamics below a threshold for the nonlinear Schrödinger equations with the Kirchhoff boundary and the repulsive Dirac delta boundary on a star graphhttps://zbmath.org/1541.354562024-09-27T17:47:02.548271Z"Hamano, Masaru"https://zbmath.org/authors/?q=ai:hamano.masaru"Ikeda, Masahiro"https://zbmath.org/authors/?q=ai:ikeda.masahiro"Inui, Takahisa"https://zbmath.org/authors/?q=ai:inui.takahisa"Shimizu, Ikkei"https://zbmath.org/authors/?q=ai:shimizu.ikkeiSummary: We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetric decomposition.Periodic peakons and compacton families of the Hirota-type peakon equationhttps://zbmath.org/1541.354572024-09-27T17:47:02.548271Z"Han, Maoan"https://zbmath.org/authors/?q=ai:han.maoan"Chen, Guanrong"https://zbmath.org/authors/?q=ai:chen.guanrong"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin(no abstract)The large time asymptotic solutions of nonlinear Schrödinger type equationshttps://zbmath.org/1541.354582024-09-27T17:47:02.548271Z"Liu, Baoping"https://zbmath.org/authors/?q=ai:liu.baoping"Soffer, Avy"https://zbmath.org/authors/?q=ai:soffer.avrahamSummary: We give a short description of the proof of asymptotic-completeness for NLS-type equations with radial data in three dimensions. We also show some aspects of the method by giving a new proof of Asymptotic Completeness for the two-body Quantum Scattering case.Modulation instability and collision dynamics of solitons in birefringence optical fibershttps://zbmath.org/1541.354592024-09-27T17:47:02.548271Z"Liu, Fei-Fei"https://zbmath.org/authors/?q=ai:liu.feifei"Lü, Xing"https://zbmath.org/authors/?q=ai:lu.xing"Wang, Jian-Ping"https://zbmath.org/authors/?q=ai:wang.jianping.1"Zhou, Xian-Wei"https://zbmath.org/authors/?q=ai:zhou.xianweiThe authors investigate soliton modulation instability and collision dynamics in birefringence optical fibers. The governing equations are coupled hybrid nonlinear Schrödinger equations. In addition to discussion of modulation instability, the authors derive three-soliton solutions of the nonlinear Schrödinger equations, and the asymptotic behavior (before and after soliton collisions) of solutions is provided. During three-soliton inelastic collisions, shape restoration of one or two solitons is possible.
Reviewer: Eric Stachura (Marietta)On the non-integrable discrete focusing Hirota equation: spatial properties, discrete solitons and stability analysishttps://zbmath.org/1541.354602024-09-27T17:47:02.548271Z"Ma, Liyuan"https://zbmath.org/authors/?q=ai:ma.liyuan"Song, Haifang"https://zbmath.org/authors/?q=ai:song.haifang"Jiang, Qiuyue"https://zbmath.org/authors/?q=ai:jiang.qiuyue"Shen, Shoufeng"https://zbmath.org/authors/?q=ai:shen.shoufengSummary: This paper focuses on various properties of the non-integrable discrete focusing Hirota (\(\mathrm{Hirota}^+\)) equation, encompassing spatial structure, discrete solitons, and linear stability analysis. Through a planar nonlinear discrete dynamical map method, we construct the spatially periodic solutions of the non-integrable discrete stationary \(\mathrm{Hirota}^+\) equation under special conditions. From the area-preserving property of the two-dimensional map, the types of fixed points are classified based on the defined residue, which depends on the feature of the linearized map. It is emphasized that there is a great difference between the periodic solution of the non-integrable discrete focusing Hirota equation and that of the non-integrable discrete defocusing Hirota equation. We numerically analyze the influence of the distinct parameters and the initial points on general orbits of the map. In addition, the comparison between spatial properties of the non-integrable discrete focusing Hirota equation and that of the non-integrable discrete focusing nonlinear Schrödinger (\(\mathrm{NLS}^+\)) equation suggests that the former has more plentiful properties. It is worth mentioning that the more general period-2 solutions of the non-integrable discrete stationary \(\mathrm{Hirota}^+\) equation and the period-3 solutions of the non-integrable discrete stationary \(\mathrm{NLS}^+\) equation in specific situations are obtained for the first time. On the other hand, we also explore the stationary solitons and traveling wave solutions of the non-integrable discrete \(\mathrm{Hirota}^+\) equation using the discrete Fourier transformation and the Neumann iteration scheme. The effects of the parameters and the initial values on the shapes of the solitons are numerically investigated. It is revealed that the traveling solitons depend sensitively on both the parameters and the initial values. Finally, we elaborate the linear stability of the stationary solitary waves under small perturbation. Meanwhile, the corresponding results of the non-integrable discrete \(\mathrm{NLS}^+\) equation are compared numerically.Stability theory for two-lobe states on the tadpole graph for the NLS equationhttps://zbmath.org/1541.354612024-09-27T17:47:02.548271Z"Pava, Jaime Angulo"https://zbmath.org/authors/?q=ai:angulo-pava.jaimeSummary: The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering \(\delta\)-type boundary conditions at the junction and bound states with a positive two-lobe profile, the main novelty of this paper is at least twofold. Via a splitting eigenvalue method developed by the author, we identify the Morse index and the nullity index of a specific linearized operator around of an \textit{a priori} positive two-lobe state profile for every positive power; and we also obtain new results about the existence and the orbital stability of positive two-lobe states at least in the cubic NLS case. To our knowledge, the results contained in this paper are the first in studying positive bound states for the NLS on a tadpole graph by non-variational techniques. In particular, our approach has prospect of being extended to study stability properties of other bound states for the NLS on a tadpole graph or on other non-compact metric graph such as a looping edge graph, as well as, for other nonlinear evolution models on a tadpole graph.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On the existence of certain elliptic solutions of the cubically nonlinear Schrödinger equationhttps://zbmath.org/1541.354622024-09-27T17:47:02.548271Z"Schürmann, H. W."https://zbmath.org/authors/?q=ai:schurmann.hans-werner"Serov, V. S."https://zbmath.org/authors/?q=ai:serov.valery-sSummary: We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form \(\Psi(t,z)=(f(t,z)+id(z))e^{i\phi(z)}\) with \(f,\phi,d\in\mathbb{R} \), we prove that they are nonexistent in the general case \(f_z\neq 0\), \(f_t\neq 0\), \(d_z\neq 0\). In the three nongeneric cases \((f_z\neq 0)\), \((f_t\neq 0, f_t=0, d_z=0)\), and \((f_z=0, f_t\neq 0)\), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions.Deep learning soliton dynamics and complex potentials recognition for 1D and 2D \(\mathcal{PT}\)-symmetric saturable nonlinear Schrödinger equationshttps://zbmath.org/1541.354632024-09-27T17:47:02.548271Z"Song, Jin"https://zbmath.org/authors/?q=ai:song.jin"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenyaSummary: In this paper, we firstly extend the physics-informed neural networks (PINNs) to learn data-driven stationary and non-stationary solitons of 1D and 2D saturable nonlinear Schrödinger equations (SNLSEs) with two fundamental \(\mathcal{PT}\)-symmetric Scarf-II and periodic potentials in optical fibers. Secondly, the data-driven inverse problems are studied for \(\mathcal{PT}\)-symmetric potential functions discovery rather than just potential parameters in the 1D and 2D SNLSEs. Particularly, we propose a modified PINNs (mPINNs) scheme to identify directly the \(\mathcal{PT}\) potential functions of the 1D and 2D SNLSEs by the solution data. And the inverse problems about 1D and 2D \(\mathcal{PT}\)-symmetric potentials depending on propagation distance \(z\) are also investigated using mPINNs method. We also identify the potential functions by the PINNs applied to the stationary equation of the SNLSE. Furthermore, two network structures are compared under different parameter conditions such that the predicted \(\mathcal{PT}\) potentials can achieve the similar high accuracy. These results illustrate that the established deep neural networks can be successfully used in 1D and 2D SNLSEs with high accuracies. Moreover, some main factors affecting neural networks performance are discussed in 1D and 2D \(\mathcal{PT}\) Scarf-II and periodic potentials, including activation functions, structures of the networks, and sizes of the training data. In particular, twelve different nonlinear activation functions are in detail analyzed containing the periodic and non-periodic functions such that it is concluded that selecting activation functions according to the form of solution and equation usually can achieve better effect.Exact solutions and dynamics in Schrödinger-Hirota model having multiplicative white noise via Itô calculushttps://zbmath.org/1541.354642024-09-27T17:47:02.548271Z"Sun, Xianbo"https://zbmath.org/authors/?q=ai:sun.xianbo|sun.xianbo.1"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin(no abstract)Multiple pole solutions of the Hirota equation under nonzero boundary conditions by inverse scattering methodhttps://zbmath.org/1541.354652024-09-27T17:47:02.548271Z"Wang, Guixian"https://zbmath.org/authors/?q=ai:wang.guixian"Wang, Xiu-Bin"https://zbmath.org/authors/?q=ai:wang.xiubin"Han, Bo"https://zbmath.org/authors/?q=ai:han.boSummary: In this paper, we study multiple pole solutions for the focusing Hirota equation under the nonzero boundary conditions via inverse scattering method. The direct scattering problem is based on the spectral analysis and exhibits the Jost solutions, scattering matrix as well as their analyticity, symmetries and asymptotic behaviors. Compared with previous studies, we define a more complex discrete spectrum. The inverse scattering problem is explored by solving the corresponding matrix Riemann-Hilbert problems. Particularly, we solve the scattering problem by a suitable uniformization variable on the complex \(z\)-plane instead of a two-sheeted Riemann surface. Finally, we deduce general formulas of \(N\)-double pole and \(N\)-triple pole solutions with mixed discrete spectra and show some prominent characteristics of these solutions graphically. Our results should be helpful to further explore and enrich breather wave phenomena arising in nonlinear and complex systems.Multi-breathers and higher-order rogue waves on the periodic background in a fourth-order integrable nonlinear Schrödinger equationhttps://zbmath.org/1541.354662024-09-27T17:47:02.548271Z"Wei, Yun-Chun"https://zbmath.org/authors/?q=ai:wei.yun-chun"Zhang, Hai-Qiang"https://zbmath.org/authors/?q=ai:zhang.haiqiang"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiuSummary: In this paper, we present a systematic formulation of multi-breathers and higher-order rogue wave solutions of a fourth-order nonlinear Schrödinger equation on the periodic background. First of all, we compute a complete family of elliptic solution of this higher-order equation, which can degenerate into two particular cases, i.e., the dnoidal and cnoidal solutions. By using the modified squared wavefunction approach, we solve the spectral problem on the elliptic function background. Then, we derive multi-breather solutions in terms of the theta functions, particular examples of which are the Kuznetsov-Ma breather and the Akhmediev breather. Furthermore, taking the limit of the breather solutions at branch points, we construct higher-order rogue wave solutions by employing a generalized Darboux transformation technique. On the periodic background, we present the first-order, second-order and second-second-order rogue waves. With aid of the theta functions, we explicitly characterize the resulting breathers and rogue waves, and demonstrate their dynamic behaviors by illustrative examples. Finally, we discuss how the parameter of the higher-order effects affects the breathers and rogue waves.Global well-posedness for the fourth-order Hartree-type Schrödinger equation with Cauchy data in \(L^p\)https://zbmath.org/1541.354672024-09-27T17:47:02.548271Z"Xie, Jin"https://zbmath.org/authors/?q=ai:xie.jin.1"Wang, Deng"https://zbmath.org/authors/?q=ai:wang.deng"Yang, Han"https://zbmath.org/authors/?q=ai:yang.han.1Summary: This paper is concerned with the Cauchy problem of the nonlinear fourth-order Schrödinger equation on \(\mathbb{R}^n\), with the nonlinearity of Hartree-type \((|\cdot|^{-\gamma}\ast |u|^2)u\). The existence of a global solution to the Cauchy problem is established when initial data belongs to \(L^p\) (\(p<2\)). Moreover, the solution is given by using a data-decomposition argument, two kinds of generalized Strichartz estimates, and the interpolation theorem.Classification and soliton for a generalized fourth-order dispersive nonlinear Schrödinger equation in a Heisenberg spin chainhttps://zbmath.org/1541.354682024-09-27T17:47:02.548271Z"Yang, Deniu"https://zbmath.org/authors/?q=ai:yang.deniu(no abstract)On the convergence of the solution for a reduced model of the vectorial quantum Zakharov systemhttps://zbmath.org/1541.354692024-09-27T17:47:02.548271Z"Yang, Guiyu"https://zbmath.org/authors/?q=ai:yang.guiyu"Zhang, Jingjun"https://zbmath.org/authors/?q=ai:zhang.jingjun"Jiang, Zaihong"https://zbmath.org/authors/?q=ai:jiang.zaihongSummary: In this paper, we study the limit behavior of the smooth solution for a reduced vectorial quantum Zakharov system which describes the interaction between the quantum Langmuir waves and quantum ion-acoustic waves in the plasmas. We first give the local existence and uniqueness of the solution to the quantum Zakharov system. Then we derive the uniform bounds of solution with appropriate initial data, and prove that the solution of the quantum Zakharov system converges to the solution of the nonlinear Schrödinger system.
{\copyright 2024 American Institute of Physics}Fractional-order effect on soliton wave conversion and stability for the two-Lévy-index fractional nonlinear Schrödinger equation with PT-symmetric potentialhttps://zbmath.org/1541.354702024-09-27T17:47:02.548271Z"Yu, Fajun"https://zbmath.org/authors/?q=ai:yu.fajun"Li, Li"https://zbmath.org/authors/?q=ai:li.li"Zhang, Jiefang"https://zbmath.org/authors/?q=ai:zhang.jiefang"Yan, Jingwen"https://zbmath.org/authors/?q=ai:yan.jingwenSummary: We investigate a variable-coefficient fractional nonlinear Schrödinger(vc-FNLS) equation with Wadati potential and PT-symmetric potential. We find the Lévy index can be used to transition from a breather wave to a soliton as the fractional order derivative is increasing. The influences of fractional \(\alpha\) and \(\beta\) on the breather, dark and bright solitons of space-time vc-FNLS equation are analyzed in detail. Some different propagation dynamics are generated via the different parameters \(\alpha\) and \(\beta\). We find that the value of \(\alpha\) decreases, the number of oscillations or singularities increases for small time values and decreases for large time values. And the stability of fundamental soliton and asymmetric soliton is explored via the linear stability analysis and direct propagations. Interestingly stability of the soliton is also explored against collisions with Wadati potential and PT-symmetric potential. To additionally explore robustness of the solitons in the vc-FNLS model, it is also relevant to show that they keep its integrity against ``bombardment'' by impinging waves. These states indicate that the interaction acts both repulsively and attractively, and the analysis can be extended for vc-FNLS equations with more than two different fractional-diffraction terms. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials.Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensatehttps://zbmath.org/1541.354712024-09-27T17:47:02.548271Z"Zhang, C.-R."https://zbmath.org/authors/?q=ai:zhang.chunrui.1|zhang.chenrui|zhang.changrong|zhang.chaorui|zhang.chen-rong|zhang.chengzhao-richard|zhang.chengrui|zhang.changrui|zhang.cairong|zhang.chuanrong|zhang.chunrui|zhang.chunru|zhang.chaoran|zhang.canrong"Tian, B."https://zbmath.org/authors/?q=ai:tian.bowen|tian.baochuan|tian.bao|tian.bo|tian.baoyu|tian.baoxian|tian.bailing|tian.baoyuang|tian.baodan|tian.baoping|tian.bin|tian.beiyi|tian.boshi|tian.beihang|tian.boyu|tian.baoguo|tian.bing|tian.baoliang|tian.baolin|tian.baoguang|tian.boping|tian.baofeng|tian.baijun|tian.binbin|tian.beping"Qu, Q.-X."https://zbmath.org/authors/?q=ai:qu.qiuxia|qu.qixing"Yuan, Y.-Q."https://zbmath.org/authors/?q=ai:yuan.yu-qiang|yuan.yeqing|yuan.yunqiang|yuan.yin-quan|yuan.yu-quan"Wei, C.-C."https://zbmath.org/authors/?q=ai:wei.chongchong|wei.cheng-cheng|wei.changcheng|wei.chin-chung|wei.chia-chen|wei.chiu-chiSummary: The Bose-Einstein-condensation applications give rise to the superfluidity in the liquid helium and superconductivity in the metals. In this paper, we work on a three-component Gross-Pitaevskii system, which describes the matter waves in an spin-1 spinor Bose-Einstein condensate. We construct a multi-fold binary Darboux transformation with the zero seed solutions to describe the three vertical spin projection of the spin-1 spinor BEC, which is different from all the existing Darboux-type ones for the same system, and derive three types of the exponential-and-rational mixed soliton solutions associated with two conjugate complex eigenvalues. For such mixed solitons, we give their asymptotic expressions, indicating that they consist of the Ieda-Miyakawa-Wadati (IMW)-polar-state or IMW-ferromagnetic solitons but possess the time-dependent velocities. Asymptotically and graphically, interaction mechanisms between the mixed and exponential solitons are classified in six cases, and we exhibit the inelastic and elastic interactions through calculating the modifications of the polarization matrices and phase shifts for the two interacting solitons. We find that both the IMW-polar-state solitons, including the mixed and exponential solitons, can not alter the other soliton's intensity distribution during the interaction, while the mixed or exponential soliton in the IMW-ferromagnetic state does.Analysis on compact symmetric spaces: eigenfunctions and nonlinear Schrödinger equationshttps://zbmath.org/1541.354722024-09-27T17:47:02.548271Z"Zhang, Yunfeng"https://zbmath.org/authors/?q=ai:zhang.yunfengSummary: We discuss several open problems on harmonic analysis on compact globally symmetric spaces, and their applications towards nonlinear Schrödinger equations.
For the entire collection see [Zbl 1537.35003].Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitonshttps://zbmath.org/1541.354732024-09-27T17:47:02.548271Z"Zhou, Zijian"https://zbmath.org/authors/?q=ai:zhou.zijian"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.10"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenyaSummary: In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev-Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.The formation of invariant exact optical soliton solutions of Landau-Ginzburg-Higgs equation via Khater analytical approachhttps://zbmath.org/1541.354742024-09-27T17:47:02.548271Z"Faridi, Waqas Ali"https://zbmath.org/authors/?q=ai:faridi.waqas-ali"AlQahtani, Salman A."https://zbmath.org/authors/?q=ai:al-qahtani.salman-aSummary: This work aims to enhance our comprehension of the dynamical features of the nonlinear Landau-Ginzburg-Higgs evolution equation, which provides a theoretical framework for identifying various phenomena, such as the formation of superconducting states and the spontaneous breakdown of symmetries. When symmetry breaking is involved in phase transitions in particle physics or condensed matter systems, the Landau-Ginzburg-Higgs model combines the ideas of the Landau-Ginzburg theory and the Higgs mechanism. The equation plays a crucial role in characterizing the Higgs field and its related particles, including Higgs boson. In a standard model of the particle physics, Higgs mechanism explains precisely how mass is acquired. The Lie invariance requirements are taken into account by the symmetry generators. The method produces a 3-dimensional Lie algebra of the Landau-Ginzburg-Higgs model with translational symmetry (dilation or scaling) and translations in the space and the time associated with the mass and energy conservation. It is shown to be the optimal sub-algebraic system after similarity reductions are also performed. The next wave transformation method reduces the governing system to ordinary differential equations and yields a large number of exact travelling wave solutions. The Khater approach is used to solve an ordinary differential equation and investigate the closed-form analytical travelling wave solutions for the considered diffusive system. The obtained results include a singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solution. The results of the sensitivity analysis demonstrate how vulnerable the suggested equation is to various initial conditions. The findings are visually displayed in contour, three-dimensional, and two-dimensional forms to emphasize the features of pulse propagation.Analytic solutions of 2D cubic quintic complex Ginzburg-Landau equationhttps://zbmath.org/1541.354752024-09-27T17:47:02.548271Z"Tchuimmo, F. Waffo"https://zbmath.org/authors/?q=ai:tchuimmo.f-waffo"Tafo, J. B. Gonpe"https://zbmath.org/authors/?q=ai:tafo.j-b-gonpe"Chamgoue, A."https://zbmath.org/authors/?q=ai:chamgoue.andre-cheage"Mezamo, N. C. Tsague"https://zbmath.org/authors/?q=ai:mezamo.n-c-tsague"Kenmogne, F."https://zbmath.org/authors/?q=ai:kenmogne.fabien"Nana, L."https://zbmath.org/authors/?q=ai:nana.laurentSummary: The dynamical behaviour of traveling waves in a class of two-dimensional system whose amplitude obeys the two-dimensional complex cubic-quintic Ginzburg-Landau equation is deeply studied as a function of parameters near a subcritical bifurcation. Then, the bifurcation method is used to predict the nature of solutions of the considered wave equation. It is applied to reduce the two-dimensional complex cubic-quintic Ginzburg-Landau equation to the quintic nonlinear ordinary differential equation, easily solvable. Under some constraints of parameters, equilibrium points are obtained and phase portraits have been plotted. The particularity of these phase portraits obtained for new ordinary differential equation is the existence of homoclinic or heteroclinic orbits depending on the nature of equilibrium points. For some parameters, one has the orbits starting to one fixed point and passing through another fixed point before returning to the same fixed point, predicting then the existence of the combination of a pair of pulse-dark soliton. One has also for other parameters, the orbits linking three equilibrium points predicting the existence of a dark soliton pair. These results are very important and can predict the same solutions in many domains, particularly in wave phenomena, mechanical systems, or laterally heated fluid layers. Moreover, depending on the values of parameter systems, the analytical expression of the solutions predicted is found. The three-dimensional graphs of these solutions are plotted as well as their 2D plots in the propagation direction.On the blow-up of the solution of a \((1+1)\)-dimensional thermal-electrical modelhttps://zbmath.org/1541.354762024-09-27T17:47:02.548271Z"Artemeva, M. V."https://zbmath.org/authors/?q=ai:artemeva.magarita-vitalevna"Korpusov, M. O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovichSummary: We consider a \((1+1)\)-dimensional thermal-electrical model of semiconductor heating in an electric field. For the corresponding initial-boundary value problem, we prove the existence of a classical solution that cannot be continued in time and obtain sufficient conditions for the blow-up of the solution in a finite time.Long time gyrokinetic equationshttps://zbmath.org/1541.354772024-09-27T17:47:02.548271Z"Cheverry, Christophe"https://zbmath.org/authors/?q=ai:cheverry.christophe"Farhat, Shahnaz"https://zbmath.org/authors/?q=ai:farhat.shahnazSummary: The aim of this text is to elucidate the oscillating patterns
(see [\textit{C. Cheverry}, ``Mathematical perspectives in plasma turbulence'', Res. Rep. Math. 2, No. 2 (2018), see also \url{HAL:hal-01617652}])
which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency \(\varepsilon^{-1}\gg 1\). By exploiting the properties of underlying integrable systems, we can complement the KAM picture
(see [\textit{G. Benettin} and \textit{P. Sempio}, Nonlinearity 7, No. 1, 281--303 (1994; Zbl 0856.70010);
\textit{M. Braun}, SIAM Rev. 23, 61--93 (1981; Zbl 0479.76128)])
and go beyond the classical results about gyrokinetics
(see [\textit{M. Bostan}, Multiscale Model. Simul. 8, No. 5, 1923--1957 (2010; Zbl 1220.35176);
\textit{A. J. Brizard} and \textit{T. S. Hahm}, Rev. Mod. Phys. 79, No. 2, 421--468 (2007; Zbl 1205.76309)]).
The purely magnetic situation was addressed by
\textit{C. Cheverry} [Commun. Math. Phys. 338, No. 2, 641--703 (2015; Zbl 1333.35290); J. Differ. Equations 262, No. 3, 2987--3033 (2017; Zbl 1358.35194)].
We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field.Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regimehttps://zbmath.org/1541.354782024-09-27T17:47:02.548271Z"Di Fratta, Giovanni"https://zbmath.org/authors/?q=ai:di-fratta.giovanni"Slastikov, Valeriy V."https://zbmath.org/authors/?q=ai:slastikov.valeriy-v"Zarnescu, Arghir D."https://zbmath.org/authors/?q=ai:zarnescu.arghir-daniSummary: The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields \(H^1 (S, T)\), where \(S\) and \(T\) are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface \(T\) is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.Invisibility enables super-visibility in electromagnetic imaginghttps://zbmath.org/1541.354792024-09-27T17:47:02.548271Z"He, Youzi"https://zbmath.org/authors/?q=ai:he.youzi"Li, Hongjie"https://zbmath.org/authors/?q=ai:li.hongjie"Liu, Hongyu"https://zbmath.org/authors/?q=ai:liu.hongyu"Wang, Xianchao"https://zbmath.org/authors/?q=ai:wang.xianchaoSummary: This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three phases. First, we determine the interior Maxwell transmission eigenvalues of the scatterer from a family of far-field data by the mechanism of the linear sampling method. Next, we determine the corresponding transmission eigenfunctions by solving a constrained optimization problem. Finally, based on both global and local geometric properties of the transmission eigenfunctions, we design an imaging functional which can be used to determine the shape of the medium scatterer. We provide rigorous theoretical basis for our method. Numerical experiments verify the effectiveness, better accuracy and super-resolution results of the proposed scheme.Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearityhttps://zbmath.org/1541.354802024-09-27T17:47:02.548271Z"Meng, Xin"https://zbmath.org/authors/?q=ai:meng.xin"Ji, Shuguan"https://zbmath.org/authors/?q=ai:ji.shuguanThe authors consider the radially symmetric curl-curl wave equation with double power nonlinearity:
\[
\rho(x) u_{tt}+\nabla \times (M(x) \nabla \times u) +\mu(x) u+v_p(x) |u|^{p-1}u +v_q(x) |u|^{q-1}u=0
\]
where \((x,t) \in \mathbb{R}^3 \times \mathbb{R}\). Here \(M: \mathbb{R}^3\to \mathbb{R}^{3\times 3}\) and \(\rho, \mu, v_p, v_q\) are positive, radially symmetric functions with \(1<p<q\). Breather solutions, i.e. classical solutions which are periodic in time and spatially exponentially localized, are considered. In particular, solutions having the form
\[
u(x,t)=y(|x|,t)\dfrac{x}{|x|}
\]
are constructed, where \(y\) is some \(C^2\) function such that \(y(0,t)=y''(0,t)=0\). The periodic behavior of the solution is characterized, and a solution is found which can generated a continuum of phase shifted breathers.
Reviewer: Eric Stachura (Marietta)Solution structures of an electrical transmission line model with bifurcation and chaos in Hamiltonian dynamicshttps://zbmath.org/1541.354812024-09-27T17:47:02.548271Z"Qi, Jianming"https://zbmath.org/authors/?q=ai:qi.jianming"Cui, Qinghua"https://zbmath.org/authors/?q=ai:cui.qinghua"Zhang, Le"https://zbmath.org/authors/?q=ai:zhang.le"Sun, Yiqun"https://zbmath.org/authors/?q=ai:sun.yiqun(no abstract)Dynamics of damped single valued magnetic wave in inhomogeneous circularly polarized ferriteshttps://zbmath.org/1541.354822024-09-27T17:47:02.548271Z"Tchokouansi, Hermann T."https://zbmath.org/authors/?q=ai:tchokouansi.hermann-t"Tchomgo Felenou, E."https://zbmath.org/authors/?q=ai:tchomgo-felenou.e"Kuetche, Victor K."https://zbmath.org/authors/?q=ai:kuetche.victor-kamgang"Tchidjo, Robert Tamwo"https://zbmath.org/authors/?q=ai:tchidjo.robert-tamwoSummary: In this short note, we focus our attention on the dynamics of magnetic wave in polarized ferrite. Indeed, we pay particular attention to the new nonlinear system derived by \textit{B. A. Kamdem} et al. in their recent paper [Phys. Scr. 96, No. 11, Article ID 115206 (2021; \url{doi:10.1088/1402-4896/ac12e6})] describing the propagation of polarized wave guide excitation in microwave ferrites, subjected to damping and inhomogeneous exchange effects simultaneously. We provide from the well-known inverse scattering transform method the soliton solution of the new system free of damping and inhomogeneous exchange effects, and using this solution as initial condition, we verify the consistency of its analytical expression via some numerical simulations. We use this solution to study numerically first of all the effect of damping and the effect of inhomogeneity separately before studying the simultaneous effects on the moving wave. As expected, damping acts on the wave by slowing down its amplitude while its impact on the width is negligible. The inhomogeneous exchange effect acts on the amplitude of the wave while creating a shift.Bifurcations and exact solutions of optical soliton models in fifth-order weakly nonlocal nonlinear mediahttps://zbmath.org/1541.354832024-09-27T17:47:02.548271Z"Wu, Rong"https://zbmath.org/authors/?q=ai:wu.rong.1|wu.rong"Chen, Guanrong"https://zbmath.org/authors/?q=ai:chen.guanrong"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin(no abstract)On electromagnetic wave equations for a nonhomegenous microperiodic mediumhttps://zbmath.org/1541.354842024-09-27T17:47:02.548271Z"Wojnar, Ryszard"https://zbmath.org/authors/?q=ai:wojnar.ryszardSummary: We consider the equations of the electromagnetic field in a heterogeneous microperiodic medium, using the representation of the field by the vector potential and the scalar potential. The wave equation for scalar potential is separated from the equation for vector potential, but not vice versa. Thus, the system of equations loses the beautiful symmetry known for the case of a homogeneous medium. The homogenized equations and the expressions for the effective material coefficients are given. A special case of Oersted's experiment in axially nonhomogeneous medium was considered more closely.
For the entire collection see [Zbl 1531.35008].A method of moments estimator for interacting particle systems and their mean field limithttps://zbmath.org/1541.354852024-09-27T17:47:02.548271Z"Pavliotis, Grigorios A."https://zbmath.org/authors/?q=ai:pavliotis.grigorios-a"Zanoni, Andrea"https://zbmath.org/authors/?q=ai:zanoni.andreaSummary: We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction, and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.New completeness theorems on the boundary in elasticityhttps://zbmath.org/1541.354862024-09-27T17:47:02.548271Z"Cialdea, A."https://zbmath.org/authors/?q=ai:cialdea.albertoSummary: The completeness on the boundary (in the sense of Picone) of certain systems related to the III and IV BVPs for the elasticity system is proved. The completeness is obtained in both \(L^p\) (\(1 \leqslant 1 < \infty\)) and uniform norms.Propagation and dispersion of Bloch waves in periodic media with soft inclusionshttps://zbmath.org/1541.354872024-09-27T17:47:02.548271Z"Godin, Yuri A."https://zbmath.org/authors/?q=ai:godin.yuri-a"Vainberg, Boris"https://zbmath.org/authors/?q=ai:vainberg.boris-rSummary: We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and low frequency cutoff are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.
{\copyright 2024 American Institute of Physics}Analogues the Kolosov-Muskhelishvili formulas for isotropic materials with double voidshttps://zbmath.org/1541.354882024-09-27T17:47:02.548271Z"Gulua, Bakur"https://zbmath.org/authors/?q=ai:gulua.bakurSummary: Analogues of the well-known Kolosov-Muskhelishvili formulas for homogeneous equations of statics in the case of elastic materials with double voids are obtained. It is shown that in this theory the displacement and stress vector components are represented by two analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium with double voids.
For the entire collection see [Zbl 1531.35008].Solution of the Kirsch problem for the elastic materials with voids in the case of approximation \(N=1\) of Vekua's theoryhttps://zbmath.org/1541.354892024-09-27T17:47:02.548271Z"Gulua, Bakur"https://zbmath.org/authors/?q=ai:gulua.bakur"Jangava, Roman"https://zbmath.org/authors/?q=ai:jangava.roman"Kasrashvili, Tamar"https://zbmath.org/authors/?q=ai:kasrashvili.tamar"Narmania, Miranda"https://zbmath.org/authors/?q=ai:narmania.mirandaSummary: In this paper we consider a boundary value problem for an infinite plate with a circular hole. The plate is the elastic material with voids. The hole is free from stresses, while unilateral tensile stresses act at infinity. The state of plate equilibrium is described by the system of differential equations that is derived from three-dimensional equations of equilibrium of an elastic material with voids (Cowin-Nunziato model) by Vekua's reduction method. its general solution is represented by means of analytic functions of a complex variable and solutions of Helmholtz equations. The problem is solved analytically by the method of the theory of functions of a complex variable.
For the entire collection see [Zbl 1531.35008].Investigation of the transient responses of a beam on an elastic polymeric foundationhttps://zbmath.org/1541.354902024-09-27T17:47:02.548271Z"Kashcheeva, Anna Dmitrievna"https://zbmath.org/authors/?q=ai:kashcheeva.anna-dmitrievna"Zamyshlyaeva, Alena Aleksandrovna"https://zbmath.org/authors/?q=ai:zamyshlyaeva.alena-aleksandrovnaSummary: The negative impact of vibrations on various devices and mechanisms can be significant, so it is important to take this factor into account when designing, operating and maintaining various equipment and engineering systems. Various methods and technologies can be used to protect against the negative effects of vibrations. Special damping materials are often used. This research paper is devoted to the analysis of the effectiveness of vibration reduction taking into account the physical parameters of elastic polymeric materials. To conduct the study, a mathematical model describing motion of the beam resting on an elastic polymeric foundation is constructed. The model is based on a system of nonlinear differential equations. An algorithm was developed and applied for the numerical solution of this system of equations. Numerical experiments were carried out for the study of the system reaction to different cases of accelerations. As a result, the deflection structure for materials with different physical characteristics were obtained. These results can serve as a starting point for a deeper study of materials and creation of more complex structures.Decay of solutions for second gradient viscoelasticity with type II heat conductionhttps://zbmath.org/1541.354912024-09-27T17:47:02.548271Z"Magaña, Antonio"https://zbmath.org/authors/?q=ai:magana.antonio"Quintanilla, Ramón"https://zbmath.org/authors/?q=ai:quintanilla.ramonSummary: In this work we analyse the time decay of solutions for the second gradient thermoelasticity when two dissipation mechanisms are introduced in the system. First we consider that the fourth spatial derivative of the velocity is present in the dissipation and prove that the decay of the solutions cannot be of exponential type. In fact, we show that the solutions decay polynomially, and in two different ways depending on how the variables are coupled. Second we consider only the Laplacian of the velocity in the dissipation and we prove that the solutions decay exponentially. The lack of analyticity of the semigroup is also shown. The mathematical tools we use are the Lumer-Phillips corollary to the Hille-Yosida theorem, the characterization of the exponentially stable semigroups provided by Huang-Prüss and the one corresponding to polynomially stable semigroups obtained by Borichev-Tomilov.A computational procedure for exact solutions of Burgers' hierarchy of nonlinear partial differential equationshttps://zbmath.org/1541.354922024-09-27T17:47:02.548271Z"Obaidullah, U."https://zbmath.org/authors/?q=ai:obaidullah.u"Jamal, Sameerah"https://zbmath.org/authors/?q=ai:jamal.sameerahSummary: This paper considers the exact solution of Burgers' hierarchy of nonlinear evolution equations. We construct the general \(n\)th conservation law of the hierarchy and prove that these expressions may be transformed into ordinary differential equations. In particular, a coordinate transformation leads to the systematic reduction of the conservation law properties of the Burgers' hierarchy. Such an approach yields a nonlinear equation, where a second transformation is derived to linearize the expression. Consequently, this approach describes a procedure for finding the exact solutions of the hierarchy. A formula of the \(n\)th solution is provided, and to demonstrate its application, we discuss the solution to several members of the nonlinear hierarchy.On scaling properties for a class of two-well problems for higher order homogeneous linear differential operatorshttps://zbmath.org/1541.354932024-09-27T17:47:02.548271Z"Raiţă, Bogdan"https://zbmath.org/authors/?q=ai:raita.bogdan"Rüland, Angkana"https://zbmath.org/authors/?q=ai:ruland.angkana"Tissot, Camillo"https://zbmath.org/authors/?q=ai:tissot.camillo"Tribuzio, Antonio"https://zbmath.org/authors/?q=ai:tribuzio.antonioSummary: We study the scaling behavior of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in the direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order \(m \in \mathbb{N})\) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from [\textit{A. Chan} and \textit{S. Conti}, Math. Models Methods Appl. Sci. 25, No. 6, 1091--1124 (2015; Zbl 1311.49030)] which was also discussed in [\textit{B. Raiţă} et al., Acta Appl. Math. 184, Paper No. 5, 50 p. (2023; Zbl 1514.35428)] provides an example of this family of new scaling laws.Solvability of nonlinear equilibrium problems for Timoshenko-type shallow shells in curvilinear coordinateshttps://zbmath.org/1541.354942024-09-27T17:47:02.548271Z"Timergaliev, S. N."https://zbmath.org/authors/?q=ai:timergaliev.samat-n(no abstract)Soliton-like solutions supported by refined hydrodynamic-type model of an elastic medium with soft inclusionshttps://zbmath.org/1541.354952024-09-27T17:47:02.548271Z"Vladimirov, Vsevolod"https://zbmath.org/authors/?q=ai:vladimirov.vsevolod-a"Skurativskyi, Sergii"https://zbmath.org/authors/?q=ai:skurativskyi.sergiiSummary: A nonlinear elastic medium containing sharp inhomogeneities is considered. The properties of a modified model of such a medium are investigated. The modification consists in including in the asymptotic equation of state those terms that were discarded in the previously considered models. The main purpose of the ongoing research is to analyze the existence, stability, and dynamic properties of soliton-like solutions within the modified model, as well as to compare these solutions with analogous solutions obtained in the previously considered models.Local existence of solutions to the Euler-Poisson system, including densities without compact supporthttps://zbmath.org/1541.354962024-09-27T17:47:02.548271Z"Brauer, Uwe"https://zbmath.org/authors/?q=ai:brauer.uwe"Karp, Lavi"https://zbmath.org/authors/?q=ai:karp.laviSummary: Local existence and uniqueness for a class of solutions for the Euler Poisson system is shown, whose properties can be described as follows. Their density \(\rho\) either falls off at infinity or has compact support. Their mass and the energy functional is finite and they also include the static spherical solutions for \(\gamma =\frac{6}{5}\). The result is achieved by using weighted Sobolev spaces of fractional order and a new non-linear estimate that allows to estimate the physical density by the regularised non linear matter variable.
For the entire collection see [Zbl 1497.42002].Geometric background for the Teukolsky equation revisitedhttps://zbmath.org/1541.354972024-09-27T17:47:02.548271Z"Millet, Pascal"https://zbmath.org/authors/?q=ai:millet.pascalSummary: The aim of this review paper is to revisit the geometric framework of the Teukolsky equation in a form that is suitable for analysts working on this equation. We introduce spinor bundles, the Newman-Penrose formalism and the Geroch-Held-Penrose (GHP) formalism. In particular, we develop the case of Kerr spacetimes, for which we provide detailed computations.On local decay of inflaton and axion fieldshttps://zbmath.org/1541.354982024-09-27T17:47:02.548271Z"Morales, Matías"https://zbmath.org/authors/?q=ai:morales.matias"Muñoz, Claudio"https://zbmath.org/authors/?q=ai:munoz.claudioSummary: We consider the long time behavior of solutions to scalar field models appearing in the theory of cosmological inflation, oscillons and cold dark matter, in presence or absence of the cosmological constant. These models are not included in standard mathematical literature due to their unusual nonlinearities, which model different features with respect to classical fields. Here we prove that these models fit in the theory of dispersive decay by computing new virials adapted to their setting. Several examples, candidates to model both effects are studied in detail.Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetimehttps://zbmath.org/1541.354992024-09-27T17:47:02.548271Z"Wei, Changhua"https://zbmath.org/authors/?q=ai:wei.changhua"Yong, Zikai"https://zbmath.org/authors/?q=ai:yong.zikaiSummary: We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann-Lemaître-Robertson-Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.
{\copyright 2024 American Institute of Physics}Anomalous diffusion limit for a kinetic equation with a thermostatted interfacehttps://zbmath.org/1541.355002024-09-27T17:47:02.548271Z"Bogdan, Krzysztof"https://zbmath.org/authors/?q=ai:bogdan.krzysztof"Komorowski, Tomasz"https://zbmath.org/authors/?q=ai:komorowski.tomasz"Marino, Lorenzo"https://zbmath.org/authors/?q=ai:marino.lorenzoSummary: We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [\textit{T. Komorowski} et al., Ann. Probab. 48, No. 5, 2290--2322 (2020; Zbl 1452.35212)], where the case of a non-degenerate probability of killing has been considered.Erratum to: ``Evaluation of the boiled water mass fraction during its heating by a laser heating element''https://zbmath.org/1541.355012024-09-27T17:47:02.548271Z"Chudnovskiĭ, V. M."https://zbmath.org/authors/?q=ai:chudnovskii.v-m"Guzev, M. A."https://zbmath.org/authors/?q=ai:guzev.mikhail-a|guzev.mickhail-aSummary: In the article of the authors [Dal'nevost. Mat. Zh. 22, No. 2, 164--166 (2022; Zbl 1507.35283)] an error was made. In conclusion you should read: ``The experiment shows that only 2.8\% of the water heated by laser heating near the fiber tip boiled up and turned into steam''.Invariant Gibbs measures for the three dimensional cubic nonlinear wave equationhttps://zbmath.org/1541.355022024-09-27T17:47:02.548271Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjorn"Deng, Yu"https://zbmath.org/authors/?q=ai:deng.yu"Nahmod, Andrea R."https://zbmath.org/authors/?q=ai:nahmod.andrea-r"Yue, Haitian"https://zbmath.org/authors/?q=ai:yue.haitianSummary: We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic \(\Phi^4_3\)-model. This result is the hyperbolic counterpart to seminal works on the parabolic \(\Phi^4_3\)-model by \textit{M. Hairer} [Invent. Math. 198, No. 2, 269--504 (2014; Zbl 1332.60093)] and \textit{M. Hairer} and \textit{K. Matetski} [Ann. Probab. 46, No. 3, 1651--1709 (2018; Zbl 1406.60094)].
The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates.
The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate.Existence and uniqueness of mass conserving solutions to the coagulation, multi-fragmentation equations with compactly supported kernelshttps://zbmath.org/1541.355032024-09-27T17:47:02.548271Z"Das, Arijit"https://zbmath.org/authors/?q=ai:das.arijit"Saha, Jitraj"https://zbmath.org/authors/?q=ai:saha.jitrajSummary: In this article, the existence result of a solution to continuous nonlinear, initial value problem is studied. In particular, we consider a special type of problem representing the time evolution of particle number density due to the coagulation, multi-fragmentation events among the particles present in a system. The existence theorem is proved with the kinetic kernels having compact support. The proof of the main theorem is based on the contraction mapping principle. Furthermore, the mass conservation property of the existed solution is also investigated.
For the entire collection see [Zbl 1521.76009].Diffusion of a collisionless gashttps://zbmath.org/1541.355042024-09-27T17:47:02.548271Z"Kozlov, V. V."https://zbmath.org/authors/?q=ai:kozlov.valerii-vasilievich|kozlov.viktor-vladimirovich|kozlov.viktor-vyacheslavovichSummary: We study a diffusion-type equation for the density of a collisionless relativistic gas (Jüttner gas). The rate of diffusion propagation turns out to be finite. We consider problems of the existence and uniqueness of solutions of this equation, as well as some of its generalized solutions.Spin solitons in spin-1 Bose-Einstein condensateshttps://zbmath.org/1541.355052024-09-27T17:47:02.548271Z"Meng, Ling-Zheng"https://zbmath.org/authors/?q=ai:meng.ling-zheng"Qin, Yan-Hong"https://zbmath.org/authors/?q=ai:qin.yanhong"Zhao, Li-Chen"https://zbmath.org/authors/?q=ai:zhao.li-chenSummary: Vector solitons in Bose-Einstein condensates are usually investigated analytically with identical intra- and interatomic interactions (for an integrable model). We obtain six families of exact spin soliton solutions for nonintegrable cases, which can be used to describe spin-1 Bose-Einstein condensates. The stability analyses and numerical simulations indicate that three families of spin solitons are robust against spin-dependent interactions and white noise. We further investigate the motion of these stable spin solitons driven by external linear potentials. Their moving trajectories demonstrate that the spin solitons admit a negative-positive mass transition. Some splitting and diffusing behaviors can emerge during the motion of a spin soliton that are absent in spin-\(1/2\) systems. The collisions between spin solitons are exhibited with varying relative velocity and phase. The nonintegrable properties of the systems can give rise to weak amplitude and location oscillations after collision. These stable spin soliton excitations can be used to study the negative inertial mass of solitons, the dynamics of soliton-impurity systems, and the spin dynamics in Bose-Einstein condensates.Soliton based director deformation in a twist grain boundary liquid crystalhttps://zbmath.org/1541.355062024-09-27T17:47:02.548271Z"Saravanan, M."https://zbmath.org/authors/?q=ai:saravanan.moorthi"Senjudarvannan, R."https://zbmath.org/authors/?q=ai:senjudarvannan.rSummary: We investigate the director dynamics of a twist grain boundary liquid crystal under the one constant approximation for the different elastic constants representing the various deformation present in the liquid crystal medium. The free energy density is deduced to a higher-order vector nonlinear partial differential equation by balancing the torque experienced by the nematic molecules under a viscous field and the molecular field arises due to the presence of elastic constants. Upon employing the stereographic projection method we further reduced the vector nonlinear differential equation into a complex scalar nonlinear partial differential equation. We obtain a series of localized solutions for the complex scalar nonlinear partial differential equation through the standard tanh method.Formation, propagation, and excitation of matter solitons and rogue waves in chiral BECs with a current nonlinearity trapped in external potentialshttps://zbmath.org/1541.355072024-09-27T17:47:02.548271Z"Song, Jin"https://zbmath.org/authors/?q=ai:song.jin"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya(no abstract)Well-posedness and singularity formation for Vlasov-Riesz systemhttps://zbmath.org/1541.355082024-09-27T17:47:02.548271Z"Choi, Young-Pil"https://zbmath.org/authors/?q=ai:choi.young-pil"Jeong, In-Jee"https://zbmath.org/authors/?q=ai:jeong.in-jeeSummary: We investigate the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring an interaction potential generalizing previously studied cases, including the Coulomb \(\Phi = (- \Delta)^{-1}\rho \), Manev \((- \Delta)^{-1} + (- \Delta)^{-\frac12} \), and pure Manev \((- \Delta)^{-\frac12}\) potentials. For the first time, we extend the local theory of classical solutions to potentials more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known blow-up result for attractive Vlasov-Poisson for \(d\ge4 \). Our local well-posedness and singularity formation results extend to cases when linear diffusion and damping in velocity are present.The plasma-charge model in a convex domainhttps://zbmath.org/1541.355092024-09-27T17:47:02.548271Z"Wu, Jingpeng"https://zbmath.org/authors/?q=ai:wu.jingpengSummary: The aim of this paper is to study the initial-boundary value problems of a Vlasov type system in a convex domain, so called the plasma-charge model, in which there are two kinds of singular sets, one caused by the boundary effect, the other by the heavy point charges. We prove the local existence of classical solutions for the case that the point charges are moving and global existence of classical solutions for the case that the point charges are fixed away from the boundary. The crucial tools are the extended Velocity Lemma for the plasma-charge model and the Pfaffelmoser's method developed by
\textit{H. J. Hwang} and \textit{J. J. L. Velázquez} [Arch. Ration. Mech. Anal. 195, No. 3, 763--796 (2010; Zbl 1218.35235)] and
\textit{C. Marchioro} et al. [Arch. Ration. Mech. Anal. 201, No. 1, 1--26 (2011; Zbl 1321.76081)].
In the Pfaffelmoser's argument, a new idea is that the plasma particles can only be close to one of the singular sets during the time interval \([t-\delta,t]\) with small length \(\delta\), which allows us to obtain the global existence for the fixed point charges case by adapting the techniques established by
\textit{H. J. Hwang} et al. [Discrete Contin. Dyn. Syst. 33, No. 2, 723--737 (2013; Zbl 1271.82026)] and
\textit{H. J. Hwang} and \textit{J. J. L. Velázquez} [Arch. Ration. Mech. Anal. 195, No. 3, 763--796 (2010; Zbl 1218.35235)] and
Marchioro et al. [loc. cit.]
to the corresponding singular sets respectively.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}Kinetic compartmental models driven by opinion dynamics: vaccine hesitancy and social influencehttps://zbmath.org/1541.355102024-09-27T17:47:02.548271Z"Bondesan, Andrea"https://zbmath.org/authors/?q=ai:bondesan.andrea"Toscani, Giuseppe"https://zbmath.org/authors/?q=ai:toscani.giuseppe"Zanella, Mattia"https://zbmath.org/authors/?q=ai:zanella.mattiaSummary: We propose a kinetic model for understanding the link between opinion formation phenomena and epidemic dynamics. The recent pandemic has brought to light that vaccine hesitancy can present different phases and temporal and spatial variations, presumably due to the different social features of individuals. The emergence of patterns in societal reactions permits to design and predict the trends of a pandemic. This suggests that the problem of vaccine hesitancy can be described in mathematical terms, by suitably coupling a kinetic compartmental model for the spreading of an infectious disease with the evolution of the personal opinion of individuals, in the presence of leaders. The resulting model makes it possible to predict the collective compliance with vaccination campaigns as the pandemic evolves and to highlight the best strategy to set up for maximizing the vaccination coverage. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.Properties and stability analysis of the sixth-order Boussinesq equations for Rossby waveshttps://zbmath.org/1541.355112024-09-27T17:47:02.548271Z"Yang, Xiaoqian"https://zbmath.org/authors/?q=ai:yang.xiaoqian"Zhang, Zongguo"https://zbmath.org/authors/?q=ai:zhang.zongguo"Zhang, Ning"https://zbmath.org/authors/?q=ai:zhang.ning.1|zhang.ning.8|zhang.ning.4|zhang.ning|zhang.ning.2Summary: The sixth-order Boussinesq equation describing Rossby waves in the barotropic atmosphere is derived from the quasi-geostrophic vorticity equation using scale analysis and perturbation expansion. The symmetry and conservation laws of the equation are analyzed. The solution to the equation is obtained through the Jacobi elliptic function expansion method, and the effect of high-order terms and wave numbers on the waves are discussed. The results show that both the higher order terms and the numbers of waves affect the height of the amplitude of the wave. When the high-order terms are present, the height of the amplitude is lower than when the high-order terms are not present and when the wave number is larger, the height of the amplitude is also higher, and this conclusion is not affected by the presence or absence of high-order terms. Finally, the stability of the equation is analyzed by the concept of linear stability analysis. It is pointed out that when the higher order term does not exist, the stable region of the equation disappears and the equation becomes unstable.Games associated with products of eigenvalues of the Hessianhttps://zbmath.org/1541.355122024-09-27T17:47:02.548271Z"Blanc, Pablo"https://zbmath.org/authors/?q=ai:blanc.pablo"Charro, Fernando"https://zbmath.org/authors/?q=ai:charro.fernando"Manfredi, Juan J."https://zbmath.org/authors/?q=ai:manfredi.juan-j"Rossi, Julio D."https://zbmath.org/authors/?q=ai:rossi.julio-danielSummary: We introduce games associated with second-order partial differential equations given by arbitrary products of eigenvalues of the Hessian. We prove that, as a parameter that controls the step length goes to zero, the value functions of the games converge uniformly to a viscosity solution of the partial differential equation. The classical Monge-Ampère equation is an important example under consideration.Lie symmetry, exact solutions and conservation laws of time fractional Black-Scholes equation derived by the fractional Brownian motionhttps://zbmath.org/1541.355132024-09-27T17:47:02.548271Z"Yu, Jicheng"https://zbmath.org/authors/?q=ai:yu.jichengSummary: The Black-Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black-Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black-Scholes equation.Bifurcation of finger-like structures in traveling waves of epithelial tissues spreadinghttps://zbmath.org/1541.355142024-09-27T17:47:02.548271Z"Berlyand, Leonid"https://zbmath.org/authors/?q=ai:berlyand.leonid-v"Rybalko, Antonina"https://zbmath.org/authors/?q=ai:rybalko.antonina"Rybalko, Volodymyr"https://zbmath.org/authors/?q=ai:rybalko.volodymyr"Safsten, Clarke Alex"https://zbmath.org/authors/?q=ai:safsten.clarke-alexSummary: We consider a continuous active polar fluid model for the spreading of epithelial monolayers introduced by \textit{R. Alert} et al. [Phys. Rev. Lett. 122, No. 8, Article ID 088104, 7 p. (2019; \url{doi:10.1103/PhysRevLett.122.088104})]. The corresponding free boundary problem possesses flat front traveling wave solutions. Linear stability of these solutions under periodic perturbations is considered. It is shown that the solutions are stable for short-wave perturbations while exhibiting long-wave instability under certain conditions on the model parameters (if the traction force is sufficiently strong). Then, considering the prescribed period as the bifurcation parameter, we establish the emergence of nontrivial traveling wave solutions with a finger-like periodic structure (pattern). We also construct asymptotic expansions of the solutions in the vicinity of the bifurcation point and study their stability. We show that, depending on the value of the contractility coefficient, the bifurcation can be a subcritical or a supercritical pitchfork.Spatio-temporal dynamics in a diffusive Bazykin model: effects of group defense and prey-taxishttps://zbmath.org/1541.355152024-09-27T17:47:02.548271Z"Dey, Subrata"https://zbmath.org/authors/?q=ai:dey.subrata"Banerjee, Malay"https://zbmath.org/authors/?q=ai:banerjee.malay"Ghorai, S."https://zbmath.org/authors/?q=ai:ghorai.saktipadaThe paper deals with the Bazykin prey-predator model with a logistic growth rate for the prey species and a Monod-Haldane function \(F(N)=\frac{\alpha N}{1+\beta N^2}\) for the functional response, namely
\[
\left\{\begin{array}{l}\displaystyle\frac{dN}{dT}=N(\sigma-\eta N)-\displaystyle\frac{\alpha NP}{1+\beta N^2},\\
\displaystyle\frac{dP}{dT}=\displaystyle\frac{\zeta \alpha NP}{1+\beta N^2}-\gamma P-\delta P^2, \end{array}\right.\tag{1}
\]
where \(N(T)\) and \(P(T)\) are the densities of the prey and predator populations at time \(T\), \(\zeta\) is the conversion coefficient, \(\gamma\) and \(\delta\) denote the natural mortality rate and intra-species competition of the predator population, \(\sigma\) and \(\eta\) represent the intrinsic growth rate and intra-species competition of the prey species. The authors investigate the equilibria, stability, the local and global bifurcations for problem \((1)\). The long transient dynamics with numerical visualization are also studied. Then the authors prove the global existence and boundedness of the solution to the corresponding spatio-temporal model, the stability of the homogeneous steady states, and the Turing bifurcation. Numerical simulations showing the long transient dynamics, and various stationary and dynamic patterns are also presented.
Reviewer: Rodica Luca (Iaşi)Travelling waves for a fast reaction limit of a discrete coagulation-fragmentation model with diffusion and proliferationhttps://zbmath.org/1541.355162024-09-27T17:47:02.548271Z"Estavoyer, Maxime"https://zbmath.org/authors/?q=ai:estavoyer.maxime"Lepoutre, Thomas"https://zbmath.org/authors/?q=ai:lepoutre.thomasSummary: We study traveling wave solutions for a reaction-diffusion model, introduced in the article
[\textit{V. Calvez} et al., ``Regime switching on the propagation speed of travelling waves of some size-structured Myxobacteria population models'', Preprint, \url{HAL:hal-04532644}],
describing the spread of the social bacterium \textit{Myxococcus xanthus}. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by \(k\). In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter \(k\) tends to \(+ \infty\). For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio \(\theta^\star\) between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below \(\theta^\star\), the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.On the fast spreading scenariohttps://zbmath.org/1541.355172024-09-27T17:47:02.548271Z"He, Siming"https://zbmath.org/authors/?q=ai:he.siming"Tadmor, Eitan"https://zbmath.org/authors/?q=ai:tadmor.eitan"Zlatoš, Andrej"https://zbmath.org/authors/?q=ai:zlatos.andrejThe authors study the influence of fast hyperbolic flows (with velocity \(A(-x_1,\dots, -x_{d-1},x_d)\), \(A\) large, on \(\mathbb{R}^d\)) on suppression of singularities in chemotaxis advected by such flows. Similarly, fast shear flows (with velocity \(A(u(x_2,\dots,x_d),0,\dots,0)\), \(A\) large, on channel domains \(\mathbb{R}\times \mathbb{T}^{d-1}\)) can suppress chemotactic blow-ups, and may lead to quenching of solutions with compactly supported data in reaction-diffusion models (e.g. combustion) advected by those flows. The analyses are lead in two- and three-dimensional cases: \(d=2,\ 3\).
Reviewer: Piotr Biler (Wrocław)Wave propagations for dispersive variants of spatial models in epidemiology and ecologyhttps://zbmath.org/1541.355182024-09-27T17:47:02.548271Z"Koçak, Hüseyin"https://zbmath.org/authors/?q=ai:kocak.huseyin.2|kocak.huseyin.1"Pinar, Zehra"https://zbmath.org/authors/?q=ai:pinar.zehraSummary: In this paper, apart from the well-known second-order reaction-diffusion models in epidemiology and ecology, the third-order variants, called the reaction-dispersion models, are proposed. Using an ansatz method with the suitable function, the travelling wave solutions of the corresponding systems are investigated. The long-time behaviour of obtained solutions, which are actually kink type waves, are discussed. It is seen that the proposed system is able to simply display such epidemic model.Modeling insect growth regulators for pest managementhttps://zbmath.org/1541.355192024-09-27T17:47:02.548271Z"Lou, Yijun"https://zbmath.org/authors/?q=ai:lou.yijun"Wu, Ruiwen"https://zbmath.org/authors/?q=ai:wu.ruiwen.2|wu.ruiwenSummary: Insect growth regulators (IGRs) have been developed as effective control measures against harmful insect pests to disrupt their normal development. This study is to propose a mathematical model to evaluate the cost-effectiveness of IGRs for pest management. The key features of the model include the temperature-dependent growth of insects and realistic impulsive IGRs releasing strategies. The impulsive releases are carefully modeled by counting the number of implements during an insect's temperature-dependent development duration, which introduces a surviving probability determined by a product of terms corresponding to each release. Dynamical behavior of the model is illustrated through dynamical system analysis and a threshold-type result is established in terms of the net reproduction number. Further numerical simulations are performed to quantitatively evaluate the effectiveness of IGRs to control populations of harmful insect pests. It is interesting to observe that the time-changing environment plays an important role in determining an optimal pest control scheme with appropriate release frequencies and time instants.Dynamical analysis of an age-structured SEIR model with relapsehttps://zbmath.org/1541.355202024-09-27T17:47:02.548271Z"Nabti, Abderrazak"https://zbmath.org/authors/?q=ai:nabti.abderrazakSummary: Mathematical models play a crucial role in controlling and preventing the spread of diseases. Based on the communication characteristics of diseases, it is necessary to take into account some essential epidemiological factors such as the time delay that takes an individual to progress from being latent to become infectious, the infectious age which refers to the duration since the initial infection and the occurrence of reinfection after a period of improvement known as relapse, etc. Moreover, age-structured models serve as a powerful tool that allows us to incorporate age variables into the modeling process to better understand the effect of these factors on the transmission mechanism of diseases. In this paper, motivated by the above fact, we reformulate an SEIR model with relapse and age structure in both latent and infected classes. Then, we investigate the asymptotic behavior of the model by using the stability theory of differential equations. For this purpose, we introduce the basic reproduction number \(\mathcal{R}_0\) of the model and show that this threshold parameter completely governs the stability of each equilibrium of the model. Our approach to show global attractivity is based on the fluctuation lemma and Lyapunov functionals method with some results on the persistence theory. The conclusion is that the system has a disease-free equilibrium which is globally asymptotically stable if \(\mathcal{R}_0 <1\), while it has only a unique positive endemic equilibrium which is globally asymptotically stable whenever \(\mathcal{R}_0 >1\). Our results imply that early diagnosis of latent infection with decrease in both transmission and relapse rates may lead to control and restrict the spread of disease. The theoretical results are illustrated with numerical simulations, which indicate that the age variable is an essential factor affecting the spread of the epidemic.Spatial dynamics of a reaction-diffusion SIS epidemic model with mass-action-type nonlinearityhttps://zbmath.org/1541.355212024-09-27T17:47:02.548271Z"Wang, Renhu"https://zbmath.org/authors/?q=ai:wang.renhu"Wang, Xuezhong"https://zbmath.org/authors/?q=ai:wang.xuezhongSummary: This work is devoted to investigate the global asymptotic stability of equilibriums for a reaction-diffusion susceptible-infected-susceptible (SIS) epidemic model with spatial heterogeneity and mass-action-type nonlinearity. By discretizing the spatial variables of the model, first, Lyapunov functions are constructed for the corresponding ordinary differential equations (ODEs) model of the original SIS PDEs model, and then the construction method is generalized to the PDEs model in which either the susceptible or the infectious individuals are spreading in spatial heterogeneity environment. For both the cases, we obtained the standard threshold dynamics results.A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systemshttps://zbmath.org/1541.355222024-09-27T17:47:02.548271Z"Wang, Zhongjian"https://zbmath.org/authors/?q=ai:wang.zhongjian"Xin, Jack"https://zbmath.org/authors/?q=ai:xin.jack-x"Zhang, Zhiwen"https://zbmath.org/authors/?q=ai:zhang.zhiwenSummary: We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller-Segel (KS) chemotaxis system in two and three space dimensions, then further develop the DeepParticle method to learn and generate solutions under variations of physical parameters. The KS solutions are approximated as empirical measures of particles that self-adapt to the high gradient part of solutions. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at a finite time \(T\) prior to blowup without assuming the invertibility of the transforms. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. To reduce the computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. We present numerical results of the DeepParticle framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. The physical parameter in this work is either the evolution time or the flow amplitude in the advection-dominated regime.Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equationhttps://zbmath.org/1541.355232024-09-27T17:47:02.548271Z"Alabau-Boussouira, Fatiha"https://zbmath.org/authors/?q=ai:alabau-boussouira.fatiha"Cannarsa, Piermarco"https://zbmath.org/authors/?q=ai:cannarsa.piermarco"Urbani, Cristina"https://zbmath.org/authors/?q=ai:urbani.cristinaSummary: We study the exact controllability of the evolution equation
\[
u'(t)+Au(t)+p(t)Bu(t)=0
\]
where \(A\) is a nonnegative self-adjoint operator on a Hilbert space \(X\) and \(B\) is an unbounded linear operator on \(X\), which is dominated by the square root of \(A\). The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function \(p\), which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of \(A\), as initial data.
The analogous problem (in a more general form) was addressed in our previous paper [NoDEA, Nonlinear Differ. Equ. Appl. 29, No. 4, Paper No. 38, 32 p. (2022; Zbl 1537.35348)] for a bounded operator \(B\). The current extension to unbounded operators allows for many more applications, including the Fokker-Planck equation in one space dimension, and a larger class of control actions.On optimal control of thermoelastic vibrations of a plate-striphttps://zbmath.org/1541.355242024-09-27T17:47:02.548271Z"Jilavyan, S. H."https://zbmath.org/authors/?q=ai:jilavyan.sami-h"Grigoryan, E. R."https://zbmath.org/authors/?q=ai:grigoryan.edmon-rSummary: The problem of optimal control of elastic vibrations of an isotropic plate-strip under the influence of temperature and force fields is studied. The function of changing the external load on the plane of the plate is represented as a control function. Optimal control is also carried out by the distribution function of the temperature of the external field over the plate. The well-known classical hypotheses of thermo-elastic bending of the plate are accepted. The equations of transverse vibrations of the plate and heat conduction in the plate are solved under the boundary conditions of heat transfer and the stress state on the planes of the plate. The method of Fourier series, the method of representing moment relations, the well-known method of minimizing the functional are used.On a class of nonlinear evolution inequalities of convolution type on Riemannian manifoldshttps://zbmath.org/1541.355252024-09-27T17:47:02.548271Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: A higher order (in time) evolution inequality on a complete noncompact Riemannian manifold is investigated. The considered problem involves a nonlinear convolution term and an inhomogeneous term depending on time and space. Under certain assumptions on the Ricci curvature of the manifold, we establish sufficient conditions for the nonexistence of weak solutions. Next, some special cases are discussed. Our obtained results are new even in the Euclidean case.Microscopic derivation of a traffic flow model with a bifurcationhttps://zbmath.org/1541.355262024-09-27T17:47:02.548271Z"Cardaliaguet, P."https://zbmath.org/authors/?q=ai:cardaliaguet.pierre"Forcadel, N."https://zbmath.org/authors/?q=ai:forcadel.nicolasSummary: The goal of the paper is a rigorous derivation of a macroscopic traffic flow model with a bifurcation or a local perturbation from a microscopic one. The microscopic model is a simple follow-the-leader with random parameters. The random parameters are used as a statistical description of the road taken by a vehicle and its law of motion. The limit model is a deterministic and scalar Hamilton-Jacobi on a network with a flux limiter, the flux-limiter describing how much the bifurcation or the local perturbation slows down the vehicles. The proof of the existence of this flux limiter -- the first one in the context of stochastic homogenization -- relies on a concentration inequality and on a delicate derivation of a superadditive inequality.The ground state solutions to a class of biharmonic Choquard equations on weighted lattice graphshttps://zbmath.org/1541.355272024-09-27T17:47:02.548271Z"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.231"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjieSummary: In this paper, we consider the biharmonic Choquard equation with the nonlocal term on the weighted lattice graph \({\mathbb{Z}}^N\), namely for any \(p>1\) and \(\alpha \in (0,N)\)
\[
\begin{aligned} \Delta^2u-\Delta u+V(x)u=\left(\sum_{y\in{\mathbb{Z}}^N,y\not =x}\frac{|u(y)|^p}{d(x,y)^{N-\alpha}}\right) |u|^{p-2}u, \end{aligned}
\]
where \(\Delta^2\) is the biharmonic operator, \(\Delta\) is the \(\mu\)-Laplacian, \(V:{\mathbb{Z}}^N\rightarrow{\mathbb{R}}\) is a function, and \(d(x,y)\) is the distance between \(x\) and \(y\). If the potential \(V\) satisfies certain assumptions, using the method of Nehari manifold, we prove that for any \(p>(N+\alpha)/N\), there exists a ground state solution of the above-mentioned equation. Compared with the previous results, we adopt a new method to finding the ground state solution from mountain-pass solutions.Fokas method for the heat equation on metric graphshttps://zbmath.org/1541.355282024-09-27T17:47:02.548271Z"Sobirov, Z. A."https://zbmath.org/authors/?q=ai:sobirov.zarifboi-akhmedovich"Eshimbetov, M. R."https://zbmath.org/authors/?q=ai:eshimbetov.mardonbek-rSummary: The paper presents a method for constructing solutions to initial-boundary value problems for the heat equation on simple metric graphs such as a star-shaped graph, a tree, and a triangle with three converging edges. The solutions to the problems are constructed by the so-called \textit{Fokas method}, which is a generalization of the Fourier transform method. In this case, the problem is reduced to a system of algebraic equations for the Fourier transform of the unknown values of the solution at the vertices of the graph.Global compactness, subcritical approximation of the Sobolev quotient, and a related concentration result in the Heisenberg grouphttps://zbmath.org/1541.355292024-09-27T17:47:02.548271Z"Palatucci, Giampiero"https://zbmath.org/authors/?q=ai:palatucci.giampiero"Piccinini, Mirco"https://zbmath.org/authors/?q=ai:piccinini.mirco"Temperini, Letizia"https://zbmath.org/authors/?q=ai:temperini.letiziaSummary: We investigate some effects of the lack of compactness in the critical Sobolev embedding in the Heisenberg group.
For the entire collection see [Zbl 1537.35003].On the absence of global weak solutions for a nonlinear time-fractional Schrödinger equationhttps://zbmath.org/1541.355302024-09-27T17:47:02.548271Z"Alotaibi, Munirah"https://zbmath.org/authors/?q=ai:alotaibi.munirah-aali"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandra"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessem(no abstract)On the energy decay of a nonlinear time-fractional Euler-Bernoulli beam problem including time-delay: theoretical treatment and numerical solution techniqueshttps://zbmath.org/1541.355312024-09-27T17:47:02.548271Z"Bentrcia, Toufik"https://zbmath.org/authors/?q=ai:bentrcia.toufik"Mennouni, Abdelaziz"https://zbmath.org/authors/?q=ai:mennouni.abdelazizSummary: In this work, an extended Euler-Bernoulli beam equation is addressed, where numerous phenomena are covered including damping, time-delay, and nonlinear source effects. A generalized fractional derivative is used to model dissipation of order less than one, which offers more flexibility for modeling tasks. Through a diffusive representation, the problem well-posedness is tackled and the exponential decay of the energy associated to global solutions is proved under some conditions. In order to validate our theoretical findings, we implement a finite difference scheme and we elucidate that the boundedness of the local propagation matrix may be inaccurate for the convergence evaluation in some situations. Furthermore, we show that deep neural networks are efficient alternatives to deal with computational and stability burdens resulting from the mesh refinement in standard numerical schemes.Study of fractional semipositone problems on \(\mathbb{R}^N\)https://zbmath.org/1541.355322024-09-27T17:47:02.548271Z"Biswas, Nirjan"https://zbmath.org/authors/?q=ai:biswas.nirjanSummary: Let \(s\in (0,1)\) and \(N>2s\). In this paper, we consider the following class of nonlocal semipositone problems:
\[
(-\Delta)^s u= g(x)f_a (u)\text{ in }\mathbb{R}^N,\quad u>0\text{ in }\mathbb{R}^N,
\]
where the weight \(g\in L^1 (\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a>0\) is a parameter, and \(f_a \in \mathcal{C}(\mathbb{R})\) is strictly negative on \((-\infty,0]\). For \(f_a\) having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution \(u_a\), provided \(a\) is near zero. To obtain the positivity of \(u_a\), we establish a Brezis-Kato type uniform estimate of \((u_a)\) in \(L^r (\mathbb{R}^N)\) for every \(r \in [\frac{2N}{N-2s},\infty]\).Singular solutions for space-time fractional equations in a bounded domainhttps://zbmath.org/1541.355332024-09-27T17:47:02.548271Z"Chan, Hardy"https://zbmath.org/authors/?q=ai:chan.hardy"Gómez-Castro, David"https://zbmath.org/authors/?q=ai:gomez-castro.david"Vázquez, Juan Luis"https://zbmath.org/authors/?q=ai:vazquez.juan-luisSummary: This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann-Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.Stochastic fractional conservation lawshttps://zbmath.org/1541.355342024-09-27T17:47:02.548271Z"Chaudhary, Abhishek"https://zbmath.org/authors/?q=ai:chaudhary.abhishekSummary: In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws with stochastic forcing. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.Maximum principle for the fractional N-Laplacian flowhttps://zbmath.org/1541.355352024-09-27T17:47:02.548271Z"Choi, Q-Heung"https://zbmath.org/authors/?q=ai:choi.q-heung"Jung, Tacksun"https://zbmath.org/authors/?q=ai:jung.tacksunSummary: We deal with a family of the fractional N-Laplacian heat flows with variable exponent time-derivative on the Orlicz-Sobolev spaces. We get the maximum principle for these problems. We use the approximating method to get this result: We first show existence of a unique family of the approximating weak solutions from the variable exponent difference fractional N-Laplacian problems. We next show the maximum principle for the family of the approximating weak solution from the variable exponent difference fractional N-Laplacian problem, show the convergence of a family of the approximating weak solutions to the limits, and then obtain the maximum principle for the weak solution of a family of the fractional N-Laplacian heat flows with the variable exponent time-derivative on the Orlicz-Sobolev spaces.Dynamic boundary conditions for time dependent fractional operators on extension domainshttps://zbmath.org/1541.355362024-09-27T17:47:02.548271Z"Creo, Simone"https://zbmath.org/authors/?q=ai:creo.simone"Lancia, Maria Rosaria"https://zbmath.org/authors/?q=ai:lancia.maria-rosariaSummary: We consider a parabolic semilinear non-autonomous problem \((\tilde{P})\) for a fractional time dependent operator \(\mathcal{B}^{s,t}_{\Omega}\) with Wentzell-type boundary conditions in a possibly non-smooth domain \(\Omega\subset\mathbb{R}^N\). We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem \((P)\) via an evolution family \(U(t,\tau)\). We then prove that the mild solution of the abstract proble \((P)\) actually solves problem \((\tilde{P})\) via a generalized fractional Green formula.Existence of solutions for a class of fractional Kirchhoff variational inequalityhttps://zbmath.org/1541.355372024-09-27T17:47:02.548271Z"Deng, Shenbing"https://zbmath.org/authors/?q=ai:deng.shenbing"Luo, Wenshan"https://zbmath.org/authors/?q=ai:luo.wenshan"Ledesma, César E. Torres"https://zbmath.org/authors/?q=ai:torres-ledesma.cesar-e"Quiroz, George W. Alama"https://zbmath.org/authors/?q=ai:quiroz.george-w-alamaSummary: We are concerned with the following fractional Kirchhoff variational inequality:
\[
\begin{aligned}
(a+b[u]^2) \int_{\mathbb{R}^3} (-\Delta)^{\frac{s}{2}} u(-\Delta)^{\frac{s}{2}} (v-u)dx+\int_{\mathbb{R}_3}(1+\lambda V(x)) u(v-u)dx \\
\geq \int_{\mathbb{R}^3} f(u)(v-u)dx \quad \forall v\in \mathbb{K},
\end{aligned}
\]
where \(s\in (\frac{3}{4},1), \lambda>0\). In this paper, by applying penalization techniques from [\textit{A. Bensoussan} and \textit{J. L. Lions}, Applications des inéquations variationnelles en contrôle stochastique. Paris: Dunod (1978; Zbl 0411.49002)] combined with mountain pass theorem, we show the existence and concentration behavior of positive solution to the cited variational inequality. This result extend some results established by \textit{C. O. Alves} et al. [J. Math. Anal. Appl. 494, No. 2, Article ID 124672, 38 p. (2021; Zbl 1459.35204)] to the fractional case.Time fractional super KdV equation: Lie point symmetries, conservation laws, explicit solutions with convergence analysishttps://zbmath.org/1541.355382024-09-27T17:47:02.548271Z"Gulsen, Selahattin"https://zbmath.org/authors/?q=ai:gulsen.selahattin"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafaSummary: In this work, one-point Lie symmetry method is applied to time fractional super KdV equation in order to obtain similarity variables and similarity transformations with Riemann-Liouville derivative. These transformations reduce the governing equation to an ordinary differential equation of fractional order. A new and effective conservation theorem based on Noether's theorem is used to obtain conserved vectors. Then, we construct power series solutions for the reduced time fractional ordinary differential equation and prove that the solutions are convergent. Lastly, some interesting graphs are given to explain physical behaviors.Tensor product technique and atomic solution of fractional partial differential equationshttps://zbmath.org/1541.355392024-09-27T17:47:02.548271Z"Hammad, Ma'mon Abu"https://zbmath.org/authors/?q=ai:hammad.mamon-abu"Ghazi Alshanti, Waseem"https://zbmath.org/authors/?q=ai:ghazi-alshanti.waseem"Alshanty, Ahmad"https://zbmath.org/authors/?q=ai:alshanty.ahmad"Khalil, Roshdi"https://zbmath.org/authors/?q=ai:khalil.roshdi-rashidSummary: In this paper, we investigate the atomic solution of a special type of fractional partial differential equations. Tensor product in Banach spaces, some properties of atom operators, and some properties of conformable fractional derivatives are utilized in such process.Properties of positive solutions for the fractional Laplacian systems with positive-negative mixed powershttps://zbmath.org/1541.355402024-09-27T17:47:02.548271Z"Lü, Zhongxue"https://zbmath.org/authors/?q=ai:lu.zhongxue"Niu, Mengjia"https://zbmath.org/authors/?q=ai:niu.mengjia"Shen, Yuanyuan"https://zbmath.org/authors/?q=ai:shen.yuanyuan"Yuan, Anjie"https://zbmath.org/authors/?q=ai:yuan.anjieSummary: In this paper, by establishing the direct method of moving planes for the fractional Laplacian system with positive-negative mixed powers, we obtain the radial symmetry and monotonicity of the positive solutions for the fractional Laplacian systems with positive-negative mixed powers in the whole space. We also give two special cases.Basic fractional nonlinear-wave models and solitonshttps://zbmath.org/1541.355412024-09-27T17:47:02.548271Z"Malomed, Boris A."https://zbmath.org/authors/?q=ai:malomed.boris-a(no abstract)Global endpoint regularity estimates for the fractional Dirichlet problemhttps://zbmath.org/1541.355422024-09-27T17:47:02.548271Z"Ma, Wenxian"https://zbmath.org/authors/?q=ai:ma.wenxian"Yang, Sibei"https://zbmath.org/authors/?q=ai:yang.sibeiSummary: Let \(n\geq 2\), \(s\in (0,1)\), and \(\Omega \subset\mathbb{R}^n\) be a bounded \(C^2\) domain. The aim of this paper is to study the global endpoint regularity estimates for the fractional Dirichlet problem
\[
\begin{cases}
(-\Delta)^su=f \ & \text{in } \Omega,\\
u=0 & \text{in } \mathbb{R}^n\setminus \Omega.
\end{cases}
\]
More precisely, the authors prove the optimal global BMO regularity estimates
\[
\left\| (-\Delta)^\frac{s}{2} u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\leq C\Vert f\Vert_{L^q(\Omega)}
\text{ and }
\left\| \nabla^s u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\leq C\Vert f\Vert_{L^q(\Omega)}
\]
for some \(q\in (\frac{n}{s},\infty)\), and the global regularity estimates
\[
\left\| (-\Delta)^\frac{s}{2} u\right\|_{L^p(\mathbb{R}^n)}\leq C\Vert f\Vert_{H^q(\Omega)}
\text{ and }
\left\| \nabla^s u\right\|_{L^p(\mathbb{R}^n)}\leq C\Vert f\Vert_{H^q(\Omega)}
\]
for any given \(q\in (\frac{n}{n+s},1]\) and \(p\in [1,\frac{qn}{n-qs})\). Here, \(\nabla^s\) denotes the Riesz gradient of order \(s\) and \(H^q(\Omega)\) denotes the Hardy space on \(\Omega\). Moreover, the authors also obtain the global regularity estimates
\[
\left\| (-\Delta)^\frac{t}{2}u\right\|_{L^p(\mathbb{R}^n)}\leq C\Vert f\Vert_{H^q(\Omega)}
\text{ and }
\left\| \nabla^t u\right\|_{L^p(\mathbb{R}^n)}\leq C\Vert f\Vert_{H^q(\Omega)}
\]
for any given \(t\in (s,\min \{1,s+\frac{s}{n}\})\), \(q\in (\frac{n}{n+s-n(t-s)}, 1]\), and \(p\in [1,\frac{qn}{n-qs+qn(t-s)})\). The global regularity estimates given in this paper are further devolvement for the corresponding results in the scale of Lebesgue spaces, established by \textit{B. Abdellaoui} et al. [``Global fractional Calderón-Zygmund type regularity'', Preprint, \url{arXiv:2107.06535}], in the endpoint case.On initial value problem for diffusion equation with Caputo-Fabrizio operator on the planehttps://zbmath.org/1541.355432024-09-27T17:47:02.548271Z"Minh, Vo Ngoc"https://zbmath.org/authors/?q=ai:minh.vo-ngoc"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinhSummary: In this paper, we are interested in studying the diffusion equation with Caputo-Fabrizio derivative. This is the first time that the Caputo-Fabrizio problem on the \(\mathbb{R}^2\) domain has been studied. Under the various assumptions of the initial datum and the source functions, we provide the upper bound of the mild solution. We also obtain the upper bound of the first derivative and Caputo-Fabrizio derivative of the mild solution. In addition, we obtain the lower bound of the mild solution and its derivative.Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical caseshttps://zbmath.org/1541.355442024-09-27T17:47:02.548271Z"Miyamoto, Yasuhito"https://zbmath.org/authors/?q=ai:miyamoto.yasuhito"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: Let \(0<\theta \leq 2, N\geq 1\) and \(T>0\). We are concerned with the Cauchy problem for the fractional semilinear parabolic equation
\[
\begin{cases}
\partial_t u+(-\Delta)^{\theta/2}u=f(u) & \text{in } \mathbb{R}^N \times (0,T), \\
u(x,0)=u_0 (x)\geq 0 & \text{in } \mathbb{R}^N.
\end{cases}
\]
Here, \(f\in C[0,\infty)\) denotes a rather general growing nonlinearity and \(u_0\) may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when \(f(u)=u^{1+\theta/N}\left[ \log (u+e)\right]^a\), we obtain a sharp integrability condition on \(u_0\) which explicitly determines local in time existence/nonexistence of a nonnegative solution.Multiplicity and concentration of nontrivial solutions for a class of fractional Kirchhoff equations with steep potential wellhttps://zbmath.org/1541.355452024-09-27T17:47:02.548271Z"Shao, Liuyang"https://zbmath.org/authors/?q=ai:shao.liuyang"Chen, Haibo"https://zbmath.org/authors/?q=ai:chen.haibo(no abstract)On the fractional powers of a Schrödinger operator with a Hardy-type potentialhttps://zbmath.org/1541.355462024-09-27T17:47:02.548271Z"Siclari, Giovanni"https://zbmath.org/authors/?q=ai:siclari.giovanniSummary: Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.Some new results on the convergence of solutions for time and space fractional Sobolev equationhttps://zbmath.org/1541.355472024-09-27T17:47:02.548271Z"Tri, Vo Viet"https://zbmath.org/authors/?q=ai:tri.vo-viet"Huy, Tuan Nguyen"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: In this article, we are interested in considering the time and space fractional Sobolev equation. The derivative in this equation is understood in the sense that Caputo sense
\[
D^\alpha_t y + \mathcal{L}^s y+ m D^\alpha_t \mathcal{L} y = G(t, x, y(t, x)), \quad \text{in} \quad (0, T] \times \Omega \tag{1}
\]
where \(m>0\) and \(0<s<1\). The main objective of the paper is to investigate the two convergence problems of the mild solution when \(m \to 0^+\) and \(s \to 1^-\) respectively. In the linear case, namely, \( G = G(t, x) \), we express the solution in terms of the Fourier series of the Mittag-Leffler functions. We also provide some upper estimates on the Hilbert scale spaces using the techniques of the Wright functions. In the nonlinear case, we prove two key results. The first result concerns the existence of a global solution when the function \(G\) Lipschitz is global in the space \(L^p \). The second result is related to the convergence of the mild solution to a nonlinear problem when \(m \to 0^+\) and \(s \to 1^- \). The main technique for studying nonlinear problems is the flexible application of embeddings between Hilbert scales and Sobolev spaces.Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensionshttps://zbmath.org/1541.355482024-09-27T17:47:02.548271Z"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitali"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in \(\mathbb{R}^d\), \(d=4, 5\). The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used.Effects of multiplicative noise on the fractional Hartree equationhttps://zbmath.org/1541.355492024-09-27T17:47:02.548271Z"Xie, J."https://zbmath.org/authors/?q=ai:xie.jin.1"Yang, H."https://zbmath.org/authors/?q=ai:yang.han.1"Wang, F."https://zbmath.org/authors/?q=ai:wang.fan.3Summary: This paper is dedicated to radial solutions to the Cauchy problem for the fractional Hartree equation with multiplicative noise. First, we establish a stochastic Strichartz estimate related to the fractional Schrödinger propagator. Local well-posedness for the Cauchy problem is proved by using stochastic and radial deterministic Strichartz estimates. Then, based on Itô's formula and stopping time arguments, the existence of a global solution is studied. Finally, we investigate the blow-up phenomenon and give a criterion via localized virial estimates.
{\copyright 2024 American Institute of Physics}A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equationhttps://zbmath.org/1541.355502024-09-27T17:47:02.548271Z"Zeng, Shengda D."https://zbmath.org/authors/?q=ai:zeng.shengda-d"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weiminThe authors consider a fractional differential hemivariational inequality of parabolic-parabolic type. The fractional derivative operator is in the sense of Caputo. The aim is to establish a ``mild'' solution and an iterative method for it. The second aim is to study a nonstationary incompressible Navier-Stokes system described by a fractional reaction-diffusion equation: this is an example of the application of the proposed technique.
Reviewer: Ilya A. Chernov (Petrozavodsk)Dynamics of a diffusive viral infection model with impulsive CTL immune responsehttps://zbmath.org/1541.355512024-09-27T17:47:02.548271Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie.24|wang.jie.8|wang.jie.5|wang.jie.82|wang.jie.15|wang.jie.13|wang.jie.10|wang.jie.7|wang.jie.9|wang.jie.1|wang.jie.16|wang.jie.3|wang.jie"Yang, Ruirui"https://zbmath.org/authors/?q=ai:yang.ruirui"Huo, Haifeng"https://zbmath.org/authors/?q=ai:huo.hai-feng(no abstract)Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heathttps://zbmath.org/1541.355522024-09-27T17:47:02.548271Z"Bollati, Julieta"https://zbmath.org/authors/?q=ai:bollati.julieta"Natale, María F."https://zbmath.org/authors/?q=ai:natale.maria-fernanda"Semitiel, José A."https://zbmath.org/authors/?q=ai:semitiel.jose-a"Tarzia, Domingo A."https://zbmath.org/authors/?q=ai:tarzia.domingo-albertoSummary: Two different two-phase free boundary Stefan problems in an angular domain with temperature-dependent thermal coefficients are considered. Analytical similarity solutions are obtained imposing a Dirichlet or Neumann type boundary condition, respectively, by solving functional problems. Moreover, formulas are obtained for the determination of one unknown thermal coefficient in the overspecified problem that consists in adding a Neumann condition to the problem with a Dirichlet one if and only if some restrictions on data are verified.Spectral geometry of unduloidshttps://zbmath.org/1541.355532024-09-27T17:47:02.548271Z"Daily, Bill B."https://zbmath.org/authors/?q=ai:daily.bill-bSummary: This paper examines the eigenvalues and eigenfunctions of the Laplace operator associated with a set of mathematically defined surfaces which can be produced experimentally by attaching two equally sized rings to opposite poles of a soap bubble and separating the rings. The shapes produced are called unduloids. These calculations show 1) for a range of ring sizes, as a function of ring separation, the first indexed eigenvalue has a minimum, pointing to an ``optimum'' shape, and 2) given the eigenfunctions in the form of their differential equations and a preference for symmetry, the underlying unduloid geometry may be deduced.Convolution kernel determination problem in the third order Moore-Gibson-Thompson equationhttps://zbmath.org/1541.355542024-09-27T17:47:02.548271Z"Durdiev, D. K."https://zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovich|durdiev.durdimurod-kalanddarovich"Boltaev, A. A."https://zbmath.org/authors/?q=ai:boltaev.asliddin-askar-ugli"Rahmonov, A. A."https://zbmath.org/authors/?q=ai:rakhmonov.askar-akhmadovichSummary: This article is concerned with the study of the inverse problem of determining the difference kernel in a Volterra type integral term function in the third-order Moore-Gibson-Thompson (MGT) equation. First, the initial-boundary value problem is reduced to an equivalent problem. Using the Fourier spectral method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the integral equations are proved. The obtained solution to the integral equations of Volterra-type is also the unique solution to the equivalent problem. Based on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original inverse problem is proved.Uniqueness of the kernel determination problem in a integro-differential parabolic equation with variable coefficientshttps://zbmath.org/1541.355552024-09-27T17:47:02.548271Z"Durdiev, D. K."https://zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovich|durdiev.durdimurod-kalanddarovich"Nuriddinov, J. Z."https://zbmath.org/authors/?q=ai:nuriddinov.j-zSummary: We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the \(n\)-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem in which an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions \({{k}_1}(x,t)\) and \({{k}_2}(x,t)\) of the stated problem, an equation is formed for the difference of this solution. Further research is being conducted for the difference \({{k}_1}(x,t) - {{k}_2}(x,t)\) of solutions of the problem and using the techniques of integral equations estimates. It is shown that, if the unknown kernel \(k(x,t)\) can be represented as \(k(x,t) = \sum\limits_{i = 0}^N {{a}_i}(x){{b}_i}(t)\), then \({{k}_1}(x,t) \equiv{{k}_2}(x,t)\). Thus, the theorem on the uniqueness of the solution of the problem is proved.The backward problem of a stochastic PDE with bi-harmonic operator driven by fractional Brownian motionhttps://zbmath.org/1541.355562024-09-27T17:47:02.548271Z"Feng, Xiaoli"https://zbmath.org/authors/?q=ai:feng.xiaoli"Chen, Chen"https://zbmath.org/authors/?q=ai:chen.chen(no abstract)A unified Bayesian inversion approach for a class of tumor growth models with different pressure lawshttps://zbmath.org/1541.355572024-09-27T17:47:02.548271Z"Feng, Yu"https://zbmath.org/authors/?q=ai:feng.yu"Liu, Liu"https://zbmath.org/authors/?q=ai:liu.liu"Zhou, Zhennan"https://zbmath.org/authors/?q=ai:zhou.zhennanSummary: In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter \(m\), which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, \(m \rightarrow \infty\). Since the posterior distribution across the index regime \(m \in [2, \infty)\) can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.An inverse source problem for the Schrödinger equation with variable coefficients by data on flat subboundaryhttps://zbmath.org/1541.355582024-09-27T17:47:02.548271Z"Gölgeleyen, Fikret"https://zbmath.org/authors/?q=ai:golgeleyen.fikret"Gölgeleyen, İsmet"https://zbmath.org/authors/?q=ai:golgeleyen.ismet"Yamamoto, Masahiro"https://zbmath.org/authors/?q=ai:yamamoto.masahiroSummary: We consider an inverse source problem for the Schrödinger equation with variable coefficients. We prove the uniqueness of solution of the problem by data on a flat subboundary over time interval, under a certain condition of the coefficients of the principal terms. We first reduce the inverse problem to a Cauchy problem for a system of integro-differential equations by using Fourier transform. Next, we establish a pointwise Carleman type inequality which is the key tool in the proof of our main result.
{\copyright 2024 American Institute of Physics}Estimation of the Born data in inverse scattering of layered mediahttps://zbmath.org/1541.355592024-09-27T17:47:02.548271Z"Jia, Zekui"https://zbmath.org/authors/?q=ai:jia.zekui"Li, Maokun"https://zbmath.org/authors/?q=ai:li.maokun"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.4"Xu, Shenheng"https://zbmath.org/authors/?q=ai:xu.shenhengThis paper develops a scheme to map the total field data to the Born data in the layered media by the single-into single-output setup. Techniques be given in [\textit{V. Druskin} et al., SIAM J. Imaging Sci. 11, No. 1, 164--196 (2018; Zbl 1401.86006); \textit{L. Borcea} et al., Inverse Probl. 34, No. 6, Article ID 065008, 35 p. (2018; Zbl 1507.65160)] are applied to calculate the first term in the Born series with the reduced order model approach, and estimate recordings at fictitious sensors by using the time-domain Green's function. Since the estimated Born data only contain the single-scattering component, the nonlinearity of the inversion problem is reduced, and thus the direct imaging for high-dimensional layered media works. The effectiveness and comparison with MIMO data are illustrated by numerical experiments.
Reviewer: Fangfang Dou (Chengdu)Inverse problems for Kelvin-Voigt system with memory: global existence and uniquenesshttps://zbmath.org/1541.355602024-09-27T17:47:02.548271Z"Khompysh, Kh."https://zbmath.org/authors/?q=ai:khompysh.khonatbek"Shakir, A. G."https://zbmath.org/authors/?q=ai:shakir.aidos-ganizhanuly(no abstract)Inverse random source problem for the helium production-diffusion equationhttps://zbmath.org/1541.355612024-09-27T17:47:02.548271Z"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.199"Cheng, Hao"https://zbmath.org/authors/?q=ai:cheng.hao"Geng, Xiaoxiao"https://zbmath.org/authors/?q=ai:geng.xiaoxiao(no abstract)Stability for inverse source problems of the stochastic Helmholtz equation with a white noisehttps://zbmath.org/1541.355622024-09-27T17:47:02.548271Z"Li, Peijun"https://zbmath.org/authors/?q=ai:li.peijun.1"Liang, Ying"https://zbmath.org/authors/?q=ai:liang.yingSummary: This paper is concerned with the stability estimates for inverse source problems of the stochastic Helmholtz equation driven by white noise. The well-posedness is established for the direct source problems, which ensures the existence and uniqueness of solutions. The stability estimates are deduced for the inverse source problems, which aim to determine the strength of the random source. To enhance the stability of the inverse source problems, we incorporate a priori information regarding the regularity and support of the strength. In the case of homogeneous media, a Hölder stability estimate is established, providing a quantitative measure of the stability for reconstructing the source strength. For the more challenging scenario of inhomogeneous media, a logarithmic stability estimate is presented, capturing the intricate interactions between the source and the varying medium properties.On inverse problems in predator-prey modelshttps://zbmath.org/1541.355632024-09-27T17:47:02.548271Z"Li, Yuhan"https://zbmath.org/authors/?q=ai:li.yuhan"Liu, Hongyu"https://zbmath.org/authors/?q=ai:liu.hongyu"Lo, Catharine W. K."https://zbmath.org/authors/?q=ai:lo.catharine-wing-kwanUnder consideration is the following general coupled semi-linear system of the predator-prey model:
\[
\partial_t u - d_1 \Delta u = F(x,t,u,v), \ \partial_t v - d_2 \Delta v = G(x,t,u,v), \ (t,x) \in Q=(0,T)\times G,\ G\subset \mathbb{R}^n,
\]
\[
\partial_\nu u = \partial_\nu v = 0\ on\ S=(0,T)\times \partial G,\ u(x, 0) = f (x),\ v(x, 0) = g(x).
\]
Here, \(F(x,t,u,v)\) and \(G(x,t,u,v)\) are real-valued analytic functions. Both \(u\) and \(v\) are required to be non-negative in view of their physical meaning. Given the mapping \((f,g)\to (u|_S,v|_S,u(T,x),v(T,x))\), the problem is to restore the functions \(F,G\) provided that we know this mapping. The main result is the uniqueness theorem of this inverse problem.
Reviewer: Sergey G. Pyatkov (Khanty-Mansiysk)On the uniqueness of solutions to inverse problems for equations of various types in finding their right-hand sidehttps://zbmath.org/1541.355642024-09-27T17:47:02.548271Z"Sabitov, K. B."https://zbmath.org/authors/?q=ai:sabitov.kamil-basirovich(no abstract)A free boundary mathematical model of atherosclerosishttps://zbmath.org/1541.355652024-09-27T17:47:02.548271Z"Abi Younes, G."https://zbmath.org/authors/?q=ai:abi-younes.g"El Khatib, N."https://zbmath.org/authors/?q=ai:el-khatib.nader|el-khatib.noaman-a-f"Volpert, V."https://zbmath.org/authors/?q=ai:volpert.vladimir-a|volpert.vitaly-a(no abstract)Geometric properties for a Finsler Bernoulli exterior problemhttps://zbmath.org/1541.355662024-09-27T17:47:02.548271Z"Bianchini, Chiara"https://zbmath.org/authors/?q=ai:bianchini.chiaraSummary: The aim of this paper is to prove convexity results for an exterior anisotropic free boundary problem of the Bernoulli type. More precisely we recover the results obtained in [\textit{A. Henrot} and \textit{H. Shahgholian}, Nonlinear Anal., Theory Methods Appl. 28, No. 5, 815--823 (1997; Zbl 0863.35117)] for the exterior problem, in the Finsler setting.
For the entire collection see [Zbl 1497.42002].Well posedness analysis of a parabolic-hyperbolic free boundary problem for necrotic tumors growthhttps://zbmath.org/1541.355672024-09-27T17:47:02.548271Z"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.78|chen.wei.69|chen.wei.1|chen.wei.26|chen.wei.64|chen.wei.25|chen.wei.61|chen.wei.42|chen.wei.23|chen.wei.24|chen.wei.14|chen.wei.22|chen.wei.49|chen.wei.66|chen.wei.58|chen.wei.30|chen.wei|chen.wei.56|chen.wei.27|chen.wei.4|chen.wei.13|chen.wei.8|chen.wei.5|chen.wei.2|chen.wei.6|chen.wei.7|chen.wei.55|chen.wei.62|chen.wei.12"Wei, Xuemei"https://zbmath.org/authors/?q=ai:wei.xuemeiSummary: In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence.A diffusive Leslie-Gower type predator-prey model with two different free boundarieshttps://zbmath.org/1541.355682024-09-27T17:47:02.548271Z"Elmurodov, A. N."https://zbmath.org/authors/?q=ai:elmurodov.a-n"Sotvoldiyev, A. I."https://zbmath.org/authors/?q=ai:sotvoldyev.a-i|sotvoldiyev.akmal-i(no abstract)Propagation dynamics of a free boundary problem in advective environmentshttps://zbmath.org/1541.355692024-09-27T17:47:02.548271Z"Fan, Xueqi"https://zbmath.org/authors/?q=ai:fan.xueqi"Sun, Ningkui"https://zbmath.org/authors/?q=ai:sun.ningkui"Zhang, Di"https://zbmath.org/authors/?q=ai:zhang.diSummary: In this paper, we study the equation \(u_t = u_{x x} - \beta u_x + f (x, u)\) in the domain \(\{(t, x) \in \mathbb{R}^2 : t \geq 0\), \(x \in (- \infty, h (t)] \}\), where \(h (t)\) is the free boundary and \(\beta \geq 0\). The influence of the advection coefficient \(- \beta\) on the propagation dynamics of the solutions is considered. We find two parameters 2 and \(\beta^\ast\) such that when \(0 \leq \beta < 2\), only spreading happens; when \(2 \leq \beta < \beta^\ast\), there is a virtual spreading-transition-vanishing trichotomy result; when \(\beta \geq \beta^\ast\), only vanishing happens.Regularity results for solutions to elliptic obstacle problems in limit caseshttps://zbmath.org/1541.355702024-09-27T17:47:02.548271Z"Farroni, Fernando"https://zbmath.org/authors/?q=ai:farroni.fernando"Manzo, Gianluigi"https://zbmath.org/authors/?q=ai:manzo.gianluigiSummary: We prove the Lewy-Stampacchia's inequality for elliptic variational inequalities with obstacle involving Leray-Lions type operator whose simpler model case is given by the following
\[
u \in W^{1, N}_0(\Omega)\mapsto - \Delta_N u-\operatorname{div}\left(B(x) |u|^{N-2}u \right)
\]
where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^N\) with \(N \geqslant 2\), \(\Delta_N u\) denotes the classical \(N\)-Laplacian operator and the coefficient \(B:\Omega\rightarrow\mathbb{R}^N\) belongs to a suitable Lorentz-Zygmund space. For this kind of obstacle problems, we also provide regularity results and amongst them we give sufficient conditions to get boundedness of solutions.A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions. III: General casehttps://zbmath.org/1541.355712024-09-27T17:47:02.548271Z"Kaneko, Yuki"https://zbmath.org/authors/?q=ai:kaneko.yuki"Matsuzawa, Hiroshi"https://zbmath.org/authors/?q=ai:matsuzawa.hiroshi"Yamada, Yoshio"https://zbmath.org/authors/?q=ai:yamada.yoshioSummary: We consider the Stefan problem of nonlinear diffusion equation \(u_t = \Delta u+f(u)\) for \(t>0\) and \(x\in\Omega(t)(\subset\mathbb{R}^N)\) with positive bistable nonlinearity \(f \). We first prove that for any initial data, the long-time behavior of the solution is classified into four cases: vanishing, small spreading, big spreading and transition. In particular, we show that when transition occurs for the solution \(u \), there exists an \(x_0\in\mathbb{R}^N\) such that \(u(t,\cdot)\) converges as \(t\to\infty\) to an equilibrium solution which is radially symmetric and radially decreasing with center \(x_0 \).
We next give some results about large-time estimates of the expanding speed of \(\Omega(t)\) for small and big spreading cases. As in our previous paper [J. Math. Pures Appl. (9) 178, 1--45 (2023; Zbl 1522.35074)], it can be expected that, under a certain condition, every big spreading solution accompanies a propagating terrace. We have succeeded in understanding the large-time behavior of such a solution with terrace in terms of its level set.On the free boundary in a diffusion equation rushing across a desert zonehttps://zbmath.org/1541.355722024-09-27T17:47:02.548271Z"Lai, Pengchao"https://zbmath.org/authors/?q=ai:lai.pengchao"Lou, Bendong"https://zbmath.org/authors/?q=ai:lou.bendong"Lu, Junfan"https://zbmath.org/authors/?q=ai:lu.junfanSummary: We consider a one dimensional reaction diffusion equation with a free boundary. Suppose the free boundary will encounter a desert zone during its propagation process, and the population loss is serious in such a zone. We will prove that any solution of the problem converges to a stationary one, which must be one of four types with clear features. Our results provide sufficient conditions for the free boundary to be able or unable to rush successfully across the desert zone.Tangential contact between free and fixed boundaries for variational solutions to variable-coefficient Bernoulli-type free boundary problemshttps://zbmath.org/1541.355732024-09-27T17:47:02.548271Z"Moreira, Diego"https://zbmath.org/authors/?q=ai:moreira.diego-r"Shrivastava, Harish"https://zbmath.org/authors/?q=ai:shrivastava.harishSummary: In this paper, we show that, given appropriate boundary data, the free boundaries of minimizers of functionals of type \(J(v; A, \lambda_+, \lambda_-, \Omega)= \int_{\Omega} (\langle A(x)\nabla v,\nabla v\rangle + \Lambda (v))dx\) and the fixed boundary touch each other in a tangential fashion. We extend the results of \textit{A. L. Karakhanyan} et al. [Calc. Var. Partial Differ. Equ. 28, No. 1, 15--31 (2007; Zbl 1111.35135)] to the case of variable coefficients. We prove this result via classification of the global profiles, as per [loc. cit.].Nonexistence for higher order evolution inequalities with Hardy potential in \(\mathbb{R}^N\)https://zbmath.org/1541.355742024-09-27T17:47:02.548271Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We study the nonexistence of nontrivial weak solutions to evolution inequalities of the form
\[
\begin{cases}
\partial_t^k u -\Delta u + \frac{\lambda}{|x|^2} u \geq |u|^p \quad\text{in } (0, \infty) \times \mathbb{R}^N, \\
\partial_t^i u (0, x) = u_i (x), i = 0, 1, \cdots, k-1, \;\text{in } \mathbb{R}^N,
\end{cases}
\]
where \(u = u(t, x)\), \(k \geq 2\) is a natural number, \(\partial_t^i = \frac{\partial^i}{\partial t^i}\), \(N \geq 3\), \(\frac{2}{k} (N-2 + \frac{2}{k}) \leq \lambda <2N\) and \(p>1\). Namely, we show that, if \(u_{k-1} \geq 0\), then for all \(p>1\), there is no nontrivial weak solution. Our result improves a previous result obtained in [\textit{A. El Hamidi} and \textit{G. G. Laptev}, J. Math. Anal. Appl. 304, No. 2, 451--463 (2005; Zbl 1067.35159)], where the nonexistence of nontrivial weak solution has been established only for a certain range of \(p\).Quasi-ergodicity of transient patterns in stochastic reaction-diffusion equationshttps://zbmath.org/1541.355752024-09-27T17:47:02.548271Z"Adams, Zachary P."https://zbmath.org/authors/?q=ai:adams.zachary-pSummary: We study transient patterns appearing in a class of SPDE using the framework of quasi-stationary and quasi-ergodic measures. In particular, we prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion systems perturbed by additive cylindrical noise. We obtain convergence results in \(L^2\) and almost surely, and demonstrate an exponential rate of convergence to the quasi-stationary measure in an \(L^2\) norm. These results allow us to qualitatively characterize the behaviour of these systems in neighbourhoods of an invariant manifold of the corresponding deterministic systems at some large time \(t>0\), conditioned on remaining in the neighbourhood up to time \(t\). The approach we take here is based on spectral gap conditions, and is not restricted to the small noise regime.Multipoint initial-final value problem for the model of Davis with additive white noisehttps://zbmath.org/1541.355762024-09-27T17:47:02.548271Z"Konkina, Aleksandra Sergeevna"https://zbmath.org/authors/?q=ai:konkina.aleksandra-sergeevnaSummary: The evolution of the free surface of the filtering fluid in a reservoir of limited power is modeled by the Davis equation with homogeneous Dirichlet conditions. Depending on the nature of the free term describing the internal source of the liquid, the model will be deterministic or stochastic. The deterministic model has been studied in various aspects by many researchers with different initial (initial-final value conditions). The stochastic model is studied for the first time. The main result is the proof of the unique solvability of the evolutionary model with an additive white noise and a multipoint initial-final value condition.Existence and ergodicity for the two-dimensional stochastic Allen-Cahn-Navier-Stokes equationshttps://zbmath.org/1541.355772024-09-27T17:47:02.548271Z"Ndongmo Ngana, Aristide"https://zbmath.org/authors/?q=ai:ndongmo-ngana.aristide"Tachim Medjo, Theodore"https://zbmath.org/authors/?q=ai:tachim-medjo.theodoreSummary: We study in this article a stochastic version of a coupled Allen-Cahn-Navier-Stokes model in a two-dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter, both endowed with suitable boundary conditions. We prove the existence of solutions via a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods.Matrix representation of magnetic pseudo-differential operators via tight Gabor frameshttps://zbmath.org/1541.355782024-09-27T17:47:02.548271Z"Cornean, Horia D."https://zbmath.org/authors/?q=ai:cornean.horia-d"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernard"Purice, Radu"https://zbmath.org/authors/?q=ai:purice.raduSummary: In this paper we use some ideas from
[\textit{H. G. Feichtinger} and \textit{K. Gröchenig}, J. Funct. Anal. 146, No. 2, 464--495 (1997; Zbl 0887.46017);
\textit{K. Gröchenig}, Rev. Mat. Iberoam. 22, No. 2, 703--724 (2006; Zbl 1127.35089)]
and consider the description of Hörmander type pseudo-differential operators on \(\mathbb{R}^d\) (\(d \geq 1\)), including the case of the magnetic pseudo-differential operators introduced in
[\textit{V. Iftimie} et al., Publ. Res. Inst. Math. Sci. 43, No. 3, 585--623 (2007; Zbl 1165.35056);
Rev. Roum. Math. Pures Appl. 64, No. 2--3, 197--223 (2019; Zbl 1463.35517)],
with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.A maximal \(L_p\)-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processeshttps://zbmath.org/1541.355792024-09-27T17:47:02.548271Z"Choi, Jae-Hwan"https://zbmath.org/authors/?q=ai:choi.jae-hwan"Kim, Ildoo"https://zbmath.org/authors/?q=ai:kim.ildooSummary: Let \(Z=(Z_t)_{t\ge 0}\) be an additive process with a bounded triplet \((0,0,\varLambda_t)_{t\ge 0}\). Then the infinitesimal generators of \(Z\) is given by time dependent nonlocal operators as follows:
\[
\begin{aligned}
{\mathcal{A}}_Z(t)u(t,x) &=\lim_{h \downarrow 0} \frac{{\mathbb{E}}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h}\\
&=\int_{{\mathbb{R}}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla_x u(t,x)1_{|y|\le 1})\varLambda_t(dy).
\end{aligned}
\]
Suppose that for any Schwartz function \(\varphi\) on \({\mathbb{R}}^d\) whose Fourier transform is in \(C_c^{\infty}(B_{c_s} {\setminus} B_{c_s^{-1}})\), there exist positive constants \(N_0\), \(N_1\), and \(N_2\) such that
\[
\int_{{\mathbb{R}}^d}|{\mathbb{E}}[\varphi (x+r^{-1}Z_t)]|dx\le N_0 e^{- \frac{ N_1 t}{s(r)}},\quad \forall (r,t)\in (0,1)\times [0,T],
\]
and
\[
\Vert \psi^{\mu}(r^{-1}D)\varphi \Vert_{L_1({\mathbb{R}}^d)}\le \frac{N_2}{s(r)},\quad \forall r\in (0,1).
\]
where \(s\) is a scaling function (Definition 2.4), \(c_s\) is a positive constant related to \(s\), \(\mu\) is a symmetric Lévy measure on \({\mathbb{R}}^d\), \(\psi^{\mu}(r^{-1}D)\varphi (x)= {\mathcal{F}}^{-1} \left[ \psi^{\mu}(r^{-1}\xi) {\mathcal{F}}[\varphi]\right] (x)\) and \(\psi^{\mu}(\xi){:=}\int_{{\mathbb{R}}^d}(e^{iy\cdot \xi}-1-iy\cdot \xi 1_{|y|\le 1})\mu (dy)\). In particular, above assumptions hold for Lévy measures \(\varLambda_t\) having a nice lower bound and \(\mu\) satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures \(\varLambda_t\) and they do not have to be symmetric. In this paper, we establish the \(L_p\)-solvability to the initial value problem
\[
\frac{\partial u}{\partial t}(t,x)={\mathcal{A}}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in (0,T)\times{\mathbb{R}}^d,
\tag{0.2}
\]
where \(u_0\) is contained in a scaled Besov space \(B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb{R}}^d)\) (see Definition 2.8) with a scaling function \(s\), exponent \(p \in (1,\infty)\), \(q\in [1,\infty)\), and order \(\gamma \in [0,\infty)\). We show that equation (0.2) is uniquely solvable and the solution \(u\) obtains full-regularity gain from the diffusion generated by a stochastic process \(Z\). In other words, there exists a unique solution \(u\) to equation (0.2) in \(L_q((0,T); H_p^{\mu ;\gamma}({\mathbb{R}}^d))\), where \(H_p^{\mu ;\gamma}({\mathbb{R}}^d)\) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution \(u\) satisfies
\[
\Vert u\Vert_{L_q((0,T);H_p^{\mu ;\gamma}({\mathbb{R}}^d))}\le N\Vert u_0\Vert_{B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb{R}}^d)},
\]
where \(N\) is independent of \(u\) and \(u_0\). We finally remark that our operators \({\mathcal{A}}_Z(t)\) include logarithmic operators such as \(-a(t)\log (1-\varDelta)\) (Corollary 3.2) and operators whose symbols are non-smooth such as \(-\sum_{j=1}^dc_j(t)(-\varDelta)^{\alpha /2}_{x^j}\) (Corollary 3.9).Wigner analysis of Fourier integral operators with symbols in the Shubin classeshttps://zbmath.org/1541.355802024-09-27T17:47:02.548271Z"Cordero, Elena"https://zbmath.org/authors/?q=ai:cordero.elena"Giacchi, Gianluca"https://zbmath.org/authors/?q=ai:giacchi.gianluca"Rodino, Luigi"https://zbmath.org/authors/?q=ai:rodino.luigi"Valenzano, Mario"https://zbmath.org/authors/?q=ai:valenzano.marioSummary: We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes \(\Gamma^m(\mathbb{R}^{2d})\), with negative order \(m\). The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander's class \(S^0_{0, 0}(\mathbb{R}^{2d})\). Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order \(m\).Classification of the two-component Benjamin-Ono systemshttps://zbmath.org/1541.370772024-09-27T17:47:02.548271Z"Zhao, Min"https://zbmath.org/authors/?q=ai:zhao.min.2"Qu, Changzheng"https://zbmath.org/authors/?q=ai:qu.changzhengSummary: The Benjamin-Ono equation involving the Hilbert transformation has been studied extensively from different standpoints. Its variant forms and multi-component extensions have been proposed. In this paper, we study the classification of two-component Benjamin-Ono-type systems of the general form. Our classification is carried out by developing the perturbative symmetry approach due to \textit{A. V. Mikhailov} and \textit{V. S. Novikov} [J. Phys. A, Math. Gen. 35, No. 22, 4775--4790 (2002; Zbl 1039.35008)]. As a result, new two-component integrable Benjamin-Ono type systems are obtained.Solitons in a semi-infinite ferromagnet with anisotropy of the easy axis typehttps://zbmath.org/1541.370782024-09-27T17:47:02.548271Z"Kiselev, V. V."https://zbmath.org/authors/?q=ai:kiselev.v-v|kiselev.vladimir-valerievich|kiselev.valery-vSummary: We propose a special variant of the inverse scattering transform method to construct and analyze soliton excitations in a semi-infinite sample of an easy-axis ferromagnet in the case of a partial pinning of spins at its surface. We consider the limit cases of free edge spins and spins that are fully pinned at the sample boundary. We find frequency and modulation characteristics of solitons localized near the sample surface. In the case of different degrees of edge spin pinning, we study changes in the cores of moving solitons as a result of their elastic reflection from the sample boundary. We obtain integrals of motion that control the dynamics of magnetic solitons in a semi-infinite sample.Soliton solutions of the negative-order nonlinear Schrödinger equationhttps://zbmath.org/1541.370792024-09-27T17:47:02.548271Z"Urazboev, G. U."https://zbmath.org/authors/?q=ai:urazboev.gayrat"Baltaeva, I. I."https://zbmath.org/authors/?q=ai:baltaeva.iroda-i|baltaeva.idora-ismailovna"Babadjanova, A. K."https://zbmath.org/authors/?q=ai:babadjanova.aygul-kamildjanovnaSummary: We discuss the integration of the Cauchy problem for the negative-order nonlinear Schrödinger equation in the class of rapidly decreasing functions via the inverse scattering problem method. In particular, we obtain the time dependence of scattering data of the Zakharov-Shabat system with the potential that is a solution of the considered problem. We give an explicit representation of the one-soliton solution of the negative-order nonlinear Schrödinger equation based on the obtained results.mKdV-related flows for Legendrian curves in the pseudohermitian 3-spherehttps://zbmath.org/1541.370802024-09-27T17:47:02.548271Z"Calini, Annalisa"https://zbmath.org/authors/?q=ai:calini.annalisa-m"Ivey, Thomas"https://zbmath.org/authors/?q=ai:ivey.thomas-a"Musso, Emilio"https://zbmath.org/authors/?q=ai:musso.emilioSummary: We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group \(\mathrm{U}(2)\). We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in \(\mathrm{U}(2)\). Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.Parity-time symmetric solitons of the complex KP equationhttps://zbmath.org/1541.370822024-09-27T17:47:02.548271Z"Chang, Jen-Hsu"https://zbmath.org/authors/?q=ai:chang.jen-hsuSummary: We construct the parity-time symmetric solitons of the complex KP equation using the totally nonnegative Grassmannian. We obtain that every element in the totally nonnegative orthogonal Grassmannian corresponds to a parity-time symmetric soliton solution.Hamiltonian theory of motion of dark solitons in the theory of nonlinear Schrödinger equationhttps://zbmath.org/1541.370832024-09-27T17:47:02.548271Z"Kamchatnov, A. M."https://zbmath.org/authors/?q=ai:kamchatnov.anatoly-mSummary: We develop a method for deriving Hamilton's equations describing the dynamics of solitons when they move along an inhomogeneous and time-varying large-scale background for nonlinear wave equations that are completely integrable in the Ablowitz-Kaup-Newell-Segur (AKNS) scheme. The method is based on the development of old Stokes' ideas that allow analytically continuing the relations for linear waves into the soliton region, and is practically implemented in the example of the defocusing nonlinear Schrödinger equation. A condition is formulated under which the external potential is only to be taken into account when describing the evolution of the background, and that this case, the Newton equation is obtained for the soliton dynamics in an external potential.Long time stability result for \(d\)-dimensional nonlinear Schrödinger equationhttps://zbmath.org/1541.370842024-09-27T17:47:02.548271Z"Cong, Hongzi"https://zbmath.org/authors/?q=ai:cong.hongzi"Li, Siming"https://zbmath.org/authors/?q=ai:li.siming"Wu, Xiaoqing"https://zbmath.org/authors/?q=ai:wu.xiaoqingSummary: In this paper, we study the long time dynamical behavior of the solutions for \(d\)-dimensional nonlinear Schödinger equation with general nonlinearity by using Birkhoff normal form technique and the so-called \textbf{tame} property in Gevrey space and modified Sobolev space.Dynamical discussion and diverse soliton solutions via complete discrimination system approach along with bifurcation analysis for the third order NLSEhttps://zbmath.org/1541.370852024-09-27T17:47:02.548271Z"Rizvi, S. T. R."https://zbmath.org/authors/?q=ai:rizvi.syed-tahir-raza"Seadawy, A. R."https://zbmath.org/authors/?q=ai:seadawy.aly-r"Mustafa, B."https://zbmath.org/authors/?q=ai:mustafa.balsamSummary: The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schrödinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equationshttps://zbmath.org/1541.370882024-09-27T17:47:02.548271Z"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrong"Xia, Huan"https://zbmath.org/authors/?q=ai:xia.huanSummary: We consider the stochastic non-autonomous Kuramoto-Sivashinsky equation with multiplicative white noise and colored coefficients. Due to the anti-dissipative term in the equation, we restrict the state space on the Lebesgue space of odd functions and then obtain a pullback random attractor \(\mathcal{A}\) in the odd Lebesgue space, where a special bridge function plays a crucial role in an a priori estimate. Moreover, we prove the continuity in expectation for the set-valued mapping \((t ,s) \to \mathcal{A}(t,\theta_s\omega)\) with respect to the Hausdorff metric and establish the residual dense continuity in the pathwise sense. In the application, we verify four conditions including the joint continuity of the non-autonomous random dynamical system, the union closedness of a tempered universe, local compactness and boundedness in expectation for the random attractor.Sufficient and necessary criteria for backward asymptotic autonomy of pullback attractors with applications to retarded sine-Gordon lattice systemshttps://zbmath.org/1541.370892024-09-27T17:47:02.548271Z"Yang, Shuang"https://zbmath.org/authors/?q=ai:yang.shuang"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Zhang, Qiangheng"https://zbmath.org/authors/?q=ai:zhang.qianghengSummary: In this paper, we investigate the backward asymptotic autonomy of pullback attractors for asymptotically autonomous processes. Namely, time-components of the pullback attractors semi-converge to the global attractors of the corresponding limiting semigroups as the time-parameter goes to negative infinity. The present article is divided into two parts: theories and applications. In the theoretical part, we establish a sufficient and necessary criterion with respect to the backward asymptotic autonomy via backward compactness of pullback attractors. Moreover, this backward asymptotic autonomy is considered by the periodicity of pullback attractors. As for the applications part, we apply the abstract results to non-autonomous retarded sine-Gordon lattice systems. By backward uniform tail-estimates of solutions, we prove the existence of a pullback and global attractor for such lattice systems such that the backward asymptotic autonomy is satisfied. Furthermore, it is also fulfilled under the assumptions of the periodicity for the non-delay forcing and the convergence for processes.
{\copyright 2024 American Institute of Physics}Invasion analysis of a reaction-diffusion-advection predator-prey model in spatially heterogeneous environmenthttps://zbmath.org/1541.371022024-09-27T17:47:02.548271Z"Sun, Yihuan"https://zbmath.org/authors/?q=ai:sun.yihuanSummary: In this paper, we study a reaction-diffusion-advection predator-prey model with Holling type-II functional response in heterogeneous environment is considered, where the prey are subject to both random and directed movements. We show the effect of changes in diffusion rates of predator and prey on whether prey can be invaded when advection rate is small. Moreover, we show that the predator with a large advection rate is driven to extinction.Wave operator on compact manifoldshttps://zbmath.org/1541.420102024-09-27T17:47:02.548271Z"Pan, Yali"https://zbmath.org/authors/?q=ai:pan.yali"Fan, Dashan"https://zbmath.org/authors/?q=ai:fan.dashanSummary: On an \(n\)-dimensional \((n\geq 2)\) compact connected manifold without boundary, we obtain the sharp range of \(p\) to ensure the \(L^p\) convergence of the wave operator. We are not able to show the optimum result at the end point \(|\frac{1}{p} -\frac {1}{2}| = \frac{1}{n-1}\) if \(n\neq 3\), but an interesting result for \(n=3\) is given. The main result is an extension of a result on \(n\)-dimensional Euclidean space \(\mathbb{R}^n\).Compactness of localization operators on modulation spaces of \(\omega\)-tempered distributionshttps://zbmath.org/1541.420222024-09-27T17:47:02.548271Z"Boiti, Chiara"https://zbmath.org/authors/?q=ai:boiti.chiara"DeMartino, Antonino"https://zbmath.org/authors/?q=ai:de-martino.antoninoSummary: We give sufficient conditions for compactness of localization operators on modulation spaces \({\mathcal{M}}^{p,q}_{m_{\lambda}}(\mathbb{R}^d)\) of \(\omega\)-tempered distributions whose short-time Fourier transform is in the weighted mixed space \(L^{p,q}_{m_\lambda}\) for \(m_\lambda(x) = e^{ \lambda \omega (x)}\).
For the entire collection see [Zbl 1497.42002].Estimates for the commutators of Riesz transforms related to Schrödinger-type operatorshttps://zbmath.org/1541.420262024-09-27T17:47:02.548271Z"Wang, Yanhui"https://zbmath.org/authors/?q=ai:wang.yanhui"Wang, Kang"https://zbmath.org/authors/?q=ai:wang.kang.2Summary: Let \(\mathcal{L}_2=(-\Delta)^2+V^2\) be the Schrödinger-type operator on \(\mathbb{R}^n\) \((n \geq 5)\), let \(H^1_{\mathcal{L}_2}(\mathbb{R}^n)\) be the Hardy space related to \(\mathcal{L}_2\), and let \(\mathrm{BMO}_{\theta}(\rho)\) be the BMO-type space introduced by \textit{B. Bongioanni} et al. [J. Fourier Anal. Appl. 17, No. 1, 115--134 (2011; Zbl 1213.42075)]. In this paper, we investigate the boundedness of commutator \([b, T_{\alpha, \beta, j}]\), which is generated by the Riesz transform \(T_{\alpha, \beta, j}=V^{2\alpha} \nabla^j \mathcal{L}_2^{-\beta}\), \(j=1,2,3\), and \(b \in \mathrm{BMO}_{\theta}(\rho)\). Here, \(0 < \alpha \leq 1 - \frac{j}{4}\), \(\frac{j}{4}<\beta\leq 1\), \(\beta-\alpha=\frac{j}{4}\), and the nonnegative potential \(V\) belongs to both the reverse Hölder class \(\mathrm{RH}_s\) with \(s \geq \frac{n}{2}\) and the Gaussian class associated with \((-\Delta)^2\). The \(L^p\) boundedness of \([b, T_{\alpha, \beta, j}]\) is obtained, and it is also shown that \([b, T_{\alpha, \beta, j}]\) is bounded from \(H^1_{\mathcal{L}_2}(\mathbb{R}^n)\) to weak \(L^1 (\mathbb{R}^n)\).On Fourier restriction type problems on compact Lie groupshttps://zbmath.org/1541.430132024-09-27T17:47:02.548271Z"Zhang, Yunfeng"https://zbmath.org/authors/?q=ai:zhang.yunfengSummary: In this article, we obtain new results for Fourier restriction-type problems on compact Lie groups. We first provide a sharp form of \(L^p\) estimates of irreducible characters in terms of their Laplace-Beltrami eigenvalue, and as a consequence provide some sharp \(L^p\) estimates of joint eigenfunctions for the ring of conjugate-invariant differential operators. Then, we improve upon the previous range of exponent for scale-invariant Strichartz estimates for the Schrödinger equation, and provide new \(L^p\) bounds of Laplace-Beltrami eigenfunctions in terms of their eigenvalue similar to known bounds on tori. A key ingredient in our proof of these results is a barycentric-semiclassical subdivision of the Weyl alcove in a maximal torus. On each component of this subdivision we carry out the analysis of characters and exponential sums, and the circle method of Hardy-Littlewood and Kloosterman.Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact typehttps://zbmath.org/1541.430142024-09-27T17:47:02.548271Z"Bhowmik, Mithun"https://zbmath.org/authors/?q=ai:bhowmik.mithun"Ray, Swagato K."https://zbmath.org/authors/?q=ai:ray.swagato-kSummary: We study unique continuation inequalities for the free Schrödinger equation in the context of Riemannian symmetric spaces of noncompact type. The results imply that if the solution is small at two different times outside sets of finite measure, then the solution is small in the whole space. On the Euclidean spaces, these inequalities are equivalent to certain uncertainty principles in harmonic analysis.On a system of \(q\)-modified Laplace transform and its applicationshttps://zbmath.org/1541.440062024-09-27T17:47:02.548271Z"Kilicman, Adem"https://zbmath.org/authors/?q=ai:kilicman.adem"Sinha, Arvind Kumar"https://zbmath.org/authors/?q=ai:sinha.arvind-kumar"Panda, Srikumar"https://zbmath.org/authors/?q=ai:panda.srikumarSummary: We introduce $q$-modified Laplace transform and establish theoretical results. We also give some applications of $q$-modified Laplace transform for solving homogeneous and non-homogeneous Mboctara partial differential equations with initial and boundary values problems to show its effectiveness and performance of the proposed transform.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}The first Hadamard variation of Neumann-Poincaré eigenvalues on the spherehttps://zbmath.org/1541.450012024-09-27T17:47:02.548271Z"Ando, Kazunori"https://zbmath.org/authors/?q=ai:ando.kazunori"Kang, Hyeonbae"https://zbmath.org/authors/?q=ai:kang.hyeonbae"Miyanishi, Yoshihisa"https://zbmath.org/authors/?q=ai:miyanishi.yoshihisa"Ushikoshi, Erika"https://zbmath.org/authors/?q=ai:ushikoshi.erikaSummary: The Neumann-Poincaré operator on the two-dimensional sphere has \( \frac {1}{2(2k+1)}\), \( k=0,1,2,\dots \), as its eigenvalues and the corresponding multiplicity is \( 2k+1\). We consider the bifurcation of eigenvalues under deformation of domains, and show that the Frechét derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincaré operator.The Neumann and Dirichlet problems for the total variation flow in metric measure spaceshttps://zbmath.org/1541.460202024-09-27T17:47:02.548271Z"Górny, Wojciech"https://zbmath.org/authors/?q=ai:gorny.wojciech"Mazón, José M."https://zbmath.org/authors/?q=ai:mazon-ruiz.jose-mSummary: We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for \(L^1\) initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to \textit{N. Gigli} [Nonsmooth differential geometry -- an approach tailored for spaces with Ricci curvature bounded from below. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1404.53056)] and a version of the Gauss-Green formula.Harmonic functions with BMO traces and their limiting behaviors on metric measure spaceshttps://zbmath.org/1541.460222024-09-27T17:47:02.548271Z"Jin, Yutong"https://zbmath.org/authors/?q=ai:jin.yutong"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.5"Shen, Tianjun"https://zbmath.org/authors/?q=ai:shen.tianjunSummary: Let \((X,d,\mu)\) be a metric measure space satisfying a doubling condition and the \(L^2\)-Poincaré inequality. This paper is concerned with the boundary behavior of harmonic functions on the (open) upper half-space \(X\times\mathbb{R}_+\). We show that the traces of harmonic functions are in the bounded mean oscillation (BMO) space if and only if they satisfy the Carleson condition. This characterization is new even for uniformly elliptic operators on Euclidean spaces.The Cheeger cut and Cheeger problem in metric measure spaceshttps://zbmath.org/1541.460232024-09-27T17:47:02.548271Z"Mazón, José M."https://zbmath.org/authors/?q=ai:mazon-ruiz.jose-mSummary: In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterizations of the Cheeger constant: a variational one and another one through the eigenvalue of the 1-Laplacian. We obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by \textit{J.~Cheeger} [in: Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195--199 (1970; Zbl 0212.44903)].
We also study the Cheeger problem. Through a variational characterization of the Cheeger sets we prove the existence of Cheeger sets and obtain a characterization of the calibrable sets and a version of the Max Flow Min Cut Theorem.Existence of an optimal control for a semilinear evolution equation with unbounded operatorhttps://zbmath.org/1541.490032024-09-27T17:47:02.548271Z"Chernov, A. V."https://zbmath.org/authors/?q=ai:chernov.alexey.1|chernov.aleksei-vyacheslavovich|chernov.andrei-vladimirovich|chernov.andrew-vSummary: An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author's previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples.Optimal control of a nonsmooth PDE arising in the modeling of shear-thickening fluidshttps://zbmath.org/1541.490062024-09-27T17:47:02.548271Z"De los Reyes, Juan Carlos"https://zbmath.org/authors/?q=ai:de-los-reyes.juan-carlos"Quiloango, Paola"https://zbmath.org/authors/?q=ai:quiloango.paolaSummary: This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (\textit{Bouligand (B) stationarity}), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a \textit{weak stationarity system} for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a \textit{strong stationarity system} that includes a characterization of the Lagrange multiplier on the active and inactive sets.Optimality conditions and Lipschitz stability for non-smooth semilinear elliptic optimal control problems with sparse controlshttps://zbmath.org/1541.490122024-09-27T17:47:02.548271Z"Nhu, Vu Huu"https://zbmath.org/authors/?q=ai:nhu.vu-huu"Sang, Phan Quang"https://zbmath.org/authors/?q=ai:sang.phan-quangSummary: This paper is concerned with first- and second-order optimality conditions as well as the stability for non-smooth semilinear optimal control problems involving the \(L^1\)-norm of the control in the cost functional. In addition to the appearance of the \(L^1\)-norm leading to the non-differentiability of the objective and promoting the sparsity of the optimal controls, the non-smoothness of the nonlinear coefficient in the state equation causes the same property of the control-to-state operator. Exploiting a regularization scheme, we derive \(C\)-stationarity conditions for any local optimal control. Under a structural assumption on the associated state, we define the curvature functional for the part not including the \(L^1\)-norm of controls of the objective for which the second-order necessary and sufficient optimality conditions with minimal gap are shown. Furthermore, under a more restrictive structural assumption imposed on the mentioned state, an explicit formula for the curvature is established and thus the explicit second-order optimality conditions are stated. Finally, the Lipschitz stability of local solutions with respect to the sparsity parameter is shown.Space-time spectral method for an optimal control problem governed by a two-dimensional PDE constrainthttps://zbmath.org/1541.490172024-09-27T17:47:02.548271Z"Rezazadeh, Arezou"https://zbmath.org/authors/?q=ai:rezazadeh.arezou"Darehmiraki, Majid"https://zbmath.org/authors/?q=ai:darehmiraki.majidSummary: In this paper, we solve a two-dimensional optimal control problem with a parabolic partial differential equation (PDE) constraint. First, the space-time spectral method is used to discretize time derivative and space derivative. Then the aforementioned problem is transformed into a solvable algebraic system. Since the spectral methods converge spectrally in both space and time, they have gained a significant attention in the last few decades. We prove that our method has exponential rates of convergence in both space and time.On the continuity of the continuous Steiner symmetrizationhttps://zbmath.org/1541.490222024-09-27T17:47:02.548271Z"Buttazzo, Giuseppe"https://zbmath.org/authors/?q=ai:buttazzo.giuseppeIn this paper, the author wants to construct a continuous deformation \(\Omega_t, t\in [0,1]\) starting from an open set (or quasi-open set) \(\Omega_0\) up to the ball of same volume \(\Omega_1\) and satisfying the three conditions:
(i) the volume is preserved: \(|\Omega_t|=|\Omega|\) for any \(t\in [0,1]\),
(ii) the map \(t \mapsto \Omega_t\) is continuous for the \(\gamma\)-convergence,
(iii) the map \(t \mapsto \lambda_1( \Omega_t)\) is decreasing, where \(\lambda_1(\Omega_t)\) denotes the first eigenvalue of the Laplace operator on \(\Omega_t\) with Dirichlet boundary conditions. Alternatively, one can want that the map \(t\mapsto T(\Omega_t)\) is increasing where \(T(\Omega_t)\) denotes now the torsional rigidity of \(\Omega_t\).
A first natural idea is to use the continuous Steiner symmetrization as introduced by \textit{F. Brock} [Math. Nachr. 172, 25--48 (1995; Zbl 0886.49010)]. Indeed it satisfies conditions (i) and (iii), but unfortunately not always condition (ii) as it was already mentioned in the paper [\textit{D. Bucur} and \textit{A. Henrot}, Potential Anal. 13, No. 2, 127--145 (2000; Zbl 0977.31007)]. Therefore, in this paper, the author proposes a modification of the continuous Steiner symmetrization to circumvent this difficulty. This gives an answer to the initial question, not for any set \(\Omega_0\) but for a class of sets wider than the one we get by just using continuous Steiner symmetrization.
Reviewer: Antoine Henrot (Vandœuvre-lès-Nancy)A geometric Laplace methodhttps://zbmath.org/1541.530212024-09-27T17:47:02.548271Z"Léger, Flavien"https://zbmath.org/authors/?q=ai:leger.flavien"Vialard, François-Xavier"https://zbmath.org/authors/?q=ai:vialard.francois-xavierSummary: A classical tool for approximating integrals is the Laplace method. The first- and higher-order Laplace formulas are most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation to the first-order term of the Laplace method. The central tool is a metric introduced by \textit{Y.-H. Kim} and \textit{R. J. McCann} [J. Eur. Math. Soc. (JEMS) 12, No. 4, 1009--1040 (2010; Zbl 1191.49046)] in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two metrics arising naturally in the Kim-McCann framework. We give an explicitly quantified version of the Laplace formula, as well as examples of applications.Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric datahttps://zbmath.org/1541.530652024-09-27T17:47:02.548271Z"Isobe, Takeshi"https://zbmath.org/authors/?q=ai:isobe.takeshi"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannick"Xu, Tian"https://zbmath.org/authors/?q=ai:xu.tianSummary: Let \((M, g, \sigma)\) be an \(m\)-dimensional closed spin manifold, with a fixed Riemannian metric \(g\) and a fixed spin structure \(\sigma\); let \(\mathbb{S}(M)\) be the spinor bundle over \(M\). The spinorial Yamabe-type problems address the solvability of the following equation
\[
D_g \psi = f(x) |\psi|_g^{\frac{2}{m - 1}} \psi, \quad \psi : M \to \mathbb{S}(M), x \in M
\]
where \(D_g\) is the associated Dirac operator and \(f : M \to \mathbb{R}\) is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when \(m = 2\), a solution corresponds to a conformal isometric immersion of the universal covering \(\widetilde{M}\) into \(\mathbb{R}^3\) with prescribed mean curvature \(f\); meanwhile, for general dimensions and \(f \equiv \mathit{constant} \neq 0\), a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant.
The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold \((S^m, g)\) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential \(f\) is constant (say \(f \equiv 1\)) with the background metric \(g\) being a \(C^k\) perturbation of the canonical round metric \(g_{S^m}\), which is not conformally flat somewhere on \(S^m\); 2). \(f\) is a perturbation from constant and is of class \(C^2\), while the background metric \(g \equiv g_{S^m}\).Singularities of axially symmetric volume preserving mean curvature flowhttps://zbmath.org/1541.531102024-09-27T17:47:02.548271Z"Athanassenas, Maria"https://zbmath.org/authors/?q=ai:athanassenas.maria"Kandanaarachchi, Sevvandi"https://zbmath.org/authors/?q=ai:kandanaarachchi.sevvandiThe authors investigate the formation of singularities for surfaces evolving by volume-preserving mean curvature flow. They prove that for axially symmetric flows -- surfaces of revolution -- in \(\mathbb{R}^3\) with Neumann boundary conditions the first developing singularity is of Type I. This result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometric quantities involving gradients, curvature terms and derivatives thereof. These important results hold in arbitrary dimensions.
Reviewer: Vincenzo Vespri (Firenze)Type-0 singularities in the network flow -- evolution of treeshttps://zbmath.org/1541.531252024-09-27T17:47:02.548271Z"Mantegazza, Carlo"https://zbmath.org/authors/?q=ai:mantegazza.carlo"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Pluda, Alessandra"https://zbmath.org/authors/?q=ai:pluda.alessandraA curve \(\gamma \) of class \(C^{1}([0,1];\mathbb{R}^{2})\) is regular if \( \partial _{x}\gamma (x)\neq 0\) for every \(x\in \lbrack 0,1]\). Its unit tangent vector \(\tau =\frac{\partial _{x}\gamma }{\left\vert \partial _{x}\gamma \right\vert }\) is well defined, its unit normal vector is defined as \(\nu =R\tau \), where \(R:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}\) is the counterclockwise rotation centered in the origin of \(\mathbb{R}^{2}\) of angle \(\pi /2\). Assuming that the curve \(\gamma \) is of class \(C^{2}([0,1]; \mathbb{R}^{2})\), the curvature vector is defined as \(\overrightarrow{k}(x)= \frac{\partial _{x}^{2}\gamma }{\left\vert \partial _{x}\gamma \right\vert ^{2}}=\kappa (x)\nu (x)\), where \(\kappa (x)\) is the scalar curvature, and the arclength parameter is given by \(s(x)=\int_{0}^{x}\left\vert \partial _{x}\gamma (\xi )\right\vert d\xi \).
The authors define a network \(\mathcal{N }\) as a connected set in the Euclidean plane, composed of finitely many regular, embedded and sufficiently smooth curves that meet only at their endpoints. They call it a regular network if it possesses only triple junctions, where the unit tangent vectors form angles of 120 degrees. A network is a tree if it does not contain loops. Each curve of a network is supposed to admit a regular parametrization \(\gamma \in C^{2}([0,1];\mathbb{R}^{2})\).
Given an initial regular network \(\mathcal{N}_{0}\), composed of \(N\) curves \( \gamma _{0}^{i}\in C^{2}([0,1];\mathbb{R}^{2})\) that meet at \(m\) triple junctions \(\mathcal{O}_{1},\ldots ,\mathcal{O}_{m}\) in a bounded, convex, open, and regular subset \(\Omega \) of \(\mathbb{R}^{2}\) and possibly with \( \ell \) external vertices \(P_{1},\ldots ,P_{\ell }\in \partial \Omega \), a family of homeomorphic networks \(\mathcal{N}_{t}\) is a solution to the motion by curvature in \([0,T)\) with initial datum \(\mathcal{N}_{0}\) if there exist \(N\) regular curves parametrized by \(\gamma ^{i}\in C^{2}([0,T);[0,1])\) that satisfy
\[\left\langle \partial _{t}\gamma ^{i}(t,x),\nu ^{i}(t,x)\right\rangle =\kappa ^{i}(t,x), i=1,\ldots ,N,
\]
\[
\gamma ^{r}(1,x)=P^{r}, r=1,\ldots ,\ell , \sum_{i=1}^{3}\tau ^{p_{j}}(\mathcal{ O}^{p},t)=0,
\]
at every triple junction \(\mathcal{O}^{p}\), and \(\gamma ^{i}(0,x)=\gamma _{0}^{i}(x)\), \(i=1,\ldots ,N\), for every \(x\in \lbrack 0,1]\) and \(t\in \lbrack 0,T)\).
The authors quote from the literature a short time existence\ result for this problem: Given a regular network \(\mathcal{N}_{0}\) of class \(C^{2}\), there exists a maximal solution to the network flow with initial datum \(\mathcal{N}_{0}\) in the maximal time interval \([0,T_{max})\). It is unique up to reparametrization and it admits a smooth parametrization for all positive times.
Let \(\mathcal{N}_{t}\) be the maximal solution to the motion by curvature of networks in the time interval \([0,T)\). Then, either \( T=+\infty \) or at least one of the following properties holds: The limit inferior for \(t\rightarrow T\) of the length of at least one curve of the network \(\mathcal{N}_{t}\) is equal to zero, or the limit superior for \(t\rightarrow T\) of the integral of the squared curvature is \(+\infty \).
Assuming the multiplicity-one conjecture which says that every limit of parabolic rescalings of the flow around a fixed point in \(\mathbb{R}^{2}\) is a flow of embedded networks with multiplicity one, the authors prove that the curvature of the maximal solution \(\mathcal{N}_{t}\) to the motion by curvature in the time interval \([0,T)\), with initial datum \(\mathcal{N}_{0}\) (a regular network which is a tree), is uniformly bounded during the flow. Deeply analyzing possible singularities arising at the maximal time of existence, the authors quote from results of the literature that if \( \mathcal{N}_{0}\) is a initial regular network of class \(C^{2}\) in a strictly convex, bounded and open set \(\Omega \subset \mathbb{R}^{2}\) and \(\mathcal{N} _{t}\) a maximal solution to the motion by curvature in \([0,T)\), with initial datum \(\mathcal{N}_{0}\), and if either \(\mathcal{N}_{0}\) has at most two triple junctions, or \(\mathcal{N}_{0}\) is a tree and if for all \(t\in \lbrack 0,T)\) the triple junctions of \(\mathcal{N}_{t}\) remains uniformly far from each others and from the vertices fixed on \(\partial \Omega \), then the Multiplicity-One Conjecture holds true.
In the rest of the paper, the authors give a complete description of the evolution of tree-like networks till the first singular time.\ Let \(\mathcal{N}_{0}\) be a regular tree composed of \(N\) curves in an open, convex and regular \(\Omega \subset \mathbb{R}^{2}\), and \(\mathcal{N}_{t}\) be the maximal solution to the motion by curvature in the time interval \([0,T)\) with initial datum \(\mathcal{N}_{0} \). Assume that the multiplicity-one conjecture holds and that \( T<\infty \). Then the curves \(\{\gamma ^{i}\}_{i=1}^{N}\) of the flow \( \mathcal{N}_{t}\), up to reparametrization proportional to arc-length, converge in \(C_{loc}^{1}\) either to constant maps or to regular \(C^{2}\)-curves composing the limit network \(\widehat{\mathcal{N}}_{T}\), as \( t\rightarrow T\). The only possible singularities are given by the collapse of isolated ``inner'' curves of the network, producing a regular 4-point, or the collapse of some \textquotedblleft boundary curves\textquotedblright\ on the fixed end-points of the network, letting two concurring curves form at such an end-point an angle of 120 degrees. The authors then define a shrinking self-similar solution to the motion by curvature as a time-dependent family of networks \(\mathcal{N}_{t}\) parametrized by \( \{\gamma ^{i}\}_{i=1}^{N}\) such that each evolving curve has the form \(\gamma ^{i}(t,x)=\lambda (t)\eta ^{i}(x)\), with \(\lambda (t)>0\) and \(\lambda ^{\prime }(t)<0\). A (static) network whose curves satisfy the equation \( \kappa ^{i}(x)+\left\langle \eta ^{i}(x),\nu ^{i}(x)\right\rangle =0\) is called a shrinker. Assuming that the multiplicity-one conjecture holds true, the authors characterize such shrinkers. The paper ends with the description of an example of a type-0 singularity for the motion by curvature.
Reviewer: Alain Brillard (Riedisheim)Asymptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifoldshttps://zbmath.org/1541.580152024-09-27T17:47:02.548271Z"Mazepa, E. A."https://zbmath.org/authors/?q=ai:mazepa.elena-alexeevna"Ryaboshlykova, D. K."https://zbmath.org/authors/?q=ai:ryaboshlykova.d-kSummary: The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular, the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary noncompact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.Inverse boundary value problems for wave equations with quadratic nonlinearitieshttps://zbmath.org/1541.580192024-09-27T17:47:02.548271Z"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-a"Zhang, Yang"https://zbmath.org/authors/?q=ai:zhang.yang.14Authors' abstract: We study inverse problems for the nonlinear wave equation
\[
\square_g u + w(x, u, \nabla_g u) = 0
\]
in a Lorentzian manifold \((M, g)\) with boundary, where \(\nabla_g u\) denotes the gradient and \(w(x, u, \xi)\) is smooth and quadratic in \(\xi \). Under appropriate assumptions, we show that the conformal class of the Lorentzian metric \(g\) can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of \(w(x, u, \xi)\) with respect to \(u\) up to null forms.
Reviewer: Mohammed El Aïdi (Bogotá)The expected nodal volume of non-Gaussian random band-limited functions, and their doubling indexhttps://zbmath.org/1541.600222024-09-27T17:47:02.548271Z"Sartori, Andrea"https://zbmath.org/authors/?q=ai:sartori.andrea"Wigman, Igor"https://zbmath.org/authors/?q=ai:wigman.igorSummary: The asymptotic law for the expected nodal volume of random non-Gaussian monochromatic band-limited functions is determined in vast generality. Our methods combine microlocal analytic techniques and modern probability theory. A particularly challenging obstacle that we need to overcome is the possible concentration of nodal volume on a small portion of the manifold, requiring solutions in both disciplines and, in particular, the study of the distribution of the doubling index of random band-limited functions. As for the fine aspects of the distribution of the nodal volume, such as its variance, it is expected that the non-Gaussian monochromatic functions behave qualitatively differently compared to their Gaussian counterpart. Some conjectures pertaining to these are put forward within this manuscript.Quickest real-time detection of multiple Brownian driftshttps://zbmath.org/1541.600282024-09-27T17:47:02.548271Z"Ernst, P. A."https://zbmath.org/authors/?q=ai:ernst.philip-a"Mei, H."https://zbmath.org/authors/?q=ai:mei.hongwei"Peskir, G."https://zbmath.org/authors/?q=ai:peskir.goranSummary: Consider the motion of a Brownian particle in \(n\) dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly \(k\) of the coordinate processes get a (known) nonzero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which the \(k\) coordinate processes get the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimizes the probability of a false early detection and the expected delay of a missed late detection. The elliptic case \(k=1\) has been settled in [the authors, Ann. Appl. Probab. 32, No. 4, 2652--2670 (2022; Zbl 1499.60129)] where the hypoelliptic case \(1<k<n\) resolved in the present paper was left open (the case \(k=n\) reduces to the classic case \(n=1\) having a known solution). We also show that the methodology developed solves the problem in the general case where exactly \(k\) is relaxed to \textit{any} number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multidimensional hypoelliptic problem has been solved exactly in the literature.Existence of geometric ergodic periodic measures of stochastic differential equationshttps://zbmath.org/1541.600372024-09-27T17:47:02.548271Z"Feng, Chunrong"https://zbmath.org/authors/?q=ai:feng.chunrong"Zhao, Huaizhong"https://zbmath.org/authors/?q=ai:zhao.huaizhong"Zhong, Johnny"https://zbmath.org/authors/?q=ai:zhong.johnnyThe paper works out sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space. The main results are applied to weakly dissipative stochastic differential equations (SDEs), Langevin equations (SDEs with additive noise) and gradient systems. The Fokker-Planck equation for the associated probability density of the periodic measure with physically relevant applications is referred to.
In passing, we note that periodic measures are the time-periodic counterpart to invariant measures for stochastic dynamical systems in order to characterize long-term periodic behavior of them. The concept of stochastic resonance is related to their investigation as well.
Reviewer: Henri Schurz (Carbondale)Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity. II: Dynamicshttps://zbmath.org/1541.600422024-09-27T17:47:02.548271Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjornThe author studies the normalized three dimensional wave equation with a so called Hartree nonlinearity and random initial data. The main result is a proof of the invariance of the Gibbs measure under the flow of this equation. The novelty of the results lies in the allowance of a singularity in the Gibbs measure with respect to a Gaussian free field.
The paper is part of a series of two. In this paper the author focuses on dynamical aspects of the main result.
For Part I, see [\textit{B. Bringmann}, Stoch. Partial Differ. Equ., Anal. Comput. 10, No. 1, 1--89 (2022; Zbl 1491.60095)].
Reviewer: Denis R. Bell (Jacksonville)Invariant measures for the nonlinear stochastic heat equation with no drift termhttps://zbmath.org/1541.600432024-09-27T17:47:02.548271Z"Chen, Le"https://zbmath.org/authors/?q=ai:chen.le.1|chen.le"Eisenberg, Nicholas"https://zbmath.org/authors/?q=ai:eisenberg.nicholasSummary: This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation \(\frac{\partial u}{\partial t} - \frac{1}{2}\Delta u = b(u){\dot{W}}\), where \(b\) is assumed to be a globally Lipschitz continuous function and the noise \({\dot{W}}\) is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function \(\rho\), which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho ({\mathbb{R}}^d)\). In particular, our result covers the \textit{parabolic Anderson model} (i.e., the case when \(b(u) = \lambda u\)) starting from the Dirac delta measure.Averaging principles for multiscale stochastic Cahn-Hilliard systemhttps://zbmath.org/1541.600442024-09-27T17:47:02.548271Z"Gao, Peng"https://zbmath.org/authors/?q=ai:gao.peng.1|gao.peng.3|gao.pengSummary: In this paper, we will establish averaging principles for the multiscale stochastic Cahn-Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn-Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn-Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).
{\copyright 2024 American Institute of Physics}Hölder continuity of stochastic heat equation with rough Gaussian noisehttps://zbmath.org/1541.600452024-09-27T17:47:02.548271Z"Han, Yuecai"https://zbmath.org/authors/?q=ai:han.yuecai"Wu, Guanyu"https://zbmath.org/authors/?q=ai:wu.guanyuSummary: In this paper, we consider the stochastic heat equation \(\partial_t u (t, x) = \frac{1}{2} \Delta u (t, x) + u (t, x) \diamondsuit \dot{W} (t, x)\), where \(\dot{W} (t, x)\) is the formal derivative of a fractional Brownian sheet with \(H_0 > 1 / 2\) and \(0 < H_i < 1\) for \( i = 1, \dots, d \). We obtain the Hölder continuity of the solution in time and space.Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its applicationhttps://zbmath.org/1541.600472024-09-27T17:47:02.548271Z"Kumar, Ankit"https://zbmath.org/authors/?q=ai:kumar.ankit"Kinra, Kush"https://zbmath.org/authors/?q=ai:kinra.kush"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: In this article, we establish the \textit{Wong-Zakai approximation} result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo-Galerkin approximation method, compactness arguments, and Prokhorov's and Skorokhod's representation theorems to ensure the existence of a \textit{probabilistically weak solution} for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada-Watanabe theorem allows us to conclude the existence of a \textit{probabilistically strong solution} (analytically weak solution) for the approximating system. Subsequently, we establish the Wong-Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong-Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work's functional framework.Nonparametric estimation for independent and identically distributed stochastic differential equations with space-time dependent coefficientshttps://zbmath.org/1541.600572024-09-27T17:47:02.548271Z"Comte, Fabienne"https://zbmath.org/authors/?q=ai:comte.fabienne"Genon-Catalot, Valentine"https://zbmath.org/authors/?q=ai:genon-catalot.valentineSummary: We consider \(N\) independent and identically distributed one-dimensional inhomogeneous diffusion processes \((X_i(t),i=1,\dots,N)\) with drift \(\mu(t,x)=\sum_{j=1}^K\alpha_j(t) g_j(x)\) and diffusion coefficient \(\sigma (t,x)\), where \(K\) and the functions \(g_j(x)\) and \(\sigma(t,x)\) are known. Our concern is the nonparametric estimation of the \(K\)-dimensional unknown function \((\alpha_j(t),j=1,\dots,K)\) from the continuous observation of the sample paths \((X_i(t))\) throughout a fixed time interval \([0,\tau]\). A collection of projection estimators belonging to a product of finite-dimensional subspaces of \(\mathbb{L}^2([0,\tau])\) is built. The \(\mathbb{L}^2\)-risk is defined by the expectation of either an empirical norm or a deterministic norm fitted to the problem. Rates of convergence for large \(N\) are discussed. A data-driven choice of the dimensions of the projection spaces is proposed. The theoretical results are illustrated by numerical experiments on simulated data.First hitting time of a one-dimensional Lévy flight to small targetshttps://zbmath.org/1541.600582024-09-27T17:47:02.548271Z"Gomez, Daniel"https://zbmath.org/authors/?q=ai:gomez.daniel-armando"Lawley, Sean D."https://zbmath.org/authors/?q=ai:lawley.sean-dSummary: First hitting times (FHTs) describe the time it takes a random ``searcher'' to find a ``target'' and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order \(s\in (0,1)\) (describing a \((2s)\)-stable Lévy flight whose squared displacement scales as \(t^{1/s}\) in time \(t)\) and targets of radius \({\varepsilon}\ll 1\), we show that the MFHT is order one for \(s\in (1/2,1)\) and diverges as \(\log (1/{\varepsilon})\) for \(s=1/2\) and \({\varepsilon}^{2s-1}\) for \(s\in (0,1/2)\). We then use our asymptotic results to identify the value of \(s\in (0,1]\) which minimizes the average MFHT and find that (a) this optimal value of \(s\) vanishes for sparse targets and (b) the value \(s=1/2\) (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.Asymptotic analysis and simulation of mean first passage time for active Brownian particles in 1-Dhttps://zbmath.org/1541.600622024-09-27T17:47:02.548271Z"Iyaniwura, Sarafa A."https://zbmath.org/authors/?q=ai:iyaniwura.sarafa-a"Peng, Zhiwei"https://zbmath.org/authors/?q=ai:peng.zhiweiSummary: Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small Péclet \((Pe)\) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to \(\mathcal{O}(Pe^2)\). We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.Singular integrals and Feller semigroups with jump phenomenahttps://zbmath.org/1541.600682024-09-27T17:47:02.548271Z"Taira, Kazuaki"https://zbmath.org/authors/?q=ai:taira.kazuakiSummary: This paper is devoted to the real analysis methods for the problem of construction of Markov processes with boundary conditions in probability. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called Ventcel' (Wentzell) boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by continuous paths and by jumps in the state space and it obeys the Ventcel' boundary condition that consists of six terms corresponding to a diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first order Ventcel' boundary value problems for second order elliptic Waldenfels integro-differential operators. By using the Calderón-Zygmund theory of singular integrals in real analysis, we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces, which extend earlier theorems due to Bony-Courrège-Priouret to the VMO (vanishing mean oscillation) case. Our proof is based on various maximum principles for second order elliptic Waldenfels operators with discontinuous coefficients in the framework of Sobolev spaces.Branching stable processes and motion by mean curvature flowhttps://zbmath.org/1541.600742024-09-27T17:47:02.548271Z"Becker, Kimberly"https://zbmath.org/authors/?q=ai:becker.kimberly-e"Etheridge, Alison"https://zbmath.org/authors/?q=ai:etheridge.alison-m"Letter, Ian"https://zbmath.org/authors/?q=ai:letter.ianThe paper concerns the solution of Allen-Cohn reaction diffusion equations, \textit{S. M. Allen} and \textit{J. W. Cahn} [Acta Metall. 27, No. 6, 1085--1095 (1979; \url{doi:10.1016/0001-6160(79)90196-2})], in which the diffusion term is replaced by the generator of a pure jump process (fractional Allen-Cohn equation).
It is used to model the motion of hybrid zones in populations exhibiting long-range dispersal, i.e., narrow geographical regions where the dispersing population reproduces hybrid offspring with a residential population. It is shown that the long-term behaviour of hybrid zones maintained by selection in long-range dispersing populations converges to motion by mean curvature flow under a long range of possible spatial scalings. An explicit description of the interface width and speed of convergence is given. This goes beyond the older result of \textit{C. Imbert} and \textit{P. E. Souganidis} [``Phasefield theory for fractional diffusion-reaction equations and applications'', Preprint, \url{arXiv:0907.5524}].
The proof, related in method to \textit{A. Etheridge} et al. [Electron. J. Probab. 22, Paper No. 103, 40 p. (2017; Zbl 1386.60300)], is probabilistic: It describes the solutions of the fractional Allen-Cohn equation in terms of ternary branching \(\alpha\)-stable motions which are coupled to ternary branching Brownian motions subordinated by truncated stable subordinators.
Reviewer: Heinrich Hering (Rockenberg)An offline-online decomposition method for efficient linear Bayesian goal-oriented optimal experimental design: application to optimal sensor placementhttps://zbmath.org/1541.621972024-09-27T17:47:02.548271Z"Wu, Keyi"https://zbmath.org/authors/?q=ai:wu.keyi"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng.2"Ghattas, Omar"https://zbmath.org/authors/?q=ai:ghattas.omar-nSummary: Bayesian optimal experimental design (OED) plays an important role in minimizing model uncertainty with limited experimental data in a Bayesian framework. In many applications, rather than minimizing the uncertainty in the inference of model parameters, one seeks to minimize the uncertainty of a model-dependent quantity of interest (QoI). This is known as goal-oriented OED (GOOED). Here, we consider GOOED for linear Bayesian inverse problems governed by large-scale models represented by partial differential equations (PDE) that are computationally expensive to solve. In particular, we consider optimal sensor placement by maximizing an expected information gain (EIG) for the QoI. We develop an efficient method to solve such problems by deriving a new formulation of the goal-oriented EIG. Based on this formulation we propose an offline-online decomposition scheme that achieves significant computational reduction by computing all of the PDE-dependent quantities in an offline stage just once, and optimizing the sensor locations in an online stage without solving any PDEs. Moreover, in the offline stage we need only to compute low-rank approximations of two Hessian-related operators. The computational cost of these low-rank approximations, measured by the number of PDE solves, does not depend on the parameter or data dimensions for a large class of elliptic, parabolic, and sufficiently dissipative hyperbolic inverse problem that exhibit dimension-independent rapid spectra decay. We carry out detailed error analysis for the approximate goal-oriented EIG due to the low-rank approximations of the two operators. Furthermore, in the online stage we extend a swapping greedy method to optimize the sensor locations developed in our recent work that is demonstrated to be more efficient than a standard greedy method. We conduct a numerical experiment for a contaminant transport inverse problem with an infinite-dimensional parameter field to demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm.New approach for the chaotic dynamical systems involving Caputo-Prabhakar fractional derivative using Adams-Bashforth schemehttps://zbmath.org/1541.650532024-09-27T17:47:02.548271Z"Derakhshan, Mohammadhossein"https://zbmath.org/authors/?q=ai:derakhshan.mohammadhossein"Aminataei, Azim"https://zbmath.org/authors/?q=ai:aminataei.azimSummary: The main aim of the current paper is to propose a new numerical scheme based on the Adams-Bashforth method for solving two component time fractional differential equations involving the Caputo-Prabhakar fractional derivative of order \(\mu \). The existence and uniqueness of the solutions based on the Adams-Bashforth method reported for two component differential equations are discussed. Also, in this paper stability and convergence are demonstrated. Moreover, two numerical example are demonstrated in order to show the validity and reliability of the proposed methods.Efficient finite difference WENO scheme for hyperbolic systems with non-conservative productshttps://zbmath.org/1541.650652024-09-27T17:47:02.548271Z"Balsara, Dinshaw S."https://zbmath.org/authors/?q=ai:balsara.dinshaw-s"Bhoriya, Deepak"https://zbmath.org/authors/?q=ai:bhoriya.deepak"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wang"Kumar, Harish"https://zbmath.org/authors/?q=ai:kumar.harishSummary: Higher order finite difference weighted essentially non-oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of the applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of \textit{M. Dumbser} and \textit{D. S. Balsara} [J. Comput. Phys. 304, 275--319 (2016; Zbl 1349.76603)] as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of the applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages: (i) It can capture jumps in stationary linearly degenerate wave families exactly. (ii) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows. Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow, multiphase debris flow and two-layer shallow water equations are also shown to document the robustness of the method. For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results. Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.A stable FE-FD method for anisotropic parabolic PDEs with moving interfaceshttps://zbmath.org/1541.650662024-09-27T17:47:02.548271Z"Dong, Baiying"https://zbmath.org/authors/?q=ai:dong.baiying"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilin.1"Ruiz-Álvarez, Juan"https://zbmath.org/authors/?q=ai:ruiz-alvarez.juanSummary: In this paper, a new finite element and finite difference (FE-FD) method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes. In the spatial discretization, the standard \(P_1\) FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite, while near the interface, the maximum principle preserving immersed interface discretization is applied. In the time discretization, a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate. Correction terms are needed when the interface crosses grid lines. The moving interface is represented by the zero level set of a Lipschitz continuous function. Numerical experiments presented in this paper confirm second order convergence.Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equationhttps://zbmath.org/1541.650682024-09-27T17:47:02.548271Z"Hu, Dongdong"https://zbmath.org/authors/?q=ai:hu.dongdongSummary: This paper is concerned with high-order accurate, linearly implicit and energy-preserving schemes for the coupled nonlinear wave equation. To this end, a novel auxiliary variable approach proposed in a recent paper [\textit{L. Ju} et al., SIAM J. Numer. Anal. 60, No. 4, 1905--1931 (2022; Zbl 07572361)] is applied to reformulate the governing equation in an equivalent system, which possesses a modified energy that consists of primary functional and quadratic functional. Subsequently, we utilize the extrapolation strategy/prediction-correction technique for treating the nonlinear terms of the equivalent system to obtain a linearized energy-preserving system. Then, a series of fully decoupled high-order linear energy-preserving schemes are constructed by using the symplectic Runge-Kutta (RK) methods. We show that the modified energy functional can be precisely conserved under certain circumstances for the coefficients of the symplectic RK methods, and the unique solvability of the proposed time-stepping scheme is analyzed. In numerical implementations, a class of linear algebraic equations with constant-coefficient matrices need to be solved at each time step and only less computational costs are required. Extensive numerical experiments of the coupled nonlinear wave equation with local/nonlocal diffusion operators are provided to illustrate the energy conservation law and the computational efficiency of the schemes in long-time simulations.Asymptotically compatible energy and dissipation law of the nonuniform \(\mathrm{L}2\)-\(1_{\sigma}\) scheme for time fractional Allen-Cahn modelhttps://zbmath.org/1541.650692024-09-27T17:47:02.548271Z"Liao, Hong-lin"https://zbmath.org/authors/?q=ai:liao.honglin"Zhu, Xiaohan"https://zbmath.org/authors/?q=ai:zhu.xiaohan"Sun, Hong"https://zbmath.org/authors/?q=ai:sun.hongSummary: We build an asymptotically compatible energy of the variable-step \(\mathrm{L}2\)-\(1_{\sigma}\) scheme for the time-fractional Allen-Cahn model with the Caputo's fractional derivative of order \(\alpha \in (0,1)\), under a weak step-ratio constraint \(\tau_k /\tau_{k-1}\geq r_{\star}(\alpha)\) for \(k\geq 2\), where \(\tau_k\) is the \(k\)-th time-step size and \(r_{\star}(\alpha)\in (0.3865, 0.4037)\) for \(\alpha \in (0,1)\). It provides a positive answer to the open problem in [\textit{H.-l. Liao} et al., J. Comput. Phys. 414, Article ID 109473, 15 p. (2020; Zbl 1440.65116)], and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen-Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order \(\mathrm{L}2\)-\(1_{\sigma}\) formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy.Two new approximations for generalized Caputo fractional derivative and their application in solving generalized fractional sub-diffusion equationshttps://zbmath.org/1541.650702024-09-27T17:47:02.548271Z"Li, Xuhao"https://zbmath.org/authors/?q=ai:li.xuhao"Wong, Patricia J. Y."https://zbmath.org/authors/?q=ai:wong.patricia-j-ySummary: In this paper, we propose two new approximation methods on a general mesh for the generalized Caputo fractional derivative of order \(\alpha \in (0,1)\). The accuracy of these two methods is shown to be of order \((3-\alpha)\) which improves some previous work done to date. To demonstrate the accuracy and usefulness of the proposed approximations, we carry out experiment on test examples and apply these approximations to solve generalized fractional sub-diffusion equations. The numerical results indicate that the proposed methods perform well in practice. Our contributions lie in two aspects: (i) we propose high order approximations that work on a general mesh; (ii) we establish the well-posedness of generalized fractional sub-diffusion equations and develop numerical schemes using the new high order approximations.A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularityhttps://zbmath.org/1541.650712024-09-27T17:47:02.548271Z"Ou, Caixia"https://zbmath.org/authors/?q=ai:ou.caixia"Wang, Zhibo"https://zbmath.org/authors/?q=ai:wang.zhibo"Vong, Seakweng"https://zbmath.org/authors/?q=ai:vong.seakwengSummary: In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous \(L2\)-\(1_\sigma\) formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.Fitted tension spline scheme for a singularly perturbed parabolic problem with time delayhttps://zbmath.org/1541.650732024-09-27T17:47:02.548271Z"Tesfaye, Sisay Ketema"https://zbmath.org/authors/?q=ai:tesfaye.sisay-ketema"Duressa, Gemechis File"https://zbmath.org/authors/?q=ai:duressa.gemechis-file"Gemechu Dinka, Tekle"https://zbmath.org/authors/?q=ai:gemechu-dinka.tekle"Woldaregay, Mesfin Mekuria"https://zbmath.org/authors/?q=ai:woldaregay.mesfin-mekuriaSummary: A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter \(\varepsilon\) as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficientshttps://zbmath.org/1541.650742024-09-27T17:47:02.548271Z"Wang, Yifei"https://zbmath.org/authors/?q=ai:wang.yifei"Huang, Jin"https://zbmath.org/authors/?q=ai:huang.jin"Li, Hu"https://zbmath.org/authors/?q=ai:li.huSummary: In this paper, a second-order difference scheme is developed to solve two-dimensional two-sided space distributed-order fractional diffusion equation with variable coefficients. In the spatial direction, a second-order difference scheme is proposed, the distribution-order integral is discretized by the Gauss-Legendre quadrature formula and the space fractional derivative is approximated by the weighted and shifted Grünwald-Letnikov operators. In addition, the time direction is discretized into a second-order difference scheme by the Crank-Nicolson method. Therefore, the main numerical scheme is developed. Furthermore, a small perturbation is added to the main difference scheme to construct an alternating-direction implicit scheme. Also, the stability and convergence of the numerical scheme are proved. Finally, some numerical results are provided to show the accuracy and efficiency of the proposed method.Finite difference approximations of the spatially homogeneous Fokker-Planck-Landau equationhttps://zbmath.org/1541.650752024-09-27T17:47:02.548271Z"Wollman, Stephen"https://zbmath.org/authors/?q=ai:wollman.stephenSummary: Finite difference methods are developed for approximating the spatially homogeneous Fokker-Planck-Landau equation for Coulomb collisions. The numerical methods apply the Fast Fourier Transform to improve time efficiency. Computational work is then done to compare the numerical approximations using FFT with the numerical method of
\textit{S. Wollman} [J. Comput. Appl. Math. 324, 173--203 (2017; Zbl 1365.65199)]. In addition, some computations are done to verify the theoretical rate of convergence of the FPL equation proved in
[\textit{R. M. Strain} and \textit{Y. Guo}, Arch. Ration. Mech. Anal. 187, No. 2, 287--339 (2008; Zbl 1130.76069)].Automatic solid reconstruction from 3-D points set for flow simulation via an immersed boundary methodhttps://zbmath.org/1541.650762024-09-27T17:47:02.548271Z"Narváez, Gabriel F."https://zbmath.org/authors/?q=ai:narvaez.gabriel-f"Ferrand, Martin"https://zbmath.org/authors/?q=ai:ferrand.martin"Fonty, Thomas"https://zbmath.org/authors/?q=ai:fonty.thomas"Benhamadouche, Sofiane"https://zbmath.org/authors/?q=ai:benhamadouche.sofianeSummary: Dealing with complex geometries for industrial applications is challenging in computational fluid dynamic workflows. Current developments in scan devices offer the possibility to represent very complex solid geometries in fluid dynamic solvers. This paper proposes a novel approach for reconstructing solid geometry from 3-D scans and flow simulation. Based on a 3-D point cloud, the approach automatically reconstructs the solid surface by including local solid planes in any convex computational cell. An immersed boundary method is then used to impose appropriate boundary conditions on the solid surfaces in the co-located finite volume context. The present approach avoids the complex and time-consuming manual/assisted meshing typical of body-fitted mesh workflows while showing satisfactory robustness and accuracy.
For the entire collection see [Zbl 1529.65004].A class of IMEX schemes and their error analysis for the Navier-Stokes Cahn-Hilliard systemhttps://zbmath.org/1541.650782024-09-27T17:47:02.548271Z"Huang, Fukeng"https://zbmath.org/authors/?q=ai:huang.fukeng"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jieSummary: We construct a class of implicit-explicit (IMEX) schemes for the Navier-Stokes Cahn-Hilliard (NSCH) system and carry out a rigorous error analysis for both semi-discrete and fully discrete (with a Fourier spectral approximation in space) schemes in the space periodic case. The schemes are based on the consistent splitting approach for the Navier-Stokes equations to decouple the computation of velocity and pressure, and the generalized scalar auxiliary variable (GSAV) approach to provide uniform bound for the numerical solutions. Our IMEX schemes are fully decoupled and linear, only requiring to solve a sequence of Poisson type equations at each time step. With help of the uniform bound for the numerical solutions, we derive global error estimates in the two-dimensional case as well as local error estimates in the three-dimensional case for temporal orders one to five. We also present some numerical examples to validate the schemes.Locally conservative and flux consistent iterative methodshttps://zbmath.org/1541.650792024-09-27T17:47:02.548271Z"Linders, Viktor"https://zbmath.org/authors/?q=ai:linders.viktor"Birken, Philipp"https://zbmath.org/authors/?q=ai:birken.philippSummary: Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining \textit{locally conservative} and \textit{flux consistent} iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton-Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow.A new 2-level implicit high accuracy compact exponential approximation for the numerical solution of nonlinear fourth order Kuramoto-Sivashinsky and Fisher-Kolmogorov equationshttps://zbmath.org/1541.650802024-09-27T17:47:02.548271Z"Mohanty, R. K."https://zbmath.org/authors/?q=ai:mohanty.ranjan-kumar"Sharma, Divya"https://zbmath.org/authors/?q=ai:sharma.divyaSummary: This paper discusses about a new compact 2-level implicit numerical method in the form of exponential approximation for finding the approximate solution of nonlinear fourth order Kuramoto-Sivashinsky and Fisher-Kolmogorov equations, which have applications in chemical engineering. The described method has an accuracy of temporal order two and a spatial order three (or four) on a variable (or constant) mesh. The approach has been demonstrated to be applicable to both non-singular and singular issues. This article has established the stability of the current technique. The suggested approach is used to solve several benchmark nonlinear parabolic problems associated in chemistry and chemical engineering, and the computed results are compared with the existing results to demonstrate the proposed method's superiority.Identification of the boundary condition in the diffusion model of the hydrodynamic flow in a chemical reactorhttps://zbmath.org/1541.650822024-09-27T17:47:02.548271Z"Gamzaev, Khanlar Mekhvali oglu"https://zbmath.org/authors/?q=ai:gamzaev.khanlar-mekhvali-oglu"Baĭramova, Nushaba Khanlar gyzy"https://zbmath.org/authors/?q=ai:bairamova.nushaba-khanlar-gyzySummary: The motion of a hydrodynamic flow in a chemical reactor described by a one-dimensional one-parameter diffusion model is considered. Within the framework of this model, the task is set to identify the boundary condition at the reactor outlet containing an unknown concentration of the reagent under study leaving the reactor in a stream. In this case, the law of change in the concentration of the reagent over time at the reactor inlet is additionally set. After the introduction of dimensionless variables, a discrete analogue of the transformed inverse problem in the form of a system of linear algebraic equations is constructed by the method of difference approximation. The discrete analogue of the additional condition is written as a functional and the solution of a system of linear algebraic equations is presented as a variational problem with local regularization. A special representation is proposed for the numerical solution of the constructed variational problem. As a result, the system of linear equations for each discrete value of a dimensionless time splits into two independent linear subsystems, each of which is solved independently of each other. As a result of minimizing the functional, an explicit formula was obtained for determining the approximate concentration of the reagent under study in the flow leaving the reactor at each discrete value of the dimensionless time. The proposed computational algorithm has been tested on the data of a model chemical reactor.Dictionary-based model reduction for state estimationhttps://zbmath.org/1541.650832024-09-27T17:47:02.548271Z"Nouy, Anthony"https://zbmath.org/authors/?q=ai:nouy.anthony"Pasco, Alexandre"https://zbmath.org/authors/?q=ai:pasco.alexandreSummary: We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold \(\mathcal{M}\) of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on \(\mathcal{M}\) coming from model order reduction. Variational approaches based on linear approximation of \(\mathcal{M}\), such as PBDW, yield a recovery error limited by the Kolmogorov width of \(\mathcal{M}\). To overcome this issue, piecewise-affine approximations of \(\mathcal{M}\) have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to \(\mathcal{M}\). In this paper, we propose a state estimation method relying on dictionary-based model reduction, where space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of \(\ell_1\)-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.An arbitrarily high order and asymptotic preserving kinetic scheme in compressible fluid dynamichttps://zbmath.org/1541.650862024-09-27T17:47:02.548271Z"Abgrall, Rémi"https://zbmath.org/authors/?q=ai:abgrall.remi"Nassajian Mojarrad, Fatemeh"https://zbmath.org/authors/?q=ai:nassajian-mojarrad.fatemehSummary: We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by \textit{S. Jin} and \textit{Z. Xin} [Commun. Pure Appl. Math. 48, No. 3, 235--276 (1995; Zbl 0826.65078)]. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a ``Knudsen'' number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of \textit{R. Abgrall} and \textit{D. Torlo} [Commun. Math. Sci. 20, No. 2, 297--326 (2022; Zbl 1492.65229)] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.A comparative study of different sets of variables in a discontinuous Galerkin method with entropy balance enforcementhttps://zbmath.org/1541.650872024-09-27T17:47:02.548271Z"Alberti, Luca"https://zbmath.org/authors/?q=ai:alberti.luca"Bassi, Francesco"https://zbmath.org/authors/?q=ai:bassi.francesco"Carnevali, Emanuele"https://zbmath.org/authors/?q=ai:carnevali.emanuele"Colombo, Alessandro"https://zbmath.org/authors/?q=ai:colombo.alessandro-g"Crivellini, Andrea"https://zbmath.org/authors/?q=ai:crivellini.andrea"Nigro, Alessandra"https://zbmath.org/authors/?q=ai:nigro.alessandraSummary: This paper investigates the effectiveness of different sets of variables in solving the compressible Euler equations using a modal Discontinuous Galerkin framework. Alongside the commonly used \textit{conservative} and \textit{primitive} variables, the \textit{entropy} and \textit{logarithmic} sets are considered to enforce entropy conservation/stability and positivity preservation of the thermodynamic state, respectively. An explicit correction to enforce entropy conservation/stability at the discrete level is also considered, with a significant increase in robustness for some of the solution strategies. Several two-dimensional inviscid test cases are computed to compare the performance of the different sets of variables, adding a directional shock-capturing term to the discretised equations when necessary. The entropy and logarithmic sets proved to be the most robust, completing simulations of an astrophysical jet at Mach number 2000 up to polynomial degree seven.Stability analysis of a finite element approximation for the Navier-Stokes equation with free surfacehttps://zbmath.org/1541.650882024-09-27T17:47:02.548271Z"Audusse, Emmanuel"https://zbmath.org/authors/?q=ai:audusse.emmanuel"Barrenechea, Gabriel R."https://zbmath.org/authors/?q=ai:barrenechea.gabriel-r|barrenechea.gabriel-raul"Decoene, Astrid"https://zbmath.org/authors/?q=ai:decoene.astrid"Quemar, Pierrick"https://zbmath.org/authors/?q=ai:quemar.pierrickSummary: In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.A structure-preserving parametric finite element method for geometric flows with anisotropic surface energyhttps://zbmath.org/1541.650892024-09-27T17:47:02.548271Z"Bao, Weizhu"https://zbmath.org/authors/?q=ai:bao.weizhu"Li, Yifei"https://zbmath.org/authors/?q=ai:li.yifeiSummary: We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy density \(\gamma (\boldsymbol{n})\), where \(\boldsymbol{n} \in \mathbb{S}^{1}\) represents the outward unit normal vector. We begin with the anisotropic surface diffusion which possesses two well-known geometric structures -- area conservation and energy dissipation -- during the evolution of the closed curve. By introducing a novel surface energy matrix \(\boldsymbol{G}_{k} (\boldsymbol{n})\) depending on \(\gamma (\boldsymbol{n})\) and the Cahn-Hoffman \(\boldsymbol{\xi}\)-vector as well as a nonnegative stabilizing function \(k(\boldsymbol{n})\), we obtain a new conservative geometric partial differential equation and its corresponding variational formulation for the anisotropic surface diffusion. Based on the new weak formulation, we propose a full discretization by adopting the parametric finite element method for spatial discretization and a semi-implicit temporal discretization with a proper and clever approximation for the outward normal vector. Under a mild and natural condition on \(\gamma (\boldsymbol{n})\), we can prove that the proposed full discretization is structure-preserving, i.e. it preserves the area conservation and energy dissipation at the discretized level, and thus it is unconditionally energy stable. The proposed SP-PFEM is then extended to simulate the evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality (and thus no need to re-mesh during the evolution) of the proposed SP-PFEM for simulating anisotropic geometric flows.A new efficient hybrid method based on FEM and FDM for solving Burgers' equation with forcing termhttps://zbmath.org/1541.650902024-09-27T17:47:02.548271Z"Cakay, Aysenur Busra"https://zbmath.org/authors/?q=ai:cakay.aysenur-busra"Selim, Selmahan"https://zbmath.org/authors/?q=ai:selim.selmahanSummary: This paper presents a study on the numerical solutions of the Burgers' equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers' equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers' problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers' equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering.Analysis and approximations of an optimal control problem for the Allen-Cahn equationhttps://zbmath.org/1541.650922024-09-27T17:47:02.548271Z"Chrysafinos, Konstantinos"https://zbmath.org/authors/?q=ai:chrysafinos.konstantinos"Plaka, Dimitra"https://zbmath.org/authors/?q=ai:plaka.dimitraSummary: The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin -- in time -- scheme is considered for the approximation of the control to state and the state to adjoint mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters \(k\), \(h\) respectively in terms of the parameter \(\epsilon\) that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon \(1/ \epsilon\). Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of \(\epsilon\). These estimates and a suitable localization technique, via the second order condition (see [\textit{N. Arada} et al., Comput. Optim. Appl. 23, No. 2, 201--229 (2002; Zbl 1033.65044); \textit{E. Casas} et al., Comput. Optim. Appl. 31, No. 2, 193--219 (2005; Zbl 1081.49023); \textit{E. Casas} and \textit{J.-P. Raymond}, SIAM J. Control Optim. 45, No. 5, 1586--1611 (2006; Zbl 1123.65061); \textit{E. Casas} et al., SIAM J. Control Optim. 46, No. 3, 952--982 (2007; Zbl 1163.76028)]), allows to prove error estimates for the difference between local optimal controls and their discrete approximations as well as between the associated state and adjoint state variables and their discrete approximations.An approximation method for exact controls of vibrating systems with numerical viscosityhttps://zbmath.org/1541.650932024-09-27T17:47:02.548271Z"Cîndea, Nicolae"https://zbmath.org/authors/?q=ai:cindea.nicolae"Micu, Sorin"https://zbmath.org/authors/?q=ai:micu.sorin"Rovenţa, Ionel"https://zbmath.org/authors/?q=ai:roventa.ionel"Tudor, Mihai"https://zbmath.org/authors/?q=ai:tudor.mihai-adrianSummary: We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's ``stabilizability implies controllability'' principle and a finite elements method of order \(\theta\) with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order \(\theta\) for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.A novel finite element approximation of anisotropic curve shortening flowhttps://zbmath.org/1541.650942024-09-27T17:47:02.548271Z"Deckelnick, Klaus"https://zbmath.org/authors/?q=ai:deckelnick.klaus"Nürnberg, Robert"https://zbmath.org/authors/?q=ai:nurnberg.robertSummary: We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal \(H^1\)-error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.A space-time DG method for the Schrödinger equation with variable potentialhttps://zbmath.org/1541.650972024-09-27T17:47:02.548271Z"Gómez, Sergio"https://zbmath.org/authors/?q=ai:gomez.sergio-alejandro"Moiola, Andrea"https://zbmath.org/authors/?q=ai:moiola.andreaSummary: We present a space-time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal \(h\)-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.High-order bound-preserving local discontinuous Galerkin methods for incompressible and immiscible two-phase flows in porous mediahttps://zbmath.org/1541.650982024-09-27T17:47:02.548271Z"Guo, Xiuhui"https://zbmath.org/authors/?q=ai:guo.xiuhui"Guo, Hui"https://zbmath.org/authors/?q=ai:guo.hui.2"Tian, Lulu"https://zbmath.org/authors/?q=ai:tian.lulu"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.5Summary: In this paper, we develop high-order bound-preserving (BP) local discontinuous Galerkin methods for incompressible and immiscible two-phase flows in porous media, and employ implicit pressure explicit saturation (IMPES) methods for time discretization, which is locally mass conservative for both phases. Physically, the saturations of the two phases, \(S_w\) and \(S_n\), should belong to the range of \([0, 1]\). Nonphysical numerical approximations may result in instability of the simulation. Therefore, it is necessary to construct a BP technique to obtain physically relevant numerical approximations. However, the saturation does not satisfy the maximum principle, so most of the existing BP techniques cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both \(S_w\) and \(S_n\), respectively, and enforce \(S_w +S_n =1\) simultaneously. Numerical examples are given to demonstrate the high-order accuracy of the scheme and effectiveness of the BP technique.A mixed parameter formulation with applications to linear viscoelastic slender structureshttps://zbmath.org/1541.650992024-09-27T17:47:02.548271Z"Hernández, Erwin"https://zbmath.org/authors/?q=ai:hernandez.erwin"Lepe, Felipe"https://zbmath.org/authors/?q=ai:lepe.felipe"Vellojin, Jesus"https://zbmath.org/authors/?q=ai:vellojin.jesusSummary: We present the analysis of an abstract parameter-dependent mixed variational formulation based on Volterra integrals of second kind. Adapting the classic mixed theory in the Volterra equations setting, we prove the well posedness of the resulting system. Stability and error estimates are derived, where all the estimates are uniform with respect to the perturbation parameter. We provide applications of the developed analysis for a viscoelastic Timoshenko beam and report numerical tests for this problem. We also comment, numerically, the performance of a viscoelastic Reissner-Mindlin plate.A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive lawhttps://zbmath.org/1541.651002024-09-27T17:47:02.548271Z"Jang, Yongseok"https://zbmath.org/authors/?q=ai:jang.yongseok"Shaw, Simon"https://zbmath.org/authors/?q=ai:shaw.simon-c|shaw.simon-aSummary: We consider a fractional order viscoelasticity problem modelled by a \textit{power-law} type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.A mixed FEM for a time-fractional Fokker-Planck modelhttps://zbmath.org/1541.651012024-09-27T17:47:02.548271Z"Karaa, Samir"https://zbmath.org/authors/?q=ai:karaa.samir"Mustapha, Kassem"https://zbmath.org/authors/?q=ai:mustapha.kassem"Ahmed, Naveed"https://zbmath.org/authors/?q=ai:ahmed.naveedSummary: We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker-Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In particular, error bounds for both primary and secondary variables are derived in \(L^2\)-norm for cases with smooth and nonsmooth initial data. We further investigate a fully implicit time-stepping scheme based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal \(L^2\)-error estimates for the primary variable. Numerical examples are then presented to illustrate the theoretical contributions.Fully discrete pointwise smoothing error estimates for measure valued initial datahttps://zbmath.org/1541.651022024-09-27T17:47:02.548271Z"Leykekhman, Dmitriy"https://zbmath.org/authors/?q=ai:leykekhman.dmitriy"Vexler, Boris"https://zbmath.org/authors/?q=ai:vexler.boris"Wagner, Jakob"https://zbmath.org/authors/?q=ai:wagner.jakobThe authors study the smoothing properties of a fully discrete approximation of a classical homogeneous parabolic problem for the case where the initial condition is a regular Borel measure that is supported on some subdomain. The error estimates the authors derived are useful for initial data estimation for parabolic problems from the final time observation. This work concerns the case where the final time observation is taken at a finite number of points, specifically, for the case where a final time condition is given by a measure supported in fixed observation points. The notion of very weak solutions for parabolic homogeneous problems with initial data in the space of regular Borel measures is given. Smoothing estimates for the continuous problem and the fully discrete solutions are provided as well as a review of the pointwise smoothing error estimates for the case where the initial data is in \(L^2(\Omega)\), followed by a pointwise smoothing error estimate where the initial data is measure valued. The error estimates hold for approximate solutions with piecewise linear finite elements, and it is shown that the result can be extended to the case of quadratic Lagrange finite elements with an improved rate if additional regularity (for example, for the case of special domains such as rectangles, right, or equilateral triangles) is available.
Reviewer: Baasansuren Jadamba (Rochester)An ultra-weak local discontinuous Galerkin method with generalized numerical fluxes for the KdV-Burgers-Kuramoto equationhttps://zbmath.org/1541.651032024-09-27T17:47:02.548271Z"Lin, Guotao"https://zbmath.org/authors/?q=ai:lin.guotao"Zhang, Dazhi"https://zbmath.org/authors/?q=ai:zhang.dazhi.1"Li, Jia"https://zbmath.org/authors/?q=ai:li.jia.9"Wu, Boying"https://zbmath.org/authors/?q=ai:wu.boyingSummary: In this paper, we study an ultra-weak local discontinuous Galerkin (UWLDG) method for the KdV-Burgers-Kuramoto (KBK) type equation. While the standard UWLDG method is a powerful tool for efficiently solving high order equations, it faces challenges when applied to equations involving multiple spatial derivatives. We adopt a novel approach to discretize lower order spatial derivatives, enhancing the versatility of the UWLDG method. Additionally, we adopt generalized numerical fluxes to enhance the flexibility and extendibility of the UWLDG scheme. We introduce a class of global projections with multiple parameters to analyze the properties of these generalized numerical fluxes. With the aid of the special discretization approach and the global projections, we establish both stability and optimal error estimates of proposed method. The validity of our theoretical findings is demonstrated through numerical experiments.An entropy stable essentially oscillation-free discontinuous Galerkin method for hyperbolic conservation lawshttps://zbmath.org/1541.651042024-09-27T17:47:02.548271Z"Liu, Yong"https://zbmath.org/authors/?q=ai:liu.yong.7"Lu, Jianfang"https://zbmath.org/authors/?q=ai:lu.jianfang"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wangSummary: Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [\textit{T. Chen} and \textit{C.-W. Shu}, J. Comput. Phys. 345, 427--461 (2017; Zbl 1380.65253); \textit{J. Chan}, J. Comput. Phys. 362, 346--374 (2018; Zbl 1391.76310); \textit{T. Chen} and \textit{C.-W. Shu}, CSIAM Trans. Appl. Math. 1, No. 1, 1--52 (2020; \url{doi:10.4208/csiam-am.2020-0003})] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [\textit{J. Lu} et al., SIAM J. Numer. Anal. 59, No. 3, 1299--1324 (2021; Zbl 1467.65095); \textit{Y. Liu} et al., SIAM J. Sci. Comput. 44, No. 1, A230--A259 (2022; Zbl 1484.65226)], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm.An isoparametric finite element method for time-fractional parabolic equation on 2D curved domainhttps://zbmath.org/1541.651052024-09-27T17:47:02.548271Z"Liu, Zhixin"https://zbmath.org/authors/?q=ai:liu.zhixin.1"Song, Minghui"https://zbmath.org/authors/?q=ai:song.minghui"Liang, Hui"https://zbmath.org/authors/?q=ai:liang.huiSummary: This paper provides an efficient numerical scheme for approximating the solution of the time-fractional parabolic equation on 2D curved domain. Here, the solution of the problem exhibits a weak singularity at the initial time \(t=0\). The method is based on applying the L1 formula to approximate the Caputo time-fractional derivative and using the isoparametric finite element method to approximate the spatial direction. A fully discrete scheme is then constructed using numerical quadrature. Since \(\Omega_h\) differs from \(\Omega\), the error estimates of the boundary terms on the region between \(\Omega_h\) and \(\Omega\) and the effect of numerical quadrature are both considered. Afterward, the stability analysis and optimal error estimates for the fully discrete scheme are proved in detail. Finally, some numerical experiments are presented to verify the theoretical results.Multi-level solution strategies for implicit time discontinuous Galerkin discretizationshttps://zbmath.org/1541.651062024-09-27T17:47:02.548271Z"Lohry, Mark W."https://zbmath.org/authors/?q=ai:lohry.mark-w"Martinelli, Luigi"https://zbmath.org/authors/?q=ai:martinelli.luigiSummary: The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier-Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge-Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate.Robust scheme on 3D hybrid meshes with non-conformity for Maxwell's equations in time domainhttps://zbmath.org/1541.651082024-09-27T17:47:02.548271Z"Ritzenthaler, Valentin"https://zbmath.org/authors/?q=ai:ritzenthaler.valentin"Cantin, Pierre"https://zbmath.org/authors/?q=ai:cantin.pierre"Ferrieres, Xavier"https://zbmath.org/authors/?q=ai:ferrieres.xavierSummary: This paper presents a low-order spatial discretization to solve the Maxwell equations in the time-domain, namely: the Compatible Discrete Operator scheme. The basics to build this scheme are recalled, and it is shown how this scheme allows to efficiently deal with hybrid meshes composed of a Cartesian part and a simplicial part, with polyhedra at the interface between the two. The scheme is formulated for the Maxwell equations in the case where the computational domain is surrounded by Perfectly Matched Layers. Finally, the paper proposes some numerical examples on meshes with non-conformities, to emphasize the robustness and the interest of such a scheme.Truncation error-based anisotropic \(p\)-adaptation for unsteady flows for high-order discontinuous Galerkin methodshttps://zbmath.org/1541.651092024-09-27T17:47:02.548271Z"Rueda-Ramírez, Andrés M."https://zbmath.org/authors/?q=ai:rueda-ramirez.andres-mauricio"Ntoukas, Gerasimos"https://zbmath.org/authors/?q=ai:ntoukas.gerasimos"Rubio, Gonzalo"https://zbmath.org/authors/?q=ai:rubio.gonzalo"Valero, Eusebio"https://zbmath.org/authors/?q=ai:valero.eusebio"Ferrer, Esteban"https://zbmath.org/authors/?q=ai:ferrer.estebanSummary: We extend the \(\tau\)-estimation method to unsteady problems and use it to adapt the polynomial degree for high-order discontinuous Galerkin simulations of unsteady flows. The adaptation is local and anisotropic and allows capturing relevant unsteady flow features while enhancing the accuracy. We first revisit the definition of the truncation error, studying the effect of the treatment of the mass matrix. Secondly, we extend the \(\tau\)-estimation strategy to unsteady problems. Finally, we present and compare two adaptation strategies for unsteady problems: the dynamic and static methods. We test the efficiency of the \(p\)-adaptation strategies with unsteady two-dimensional simulations using the Euler and Navier-Stokes equations. Since the method relies on the exponential convergence of the scheme, we focus in laminar test cases. The adaptation methods enable reductions in the number of degrees of freedom with respect to uniform refinement, leading to speed-ups of up to \(\times 4.5\) and \(\times 4.5\) for Euler and Navier-Stokes.Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problemshttps://zbmath.org/1541.651102024-09-27T17:47:02.548271Z"Shakya, Pratibha"https://zbmath.org/authors/?q=ai:shakya.pratibha"Sinha, Rajen Kumar"https://zbmath.org/authors/?q=ai:sinha.rajen-kumarSummary: This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the \(L^{\infty}(L^2)\)-norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests.Finite element method combined with time graded meshes for the time-fractional coupled Burgers' equationshttps://zbmath.org/1541.651112024-09-27T17:47:02.548271Z"Sheng, Zhihao"https://zbmath.org/authors/?q=ai:sheng.zhihao"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.11"Li, Yonghai"https://zbmath.org/authors/?q=ai:li.yonghaiSummary: In this article, we develop a numerical method to solve the Caputo time-fractional coupled Burgers' equations using the finite element method combined with the \(L1\) formula on graded meshes. We specifically derive an extended discrete fractional Grönwall inequality to analyze numerical methods on graded meshes for time-fractional coupled equations. With the help of this inequality, we present the stability and the error estimate of the time semi-discrete scheme, and prove the error estimate of the fully discrete scheme. We present numerical experiments to confirm our theoretical results.Nitsche-XFEM for a time fractional diffusion interface problemhttps://zbmath.org/1541.651122024-09-27T17:47:02.548271Z"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao.2|wang.tao.22|wang.tao.52|wang.tao.54|wang.tao.33|wang.tao.48|wang.tao.9|wang.tao|wang.tao.11|wang.tao.51|wang.tao.10|wang.tao.5|wang.tao.15|wang.tao.34|wang.tao.3|wang.tao.53|wang.tao.47|wang.tao.1|wang.tao.55|wang.tao.12|wang.tao.46|wang.tao.19|wang.tao.14|wang.tao.4"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.2|chen.yanping.1|chen.yanping.3Summary: In this paper, we propose a space-time finite element method for a time fractional diffusion interface problem. This method uses the low-order discontinuous Galerkin (DG) method and the Nitsche extended finite element method (Nitsche-XFEM) for temporal and spatial discretization, respectively. Sharp pointwise-in-time error estimates in graded temporal grids are derived, without any smoothness assumptions on the solution. Finally, three numerical examples are provided to verify the theoretical results.Superconvergence analysis of the nonconforming FEM for the Allen-Cahn equation with time Caputo-Hadamard derivativehttps://zbmath.org/1541.651142024-09-27T17:47:02.548271Z"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.17"Sun, Luhan"https://zbmath.org/authors/?q=ai:sun.luhan"Wei, Yabing"https://zbmath.org/authors/?q=ai:wei.yabingSummary: This paper first introduces a high-order approximate formula for the Caputo-Hadamard fractional derivative using interpolation theory, and analyzes its convergence and accuracy. Although the formula bears resemblance to the logarithmic L2-\(1_\sigma\) formula in form, it differs in the selection of mesh points, making it suitable for more complex models in numerical computations. Subsequently, a numerical method for the Allen-Cahn equation with time Caputo-Hadamard derivative is proposed. The temporal direction is discretized using the newly derived difference formula, while the spatial direction is approximated by the anisotropic nonconforming quasi-Wilson finite element method (FEM). It is proven that the derived scheme has optimal convergence accuracy in the \(L^2\)-norm and global superconvergence property in the \(H^1\)-norm. Numerical examples are provided to corroborate the theoretical claims.A unified framework of the SAV-ZEC method for a mass-conserved Allen-Cahn type two-phase ferrofluid flow modelhttps://zbmath.org/1541.651172024-09-27T17:47:02.548271Z"Zhang, Guo-Dong"https://zbmath.org/authors/?q=ai:zhang.guodong.1"He, Xiaoming"https://zbmath.org/authors/?q=ai:he.xiaoming.1"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: This article presents a mass-conserved Allen-Cahn type two-phase ferrofluid flow model and establishes its corresponding energy law. The model is a highly coupled, nonlinear saddle point system consisting of the mass-conserved Allen-Cahn equation, the Navier-Stokes equation, the magnetostatic equation, and the magnetization equation. We develop a unified framework of the scalar auxiliary variable (SAV) method and the zero energy contribution (ZEC) approach, which constructs a mass-conserved, fully decoupled, second-order accurate in time, and unconditionally energy-stable linear scheme. We incorporate several distinct numerical techniques, including reformulations of the equations to remove the linear couplings and implicit nonlocal integration, the projection method to decouple the velocity and pressure, a symmetric implicit-explicit format for symmetric positive definite nonlinearity, and the continuous finite element method discretization. We also analyze the mass-conserved property, unconditional energy stability, and well-posedness of the scheme. To demonstrate the effectiveness, stability, and accuracy of the developed model and numerical algorithm, we implemented several numerical examples, involving a ferrofluid hedgehog in 2D and a ferromagnetic droplet in 3D. It is worth mentioning that the proposed unified framework of the SAV-ZEC method is also applicable to designing efficient schemes for other coupled-type fluid flow phase-field systems.Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection-diffusion-reaction problems with two small parametershttps://zbmath.org/1541.651182024-09-27T17:47:02.548271Z"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamal"Izadi, Mohammad"https://zbmath.org/authors/?q=ai:izadi.mohammad-a|izadi.mohammad"Noeiaghdam, Samad"https://zbmath.org/authors/?q=ai:noeiaghdam.samadSummary: This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection-diffusion-reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order \(\mathcal{O}(\Delta\tau^s + M^{-\frac{1}{2}})\) for \(s = 1, 2\), where \(\Delta\tau\) is the time step and \(M\) is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.Spectral Galerkin method for Cahn-Hilliard equations with time periodic solutionhttps://zbmath.org/1541.651192024-09-27T17:47:02.548271Z"Chai, Shimin"https://zbmath.org/authors/?q=ai:chai.shimin"Zhou, Chenguang"https://zbmath.org/authors/?q=ai:zhou.chenguangSummary: This paper is concerned with the numerical approximation to the one-dimensional Cahn-Hilliard equation with time periodic solution. We adopt the implicit Euler method and the spectral Galerkin method for the temporal and spatial discretization, respectively. Then the error estimates are proved for both the semi-discrete and the fully discrete schemes. Numerical experiments are carried out to confirm our theoretical results.On the anti-aliasing properties of entropy filtering for discontinuous spectral element approximations of under-resolved turbulent flowshttps://zbmath.org/1541.651202024-09-27T17:47:02.548271Z"Dzanic, Tarik"https://zbmath.org/authors/?q=ai:dzanic.tarik"Trojak, Will"https://zbmath.org/authors/?q=ai:trojak.will"Witherden, Freddie"https://zbmath.org/authors/?q=ai:witherden.freddie-dSummary: For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In [\textit{T. Dzanic} and \textit{F. D. Witherden}, J. Comput. Phys. 468, Article ID 111501, 21 p. (2022; Zbl 07578908)], an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.Numerical study of transient Wigner-Poisson model for RTDs: numerical method and its applicationshttps://zbmath.org/1541.651212024-09-27T17:47:02.548271Z"Jiang, Haiyan"https://zbmath.org/authors/?q=ai:jiang.haiyan"Lu, Tiao"https://zbmath.org/authors/?q=ai:lu.tiao"Yao, Wenqi"https://zbmath.org/authors/?q=ai:yao.wenqi"Zhang, Weitong"https://zbmath.org/authors/?q=ai:zhang.weitongSummary: The system of transient Wigner-Poisson equations (TWPEs) is a common model to describe carrier transport in quantum devices. In this paper, we design a second-order semi-implicit time integration scheme for TWPEs with the inflow boundary conditions, and a hybrid sinc-Galerkin/finite-difference method [\textit{H. Jiang} et al., J. Comput. Appl. Math. 409, Article ID 114152, 12 p. (2022; Zbl 1487.81086)] is applied for the spatial discretization. The fully-discretized system is rigorously proved to be unconditionally \(L^2\)-stable, and the computational cost is comparable with that of the second-order explicit Runge-Kutta scheme (ERK2). The numerical method is applied to study a double-barrier resonant tunneling diode (RTD), where representative characteristics of RTDs, including the resonant tunneling effect, bistability and the intrinsic high-frequency current oscillation, are simulated successfully.Numerical solutions of Schrödinger-Boussinesq system by orthogonal spline collocation methodhttps://zbmath.org/1541.651222024-09-27T17:47:02.548271Z"Liao, Feng"https://zbmath.org/authors/?q=ai:liao.feng"Geng, Fazhan"https://zbmath.org/authors/?q=ai:geng.fazhan"Yao, Lingxing"https://zbmath.org/authors/?q=ai:yao.lingxingSummary: This paper is concerned with the numerical solutions of Schrödinger-Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank-Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order \(O (\tau^2 + h^4)\) with mesh-size \(h\) and time step \(\tau\) in the discrete \(L^2\)-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.A stable time-space Jacobi pseudospectral method for two-dimensional sine-Gordon equationhttps://zbmath.org/1541.651232024-09-27T17:47:02.548271Z"Mittal, A. K."https://zbmath.org/authors/?q=ai:mittal.ajay-kumar|mittal.avinash-kumar|mittal.ashok-kumarSummary: In this article, we present the stability analysis and approximate solutions of circular ring solitary solitons wave for \((2 + 1)\)-dimensional nonlinear sine-Gordon equation using time-space pseudospectral method. Error bounds on discrete \(L_2\)-norm and Sobolev norm \((H^p)\) are presented. The discretization of the problem using proposed method leads a system of nonlinear equations, which are solved using Newton-Raphson method. Further, a study is carried out to investigate the effects of dissipative term in sine-Gordon equation, which presence forms circular ring solitons. To support the theoretical results, the proposed method is tested for two problems, single circular ring solitary soliton waves and the collision of four circular ring solitary soliton waves and numerical results are presented.Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equationhttps://zbmath.org/1541.651242024-09-27T17:47:02.548271Z"Pan, Zhenlan"https://zbmath.org/authors/?q=ai:pan.zhenlan"An, Jing"https://zbmath.org/authors/?q=ai:an.jingSummary: We propose in this paper an efficient differential-spectral approximation based on a reduced-dimension scheme for a fourth-order parabolic equation in a circular domain. First, we decompose the original problem into a series of equivalent one-dimensional fourth-order parabolic problems, based on which a fully discrete scheme based on differential-spectral approximation is established, and its stability and corresponding error estimation are also proved. Then, we utilized the orthogonality of Legendre polynomials to construct a set of effective basis functions and derived the matrix form associated with the full discrete scheme. Finally, several numerical examples are performed, and the numerical results account for the effectiveness and high accuracy of our algorithm.An efficient spectral method for the fractional Schrödinger equation on the real linehttps://zbmath.org/1541.651252024-09-27T17:47:02.548271Z"Shen, Mengxia"https://zbmath.org/authors/?q=ai:shen.mengxia"Wang, Haiyong"https://zbmath.org/authors/?q=ai:wang.haiyongSummary: The fractional Schrödinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method.Numerical solution of distributed-order fractional Korteweg-de Vries equation via fractional zigzag rising diagonal functionshttps://zbmath.org/1541.651262024-09-27T17:47:02.548271Z"Taghipour, M."https://zbmath.org/authors/?q=ai:taghipour.mehran"Aminikhah, H."https://zbmath.org/authors/?q=ai:aminikhah.hosseinSummary: The goal of this article is to develop a spectral collocation method for solving a distributed-order fractional Korteweg-de Vries equation using fractional Zigzag rising diagonal functions. To meet this target, we first introduce Zigzag and Jaiswal polynomials. Then, using a transformation, we find their fractional counterparts. As a linear combination of these functions, we seek a solution to the problem. We will generate operational matrices for fractional Zigzag raising diagonal functions and apply Simpson's rule to approximate the distributed fractional derivative. The resultant approximate equations are collocated to create a system of nonlinear equations. Error analysis of the numerical scheme is fully discussed. Numerical experiments have demonstrated the capability and efficiency of the method. We also demonstrate how the approach might be helpful for problems with non-smooth solutions.Temperature distribution in living tissue with two-dimensional parabolic bioheat model using radial basis functionhttps://zbmath.org/1541.651272024-09-27T17:47:02.548271Z"Verma, Rohit"https://zbmath.org/authors/?q=ai:verma.rohit-kumar|verma.rohit.1|verma.rohit-kumar.2|verma.rohit"Kumar, Sushil"https://zbmath.org/authors/?q=ai:kumar.sushilSummary: This study concerns the numerical modeling and simulation of heat distribution inside the skin tissue for cancer treatment with external exponential heating. Here, we consider the two-dimensional Pennes bioheat model for thermal therapy based on Fourier's law of heat conduction. We approximate the temporal variable using finite difference approximation and the spatial variable using the radial basis functions (RBF)-based collocation method to solve the considered model. The effect of different parameters on thermal diffusion in skin tissue has also been studied.
For the entire collection see [Zbl 1521.76009].An implicit scheme for time-fractional coupled generalized Burgers' equationhttps://zbmath.org/1541.651282024-09-27T17:47:02.548271Z"Vigo-Aguiar, J."https://zbmath.org/authors/?q=ai:vigo-aguiar.jesus"Chawla, Reetika"https://zbmath.org/authors/?q=ai:chawla.reetika"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.1"Mazumdar, Tapas"https://zbmath.org/authors/?q=ai:mazumdar.tapas.1Summary: This article presents an efficient implicit spline-based numerical technique to solve the time-fractional generalized coupled Burgers' equation. The time-fractional derivative is considered in the Caputo sense. The time discretization of the fractional derivative is discussed using the quadrature formula. The quasilinearization process is used to linearize this non-linear problem. In this work, the formulation of the numerical scheme is broadly discussed using cubic B-spline functions. The stability of the proposed method is proved theoretically through Von-Neumann analysis. The reliability and efficiency are demonstrated by numerical experiments that validate theoretical results via tables and plots.Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equationhttps://zbmath.org/1541.651302024-09-27T17:47:02.548271Z"Ye, Xiao"https://zbmath.org/authors/?q=ai:ye.xiao"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Liu, Jun"https://zbmath.org/authors/?q=ai:liu.jun.10"Liu, Yue"https://zbmath.org/authors/?q=ai:liu.yue.3|liu.yue.1|liu.yueSummary: Optimal quadratic spline collocation (QSC) method has been widely used in various problems due to its high-order accuracy, while the corresponding numerical analysis is rarely investigated since, e.g., the perturbation terms result in the asymmetry of optimal QSC discretization. We present numerical analysis for the optimal QSC method in two space dimensions via discretizing a nonlinear time-fractional diffusion equation for demonstration. The \(L2\)-\(1_{\sigma}\) formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC-\(L2\)-\(1_{\sigma}\) scheme where the nonlinear term is treated by the extrapolation. We provide the existence and uniqueness of the numerical solution, as well as the second-order temporal accuracy and fourth-order spatial accuracy with proper grading parameters. Furthermore, we consider the fast implementation based on the sum-of-exponentials technique to reduce the computational cost. Numerical experiments are performed to verify the theoretical analysis and the effectiveness of the proposed scheme.A highly efficient numerical method for the time-fractional diffusion equation on unbounded domainshttps://zbmath.org/1541.651312024-09-27T17:47:02.548271Z"Zhu, Hongyi"https://zbmath.org/authors/?q=ai:zhu.hongyi"Xu, Chuanju"https://zbmath.org/authors/?q=ai:xu.chuanjuSummary: In this paper, we propose a fast high order method for the time-fractional diffusion equation on unbounded spatial domains. The proposed numerical method is a combination of a time-stepping scheme and spectral method for the spatial discretization. First, we reformulate the unbounded domain problem into a bounded domain problem by introducing suitable artificial boundary conditions. Then the time fractional derivatives involved in the equation and the artificial boundary condition are discretized using the so-called L2 formula and sum-of-exponentials (SOE) approximation. The former has been a popular formula for discretization of the Caputo fractional derivative, while the latter is a computational cost reducing technique frequently employed in recent years for convolution integrals. The spatial discretization makes use of the standard Legendre spectral method. The stability and the accuracy of the full discrete problem are analyzed. Our obtained theoretical results include a rigorous proof of the convergence order for both uniform mesh and graded mesh, and a stability proof for the uniform mesh. Finally, several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.An efficient new technique for solving nonlinear problems involving the conformable fractional derivativeshttps://zbmath.org/1541.651322024-09-27T17:47:02.548271Z"Ahmed, Shams A."https://zbmath.org/authors/?q=ai:ahmed.shams-aSummary: In this paper, an efficient new technique is used for solving nonlinear fractional problems that satisfy specific criteria. This technique is referred to as the double conformable fractional Laplace-Elzaki decomposition method (DCFLEDM). This approach combines the double Laplace-Elzaki transform method with the Adomian decomposition method. The fundamental concepts and findings of the recently suggested transformation are presented. For the purpose of assessing the accuracy of our approach, we provide three examples and introduce the series solutions of these equations using DCLEDM. The results show that the proposed strategy is a very effective, reliable, and efficient approach for addressing nonlinear fractional problems using the conformable derivative.Numerical and analytical solution to a conformable fractional Fornberg-Whitham equationhttps://zbmath.org/1541.651332024-09-27T17:47:02.548271Z"Enyi, Cyril D."https://zbmath.org/authors/?q=ai:enyi.cyril-dennis"Nwaeze, Eze R."https://zbmath.org/authors/?q=ai:nwaeze.eze-raymond"Omaba, McSylvester E."https://zbmath.org/authors/?q=ai:omaba.mcsylvester-ejighikemeSummary: In this work, we perform a broader analytical and numerical study of a space-time conformable fractional Fornberg-Whitham equation. The concept of conformable fractional Laplace transform was infused in the well-known homotopy analysis method (HAM), to obtain a more accurate solution. We named this method \(q\)-Homotopy Analysis Conformable Fractional Laplace Transform Method \((q\)-HACFLTM). In addition, thorough numerical analysis using graphs and error analysis confirms that \(q\)-HACFLTM performs better and more accurately than the \(q\)-HAM technique previously applied to solve this problem. Moreover, the proposed method eliminates any form of restriction on the fractional order of the derivatives as was the case in the previous work of
\textit{O. S. Iyiola} and \textit{G. O. Ojo} [Pramana J. Phys. 58, No. 4, 567--575 (2005; \url{doi:10.1007/s12043-014-0915-2})].
Our method \((q\)-HACFLTM) is quite easy and highly accurate and could be adopted for use in solving other fractional differential equation models, once similar properties of Laplace transform used here could be established. The \(q\)-HACFLTM technique applied here does not require any perturbation, discretization, polynomials or Lagrange multiplier, thus giving it some advantage over methods like ADM, HPTM or VIM, etc. The Laplace transform introduced was able to handle the nonlinear terms of the equation in a more robust manner, resulting in more accurate solution and faster convergence.A FOM/ROM hybrid approach for accelerating numerical simulationshttps://zbmath.org/1541.651342024-09-27T17:47:02.548271Z"Feng, Lihong"https://zbmath.org/authors/?q=ai:feng.lihong"Fu, Guosheng"https://zbmath.org/authors/?q=ai:fu.guosheng"Wang, Zhu"https://zbmath.org/authors/?q=ai:wang.zhuSummary: The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid approach in this paper. It is developed based on an a posteriori error estimation for the output approximation of the dynamical system. By controlling the estimated error, the method dynamically switches between the full-order model and the reduced-oder model generated on the fly. Therefore, it reduces the computational cost of a high-fidelity simulation while achieving a prescribed accuracy level. Numerical tests on the non-parametric and parametric PDEs illustrate the efficacy of the proposed approach.Strong solutions for PDE-based tomography by unsupervised learninghttps://zbmath.org/1541.651362024-09-27T17:47:02.548271Z"Bar, Leah"https://zbmath.org/authors/?q=ai:bar.leah"Sochen, Nir"https://zbmath.org/authors/?q=ai:sochen.nirSummary: We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We employ an unsupervised approach such that the input to the network is a point set in an arbitrary domain, and the output is the set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit, smooth, differentiable function with a known analytical form. We solve the forward problem (observations given the underlying model's parameters), semi-inverse problem (model's parameters given the observations in the whole domain), and full tomography inverse problem (model's parameters given the observations on the boundary) by solving the forward and semi-inverse problems at the same time. The optimized loss function consists of few elements: fidelity term of \(L_2\) norm that enforces the PDE in the weak sense, an \(L_\infty\) norm term that enforces pointwise fidelity and thus promotes a strong solution, and boundary and initial conditions constraints. It further accommodates regularizers for the solution and/or the model's parameters of the differential operator. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape two dimensional (2D) second order systems with application to electrical impedance tomography (EIT) and diffusion equation. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework enables, in principle, the solution of high order and high dimensional nonlinear PDEs.Fast imaging of sources and scatterers in a stratified ocean waveguidehttps://zbmath.org/1541.651382024-09-27T17:47:02.548271Z"Liu, Keji"https://zbmath.org/authors/?q=ai:liu.kejiSummary: In this work, we have studied the asymptotic behavior of Green's function and the reciprocity relation of the far-field pattern in the stratified ocean waveguide. Moreover, two direct sampling methods (DSM) are proposed to determine the marine sources and scatterers from the far-field data. The direct approaches are fast, easy to implement, and computationally efficient since they involve only scalar product but no matrix inversion. In the numerical simulations, the DSM for the source is capable of identifying the sources from very few observation data, and the DSM for the scatterer can reconstruct the scatterers in different shapes, scales, types, and positions. The effectiveness and robustness of the novel methods are also demonstrated. Thus, the DSM can be viewed as simple and efficient numerical techniques for providing reliable initial approximate locations of the marine sources and scatterers for any existing more refined and advanced but computationally more demanding algorithms to recover the accurate physical profiles.A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental datahttps://zbmath.org/1541.651392024-09-27T17:47:02.548271Z"Tyni, Teemu"https://zbmath.org/authors/?q=ai:tyni.teemu"Stinchcombe, Adam R."https://zbmath.org/authors/?q=ai:stinchcombe.adam-r"Alexakis, Spyros"https://zbmath.org/authors/?q=ai:alexakis.spyrosSummary: We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, currents are injected through the electrodes, and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single-layer potentials; the problem is thus recast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.Guaranteed lower eigenvalue bounds for Steklov operators using conforming finite element methodshttps://zbmath.org/1541.651412024-09-27T17:47:02.548271Z"Nakano, Taiga"https://zbmath.org/authors/?q=ai:nakano.taiga"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin.2|li.qin.1|li.qin"Yue, Meiling"https://zbmath.org/authors/?q=ai:yue.meiling"Liu, Xuefeng"https://zbmath.org/authors/?q=ai:liu.xuefeng|liu.xuefeng.1Summary: For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the \textit{a priori} error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples demonstrate the efficiency of our proposed method.Efficient algorithms for Bayesian inverse problems with Whittle-Matérn priorshttps://zbmath.org/1541.651432024-09-27T17:47:02.548271Z"Antil, Harbir"https://zbmath.org/authors/?q=ai:antil.harbir"Saibaba, Arvind K."https://zbmath.org/authors/?q=ai:saibaba.arvind-krishnaSummary: This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle-Matérn Gaussian random fields. The Whittle-Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including nonstationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen-Loève expansion. We show how to incorporate this prior representation into the infinite-dimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the t