Recent zbMATH articles in MSC 35https://zbmath.org/atom/cc/352022-09-13T20:28:31.338867ZUnknown authorWerkzeugBook review of: J. Keener, Biology in time and space. A partial differential equation modeling approachhttps://zbmath.org/1491.000332022-09-13T20:28:31.338867Z"Torres, Marcella"https://zbmath.org/authors/?q=ai:torres.marcellaReview of [Zbl 1470.92003].Conformal symmetry breaking differential operators on differential formshttps://zbmath.org/1491.220042022-09-13T20:28:31.338867Z"Fischmann, Matthias"https://zbmath.org/authors/?q=ai:fischmann.matthias"Juhl, Andreas"https://zbmath.org/authors/?q=ai:juhl.andreas"Somberg, Petr"https://zbmath.org/authors/?q=ai:somberg.petrIn [\textit{A. Juhl}, Families of conformally covariant differential operators, Q-curvature and holography. Basel: Birkhäuser (2009; Zbl 1177.53001)], a program was initiated to find conformal invariants of hypersurfaces \(M\) in Riemannian manifolds \((X,g)\). The idea is to construct one-parameter families of conformally covariant differential operators \(D_{2N}(g;\lambda):C^\infty(X)\to C^\infty(M)\), \(\lambda\in\mathbb{C}\), of order \(2N\), mapping functions on the ambient manifold \(X\) to functions on \(M\). Here conformal covariance means that \[ e^{(\lambda+N)\iota^*(\varphi)}D_{2N}(e^{2\varphi}g;\lambda)(u) = D_{2N}(g;\lambda)(e^{\lambda\varphi}u) \] for all \(u,\varphi\in C^\infty(X)\), where \(\iota^*\) denotes the pull-back defined by the embedding \(\iota:M\hookrightarrow X\). Such families generalize the even-order families \(D_{2N}(\lambda):C^\infty(S^n)\to C^\infty(S^{n-1})\) of differential operators which are associated to the equatorial embedding \(S^{n-1}\hookrightarrow S^n\) of spheres, and which intertwine spherical principal series representations of the conformal group of the embedded round sphere \(S^{n-1}\), but not of the conformal group of the ambient round sphere \(S^n\). These intertwining families were first constructed in [\textit{A. Juhl}, Families of conformally covariant differential operators, Q-curvature and holography. Basel: Birkhäuser (2009; Zbl 1177.53001)] and can be regarded as differential operators \[ D_{2N}(\lambda):C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^{n-1}) \] via the stereographic projection.\\
In the paper under review, the authors consider the corresponding problem for differential operators between differential forms. They completely classify conformally covariant differential operators \[ \Omega^p(\mathbb{R}^n)\to\Omega^q(\mathbb{R}^{n-1}) \] for all degrees \(p\) and \(q\). These are intertwining operators for the conformal group of \(\mathbb{R}^{n-1}\), acting by principal series representations realized on forms, and correspond to homomorphisms of generalized Verma modules of the corresponding Lie algebra \(\mathfrak{so}(n,1)\) into Verma modules of \(\mathfrak{so}(n+1,1)\).
The classification essentially consists of two families of differential operators \[ D_N^{(p\to p)}(\lambda):\Omega^p(\mathbb{R}^n)\to\Omega^p(\mathbb{R}^{n-1}) \quad \mbox{and} \quad D_N^{(p\to p-1)}(\lambda):\Omega^p(\mathbb{R}^n)\to\Omega^{p-1}(\mathbb{R}^{n-1}), \] of degree \(N\in\mathbb{N}\), both depending on a complex parameter \(\lambda\in\mathbb{C}\), some additional sporadic operators \(\Omega^p(\mathbb{R}^n)\to\Omega^q(\mathbb{R}^{n-1})\) for \(q\in\{p-2,p-1,p,p+1\}\), as well as their compositions with the Hodge star operator \(\star\) on \(\mathbb{R}^n\) and \(\mathbb{R}^{n-1}\). The authors further obtain some relations between compositions of these operators with the exterior derivative \(d\) and its adjoint \(\delta\) on \(\mathbb{R}^n\) and \(\mathbb{R}^{n-1}\) which they refer to as factorization identities. Finally, they relate their operators to the critical \(Q\)-curvature operators and the Gauge companion operators.\\
The key technique used in the classification if the so-called F-method introduced in [\textit{T. Kobayashi}, Contemp. Math. 598, 139--146 (2013; Zbl 1290.22008) ] which characterizes the symbols of conformally covariant differential operators as (possibly vector-valued) polynomial solutions to certain differential equations. In the case at hand, the corresponding differential equations can be transformed into ordinary differential equations, explaining the occurrence of Jacobi polynomials in the explicit formulas for the two families \(D_N^{(p\to p)}(\lambda)\) and \(D_N^{(p\to p-1)}(\lambda)\).\\
We remark that the same classification was independently obtained in [\textit{T. Kobayashi} et al., Conformal symmetry breaking operators for differential forms on spheres. Singapore: Springer (2016; Zbl 1353.53002)].
Reviewer: Jan Frahm (Århus)On a discrete composition of the fractional integral and Caputo derivativehttps://zbmath.org/1491.260062022-09-13T20:28:31.338867Z"Płociniczak, Łukasz"https://zbmath.org/authors/?q=ai:plociniczak.lukaszSummary: We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.Differential equations. Theory, technique, and practicehttps://zbmath.org/1491.340012022-09-13T20:28:31.338867Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgePublisher's description: Differential equations is one of the oldest subjects in modern mathematics. It was not long after Newton and Leibniz invented the calculus that Bernoulli and Euler and others began to consider the heat equation and the wave equation of mathematical physics. Newton himself solved differential equations both in the study of planetary motion and also in his consideration of optics.
Today differential equations is the centerpiece of much of engineering, of physics, of significant parts of the life sciences, and in many areas of mathematical modeling. This text describes classical ideas and provides an entree to the newer ones. The author pays careful attention to advanced topics like the Laplace transform, Sturm-Liouville theory, and boundary value problems (on the traditional side) but also pays due homage to nonlinear theory, to modeling, and to computing (on the modern side).
This book began as a modernization of George Simmons' classic, Differential Equations with Applications and Historical Notes. Prof. Simmons invited the author to update his book. Now in the third edition, this text has become the author's own and a unique blend of the traditional and the modern. The text describes classical ideas and provides an entree to newer ones.
Modeling brings the subject to life and makes the ideas real. Differential equations can model real life questions, and computer calculations and graphics can then provide real life answers. The symbiosis of the synthetic and the calculational provides a rich experience for students, and prepares them for more concrete, applied work in future courses.
Additional Features
\begin{itemize}
\item Anatomy of an Application sections.
\item Historical notes continue to be a unique feature of this text.
\item Math Nuggets are brief perspectives on mathematical lives or other features of the discipline that will enhance the reading experience.
\item Problems for Review and Discovery give students some open-ended material for exploration and further learning. They are an important means of extending the reach of the text, and for anticipating future work.
\end{itemize}
This new edition is re-organized to make it more useful and more accessible. The most frequently taught topics are now up front. And the major applications are isolated in their own chapters. This makes this edition the most useable and flexible of any previous editions.
See the review of the second edition in [Zbl 1316.34002].A semilinear problem with a gradient term in the nonlinearityhttps://zbmath.org/1491.340362022-09-13T20:28:31.338867Z"Guerra, Ignacio"https://zbmath.org/authors/?q=ai:guerra.ignacio-aThe author considers the following semilinear problem with a gradient term in the linearity on the unit ball \(B=B\left(0,1\right)\subset\mathbb{R}
^{N},\)
\[
-\Delta u=\lambda\frac{\left(1+\left\vert \bigtriangledown u\right\vert ^{q}\right)}{\left(1-u\right)^{p}}\text{ in }B,
\]
\[
u>0~~\text{ in }B,
\]
\[
u=0\text{ on }\partial B,
\]
where \(\lambda>0,~p>0\) and \(q\geq0.\) In general, this problem can have different regimes depending on the parameters \(p,\) \(q\) and \(N.\) Here, in the radial case, the author proves that the problem has infinitely many radial solutions for \(2\leq N<2\frac{6-q+2\sqrt{8-2q}}{\left( 2-q\right) ^{2}} +1~\)and \(\lambda=\frac{N-1}{2},~\)and has a unique radial solution for \(N>2\frac{6-q+2\sqrt{8-2q}}{\left( 2-q\right) ^{2}}+1\) and \(0<\lambda <\frac{N-1}{2}.\)
Reviewer: Fatma Hıra (Atakum)Generalized wave polynomials and transmutations related to perturbed Bessel equationshttps://zbmath.org/1491.340442022-09-13T20:28:31.338867Z"Kravchenko, Vladislav V."https://zbmath.org/authors/?q=ai:kravchenko.vladislav-v"Torba, Sergii M."https://zbmath.org/authors/?q=ai:torba.sergii-m"Santana-Bejarano, Jessica Yu."https://zbmath.org/authors/?q=ai:santana-bejarano.jessica-yuSummary: The transmutation (transformation) operator associated with the perturbed Bessel equation is considered. It is shown that its integral kernel can be uniformly approximated by linear combinations of constructed here generalized wave polynomials, solutions of a singular hyperbolic partial differential equation arising in relation with the transmutation kernel. As a corollary of this result an approximation of the regular solution of the perturbed Bessel equation is proposed with corresponding estimates independent of the spectral parameter.Existence of standing pulse solutions to a skew-gradient systemhttps://zbmath.org/1491.340462022-09-13T20:28:31.338867Z"Choi, Yung-Sze"https://zbmath.org/authors/?q=ai:choi.yung-sze"Lee, Jieun"https://zbmath.org/authors/?q=ai:lee.jieun.1|lee.jieunIn the paper under review, the authors investigate the following system
\[ \begin{cases}
d u_{x x}+f(u)-v=0, \\
v_{x x}-\gamma v-B v^{3}+u=0
\end{cases}\]
on \((-\infty, \infty)\) for small \(\gamma\) and \(d\), which is related to the steady states involving a class of reaction diffusion equations. As we see, this work is the first attempt to show the existence of standing pulse solutions on \((-\infty,\infty)\) in a FitzHugh-Nagumo type system to account for nonlinear dependence of the inhibitor reaction term. Using a variational approach that involves several nonlocal terms, the authors establish the existence of standing pulse solutions with a sign change, when the parameters are restricted to some reasonable ranges. In addition, they explored some qualitative properties of the standing pulse solutions.
Reviewer: Ziheng Zhang (Tianjin)Qualitative study of effects of vorticity on traveling wave solutions to the two-component Zakharov-Itō systemhttps://zbmath.org/1491.340562022-09-13T20:28:31.338867Z"Wen, Zhenshu"https://zbmath.org/authors/?q=ai:wen.zhenshuThis paper mainly studies traveling wave solutions to the two-component Zakharov-Itō system by dynamical system method, the work extends previous results.
In this manuscript, the author obtains all possible phase portraits of the system and shows the existence of all bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, kink (antikink) wave solutions and compactons, etc.
The paper is well written and the contributions are new. I will recommend that the manuscript be accepted for publication.
Reviewer: Hang Zheng (Jinhua)Partial differentials with applications to thermodynamics and compressible flowhttps://zbmath.org/1491.350012022-09-13T20:28:31.338867Z"Braga da Costa Campos, Luis Manuel"https://zbmath.org/authors/?q=ai:braga-da-costa-campos.luis-manuel"Vilela, Luís António Raio"https://zbmath.org/authors/?q=ai:vilela.luis-antonio-raioPublisher's description: This book is part of the series ``Mathematics and Physics Applied to Science and Technology.'' It combines rigorous mathematics with general physical principles to model practical engineering systems with a detailed derivation and interpretation of results. The book presents the mathematical theory of partial differential equations and methods of solution satisfying initial and boundary conditions. It includes applications to acoustic, elastic, water, electromagnetic and other waves, to the diffusion of heat, mass and electricity, and to their interactions. The author covers simultaneously rigorous mathematics, general physical principles and engineering applications with practical interest. The book provides interpretation of results with the help of illustrations throughout and discusses similar phenomena, such as the diffusion of heat, electricity and mass. The book is intended for graduate students and engineers working with mathematical models and can be applied to problems in mechanical, aerospace, electrical and other branches of engineering.Vector fields with applications to thermodynamics and irreversibilityhttps://zbmath.org/1491.350022022-09-13T20:28:31.338867Z"Braga da Costa Campos, Luis Manuel"https://zbmath.org/authors/?q=ai:braga-da-costa-campos.luis-manuel"Vilela, Luís António Raio"https://zbmath.org/authors/?q=ai:vilela.luis-antonio-raioPublisher's description: Vector Fields with Applications to Thermodynamics and Irreversibility is part of the series ``Mathematics and Physics for Science and Technology'', which combines rigorous mathematics with general physical principles to model practical engineering systems with a detailed derivation and interpretation of results. Volume V presents the mathematical theory of partial differential equations and methods of solution satisfying initial and boundary conditions, and includes applications to: acoustic, elastic, water, electromagnetic and other waves; the diffusion of heat, mass and electricity; and their interactions. This is the first book of the volume.
The second book of volume V continues this book on thermodynamics, focusing on the equation of state and energy transfer processes including adiabatic, isothermal, isobaric and isochoric. These are applied to thermodynamic cycles, like the Carnot, Atkinson, Stirling and Barber-Brayton cycles, that are used in thermal devices, including refrigerators, heat pumps, and piston, jet and rocket engines. In connection with jet propulsion are considered adiabatic flows and normal and oblique shock waves in free space and nozzles with variable cross-section. The equations of fluid mechanics are derived for compressible two-phase flow in the presence of shear and bulk viscosity, thermal conduction and mass diffusion. The thermodynamic cycles are illustrated by detailed calculations modelling the operation of piston, turbojet and rocket engines in various ambient conditions, ranging from sea level, the atmosphere of the earth at altitude and vacuum of space, for the propulsion of land, sea, air and space vehicles.
The book is intended for graduate students and engineers working with mathematical models and can be applied to problems in mechanical, aerospace, electrical and other branches of engineering dealing with advanced technology, and also in the physical sciences and applied mathematics.
This book:
\begin{itemize}
\item Simultaneously covers rigorous mathematics, general physical principles and engineering applications with practical interest
\item Provides interpretation of results with the help of illustrations
\item Includes detailed proofs of all results
\end{itemize}Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencieshttps://zbmath.org/1491.350032022-09-13T20:28:31.338867Z"Akahori, Takafumi"https://zbmath.org/authors/?q=ai:akahori.takafumi"Ibrahim, Slim"https://zbmath.org/authors/?q=ai:ibrahim.slim"Kikuchi, Hiroaki"https://zbmath.org/authors/?q=ai:kikuchi.hiroaki"Nawa, Hayato"https://zbmath.org/authors/?q=ai:nawa.hayatoThe authors consider the nonlinear Schrödinger equation
\[
i\frac{\partial\psi}{\partial t}+\triangle\psi+\left\vert \psi\right\vert
^{p-1}\psi+\left\vert \psi\right\vert ^{\frac{4}{d-2}}\psi=0,\tag{NLS}
\]
where \(\psi=\psi\left( x,t\right) \) is a complex-valued function on
\(\mathbb{R}^{d}\times\mathbb{R}\), with \(d\geq3\) and \(p\) satisfies
\[
2+\frac{4}{d}<p+1<2^{\ast}:=2+\frac{4}{d-2}.
\]
The aim of the paper is to study the behaviour of solutions to (NLS) slightly
above the ``ground state threshold at low
frequencies'', in the spirit of \textit{K. Nakanishi} and \textit{W. Schlag} [J. Differ. Equations 250, No. 5, 2299--2333 (2011; Zbl 1213.35307); Calc. Var. Partial Differ. Equ. 44, No. 1--2, 1--45 (2012; Zbl 1237.35148)].
For this purpose, the authors need detailed information about the ground
states of the corresponding semilinear elliptic equation
\[
\omega u-\triangle u-\left\vert u\right\vert ^{p-1}u-\left\vert u\right\vert
^{\frac{4}{d-2}}u=0,\text{ \ }u\in H^{1}\left( \mathbb{R}^{d}\right)
\diagdown\{0\}.
\]
Reviewer: Ivan Naumkin (Nice)Analytic semigroups and semilinear initial boundary value problemshttps://zbmath.org/1491.350042022-09-13T20:28:31.338867Z"Taira, Kazuaki"https://zbmath.org/authors/?q=ai:taira.kazuakiPublisher's description: A careful and accessible exposition of a functional analytic approach to initial boundary value problems for semilinear parabolic differential equations, with a focus on the relationship between analytic semigroups and initial boundary value problems. This semigroup approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators, one of the most influential works in the modern history of analysis. Complete with ample illustrations and additional references, this new edition offers both streamlined analysis and better coverage of important examples and applications. A powerful method for the study of elliptic boundary value problems, capable of further extensive development, is provided for advanced undergraduates or beginning graduate students, as well as mathematicians with an interest in functional analysis and partial differential equations.A parabolic-quasilinear predator-prey model under pursuit-evasion dynamicshttps://zbmath.org/1491.350052022-09-13T20:28:31.338867Z"Telch, Bruno"https://zbmath.org/authors/?q=ai:telch.brunoSummary: For positive parameters \(\alpha\) and \(\beta\) restricted to a certain condition, we prove the global solvability of the parabolic-quasilinear system
\[
\begin{cases}
& \partial_t u - \Delta u +\nabla \cdot (u \nabla p) = \alpha u w - u \\
& \partial_t w - \nabla \cdot (\Psi (w) \nabla w) - \nabla \cdot (w \nabla q) = \beta w (1-w-u)
\end{cases}
\]
where \(p\) and \(q\) are the solutions of \(\partial_t p - \alpha_p \Delta p = \overline{\delta}_w w - \overline{\delta}_p p\) and \(\partial_t q - \alpha_q \Delta q = \overline{\delta}_u u - \overline{\delta}_q q\), respectively, \(\Psi \in C^2([0, + \infty))\) positive, and complemented with appropriated initial data and Neumann boundary condition. This system models a predator-prey interaction with pursuit and evasion, where \(u\) represents the predator density, \(w\) represents the prey density, \(p\) and \(q\) are the pheromone of the species. The result of this article extends the result of \textit{P. Amorim} and \textit{B. Telch} [J. Math. Anal. Appl. 500, No. 1, Article ID 125128, 27 p. (2021; Zbl 1470.35177)] without considering any assumption under the asymptotic behaviour of pheromone production of the predator as in there. The results presented here contribute to the results presented in
[\textit{P. Amorim}, \textit{B. Telch} and \textit{L. M. Villada}, ``A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing'', Math. Biosci. Eng. 16, No. 5, 5114--5145 (2019; \url{doi:10.3934/mbe.2019257}); Amorim and Telch, loc. cit.], in particular, it is the first result collaborating with [\textit{B. Telch}, Nonlinear Anal., Real World Appl. 59, Article ID 103269, 12 p. (2021; Zbl 1464.35145)] in the relevant case \(b \equiv 0\) in there.Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticityhttps://zbmath.org/1491.350062022-09-13T20:28:31.338867Z"Galanopoulou, Myrto"https://zbmath.org/authors/?q=ai:galanopoulou.myrto"Vikelis, Andreas"https://zbmath.org/authors/?q=ai:vikelis.andreas-p"Koumatos, Konstantinos"https://zbmath.org/authors/?q=ai:koumatos.konstantinosSummary: This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.Weak-strong uniqueness for energy-reaction-diffusion systemshttps://zbmath.org/1491.350072022-09-13T20:28:31.338867Z"Hopf, Katharina"https://zbmath.org/authors/?q=ai:hopf.katharinaSolving \((2 + 1)\)-dimensional Davey-Stewartson equation by \((G' /G)\) expansion methodhttps://zbmath.org/1491.350082022-09-13T20:28:31.338867Z"Zhu, MingXing"https://zbmath.org/authors/?q=ai:zhu.mingxing"Wu, Li"https://zbmath.org/authors/?q=ai:wu.li"Yin, ChuChu"https://zbmath.org/authors/?q=ai:yin.chuchu"Zhang, Ming Yuan"https://zbmath.org/authors/?q=ai:zhang.mingyuan(no abstract)Nonexistence of extremals for a Trudinger-Moser inequality on a Riemann surface with boundaryhttps://zbmath.org/1491.350092022-09-13T20:28:31.338867Z"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjieSummary: It is known that the Adimurthi-Druet inequality admits extremal function, when the perturbation parameter \(\alpha\) is small. However, the problem that bothered us is when extremal function of the Adimurthi-Druet inequality does not exist. Recently, Mancini-Thizy (J. Differential Equations) first solved this problem by the method of energy estimate. After that, Yang (Sci. China Math.) extended the result to a closed Riemann surface. In this paper, on a compact Riemann surface with boundary, we consider the nonexistence of extremal function for a Trudinger-Moser inequality with the Neumann boundary condition. Moreover, this result complements our work in [Math. Inequal. Appl. 24, No. 3, 775--791 (2021; Zbl 1484.46045)].Lie group analysis for a higher-order Boussinesq-Burgers systemhttps://zbmath.org/1491.350102022-09-13T20:28:31.338867Z"Liu, Fei-Yan"https://zbmath.org/authors/?q=ai:liu.feiyan"Gao, Yi-Tian"https://zbmath.org/authors/?q=ai:gao.yitianSummary: Boussinesq-Burgers (BB)-type equations have been proposed to model the shallow water waves. Under investigation in this Letter is a higher-order BB system. We obtain the Lie point symmetry generators, Lie symmetry groups and symmetry reductions for that system via the Lie group method. Hyperbolic-function, trigonometric-function and rational solutions for that system are derived.Fisher-KPP equation with Robin boundary conditions on the real half linehttps://zbmath.org/1491.350112022-09-13T20:28:31.338867Z"Suo, Jinzhe"https://zbmath.org/authors/?q=ai:suo.jinzhe"Tan, Kaiyuan"https://zbmath.org/authors/?q=ai:tan.kaiyuanSummary: We consider the Fisher-KPP equation with Robin boundary conditions on the half line. We show that this problem has even number of nonnegative stationary solutions, which are ordered, a stable one is sandwiched by two unstable ones. Then we construct several types of entire solutions between them, each entire solution connects an unstable stationary solution with a stable one. In particular, we construct two types of entire solutions \(\mathcal{U}^c (x, t)\) and \(\mathcal{U} (x, t)\) connecting 0 and the smallest positive stationary solution. In addition, \(\mathcal{U}^c\) tend to the traveling wave solution with speed \(c\) as \(t \to - \infty\) in a moving frame, and \(\mathcal{U} (x, t)\) enjoys convexity. This paper extends the recent results of \textit{B. Lou}, \textit{J. Lu} and \textit{Y. Morita} [``Entire solutions of the Fisher-KPP equation on the half line'', Eur. J. App. Math. 31, 407--422 (2020; \url{doi:10.1017/S0956792519000093})] from Dirichlet boundary condition to Robin boundary condition.Hopf bifurcation for general 1D semilinear wave equations with delayhttps://zbmath.org/1491.350122022-09-13T20:28:31.338867Z"Kmit, Irina"https://zbmath.org/authors/?q=ai:kmit.irina"Recke, Lutz"https://zbmath.org/authors/?q=ai:recke.lutzThe paper is about 1D autonomous damped and delayed semilinear wave equations of the form \[ \partial_t^2 u(t,x) = a(x,\lambda) \partial_x^2 u(t,x) + b(x,\lambda,u(t,x),\partial_t u(t,x),\partial_x u(t,x),u(t-\tau,x)), \] with boundary conditions \( u(0, t) = \partial_x u(t, 1) = 0 \), and \( b(x, \lambda, 0, 0, 0, 0) = 0 \) for all \( x \) and \( \lambda \). The goal is to prove the existence and local uniqueness of families of non-stationary time-periodic solutions which bifurcate from the stationary solution \( u = 0 \). The proof is based on a Lyapunov-Schmidt reduction where a number of technical difficulties, such as the the Fredholm property of the involved opertators, have to be overcome.
Reviewer: Guido Schneider (Stuttgart)Development of the Lomov regularization method for a singularly perturbed Cauchy problem and a boundary value problem on the half-line for parabolic equations with a ``simple'' rational turning pointhttps://zbmath.org/1491.350132022-09-13T20:28:31.338867Z"Eliseev, A. G."https://zbmath.org/authors/?q=ai:eliseev.aleksandr-georgievich"Ratnikova, T. A."https://zbmath.org/authors/?q=ai:ratnikova.tatyana-anatolevna"Shaposhnikova, D. A."https://zbmath.org/authors/?q=ai:shaposhnikova.d-aSummary: The Lomov regularization method was developed for the Cauchy problem and the mixed problem for a singularly perturbed parabolic equation in the case of a ``simple'' rational turning point of the limit operator. The maximum principle is used to prove the asymptotic convergence of the resulting series.Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equationshttps://zbmath.org/1491.350142022-09-13T20:28:31.338867Z"Han, Yuxi"https://zbmath.org/authors/?q=ai:han.yuxi"Tu, Son N. T."https://zbmath.org/authors/?q=ai:tu.son-n-tSummary: We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for non-negative Lipschitz data that vanish on the boundary, the rate of convergence is \(\mathcal{O}(\sqrt{\varepsilon})\) in the interior. Moreover, the one-sided rate can be improved to \(\mathcal{O}(\varepsilon)\) for non-negative compactly supported data and \(\mathcal{O}(\varepsilon^{1/p})\) (where \(1<p<2\) is the exponent of the gradient term) for non-negative data \(f\in \mathrm{C}^2 (\overline{\varOmega})\) such that \(f = 0\) and \(Df = 0\) on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.On thermal boundary layer in a viscous non-Newtonian mediumhttps://zbmath.org/1491.350152022-09-13T20:28:31.338867Z"Kisatov, M. A."https://zbmath.org/authors/?q=ai:kisatov.m-a"Samokhin, V. N."https://zbmath.org/authors/?q=ai:samokhin.vyacheslav-n"Chechkin, G. A."https://zbmath.org/authors/?q=ai:chechkin.gregory-aSummary: An existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological law is generalized.Existence of contrast structures in a problem with discontinuous reaction and advectionhttps://zbmath.org/1491.350162022-09-13T20:28:31.338867Z"Nefedov, N. N."https://zbmath.org/authors/?q=ai:nefedov.nikolai-nikolaevich"Nikulin, E. I."https://zbmath.org/authors/?q=ai:nikulin.e-i"Orlov, A. O."https://zbmath.org/authors/?q=ai:orlov.andrey-oSummary: In the paper, a boundary value problem for a singularly perturbed reaction-diffusion-advection equation is considered in a two-dimensional domain in the case of discontinuous coefficients of reaction and advection, whose discontinuity occurs on a predetermined curve lying in the domain. It is shown that this problem has a solution with a sharp internal transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion in a small parameter is constructed, and also sufficient conditions are obtained for the input data of the problem under which the solution exists. The proof of the existence theorem is based on the asymptotic method of differential inequalities. It is also shown that a solution of this kind is Lyapunov asymptotically stable and locally unique. The results of the paper can be used to create mathematical models of physical phenomena at the interface between two media with different characteristics, as well as for the development of numerical-analytical methods for solving singularly perturbed problems.Quasi-stationary distribution for the Langevin process in cylindrical domains. II: Overdamped limithttps://zbmath.org/1491.350172022-09-13T20:28:31.338867Z"Ramil, Mouad"https://zbmath.org/authors/?q=ai:ramil.mouadSummary: Consider the Langevin process, described by a vector (positions and momenta) in \({\mathbb{R}^d}\times{\mathbb{R}^d} \). Let \(\mathcal{O}\) be a \(({\mathcal{C}^2}\) open bounded and connected set of \({\mathbb{R}^d} \). Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain \(D:=\mathcal{O}\times{\mathbb{R}^d}\). In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain \(\mathcal{O}\).
For Part I, see [\textit{T. Lelièvre}, Stochastic Processes Appl. 144, 173--201 (2022; Zbl 1481.35294)].Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problemhttps://zbmath.org/1491.350182022-09-13T20:28:31.338867Z"Sônego, Maicon"https://zbmath.org/authors/?q=ai:sonego.maicon"do Nascimento, Arnaldo Simal"https://zbmath.org/authors/?q=ai:do-nascimento.arnaldo-simalThe authors consider the stable solutions for Allen-Cahn equations with singular perturbation. The novel results are the case that the inhomogeneous coefficients and diffusion function may be degenerate. The interesting results are the existence of stable stationary solutions which develop internal transition layer as perturbation rate is sufficiently small.
Reviewer: Jian-Wen Sun (Lanzhou)On the homogenization of an optimal control problem in a domain perforated by holes of critical size and arbitrary shapehttps://zbmath.org/1491.350192022-09-13T20:28:31.338867Z"Díaz, J. I."https://zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonso"Podolskiy, A. V."https://zbmath.org/authors/?q=ai:podolskii.alexander-v"Shaposhnikova, T. A."https://zbmath.org/authors/?q=ai:shaposhnikova.tatiana-aSummary: The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual \(L^2\)-norm of the control. We consider the so-called ``critical'' relation between the problem parameters and the period of the structure \(\varepsilon \to 0\). Two ``strange'' terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes.Homogenization of a coupled incompressible Stokes-Cahn-Hilliard system modeling binary fluid mixture in a porous mediumhttps://zbmath.org/1491.350202022-09-13T20:28:31.338867Z"Lakhmara, Nitu"https://zbmath.org/authors/?q=ai:lakhmara.nitu"Mahato, Hari Shankar"https://zbmath.org/authors/?q=ai:mahato.hari-shankarSummary: A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes-Cahn-Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter \(\varepsilon\) in the considered model. At first, the existence of solution of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach.The \(p\)-Laplacian in thin channels with locally periodic roughness and different scaleshttps://zbmath.org/1491.350212022-09-13T20:28:31.338867Z"Nakasato, Jean Carlos"https://zbmath.org/authors/?q=ai:nakasato.jean-carlos"Pereira, Marcone Corrêa"https://zbmath.org/authors/?q=ai:pereira.marcone-correaNon-uniform continuous dependence on initial data for a two-component Novikov system in Besov spacehttps://zbmath.org/1491.350222022-09-13T20:28:31.338867Z"Wu, Xing"https://zbmath.org/authors/?q=ai:wu.xing"Cao, Jie"https://zbmath.org/authors/?q=ai:cao.jieSummary: In this paper, we show that the solution map of the two-component Novikov system is not uniformly continuous on the initial data in Besov spaces \(B_{p, r}^{s - 1} (\mathbb{R}) \times B_{p, r}^s (\mathbb{R})\) with \(s > \max \{1 + \frac{1}{p}, \frac{3}{2}\}\), \(1 \leq p \leq \infty\), \(1 \leq r < \infty\). Our result covers and extends the previous non-uniform continuity in Sobolev spaces \(H^{s - 1} (\mathbb{R}) \times H^s (\mathbb{R})\) for \(s > \frac{5}{2}\) to Besov spaces.Bifurcation structure and stability of steady gravity water waves with constant vorticityhttps://zbmath.org/1491.350232022-09-13T20:28:31.338867Z"Dai, Guowei"https://zbmath.org/authors/?q=ai:dai.guowei"Li, Fengquan"https://zbmath.org/authors/?q=ai:li.fengquan"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.10|zhang.yong.12|zhang.yong.13|zhang.yong.9|zhang.yong.15|zhang.yong.2|zhang.yong.1|zhang.yong.11|zhang.yong.8|zhang.yong|zhang.yong.14|zhang.yong.4|zhang.yong.5|zhang.yong.7Summary: This paper studies the local bifurcation direction, stability properties and global structure for a nonlinear pseudodifferential equation, which describes the periodic travelling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation points. In particular, their signs can be strictly judged when constant vorticity vanishes. Furthermore, we present the stability analysis for the travelling water waves that have small vorticity and amplitude. We also show that the global bifurcation curves can't form a loop. Moreover, if the total head is bounded, the existence of waves of all amplitudes from zero up to that of Stokes' highest wave has been established.Dynamics in a diffusive predator-prey system with double Allee effect and modified Leslie-Gower schemehttps://zbmath.org/1491.350242022-09-13T20:28:31.338867Z"Li, Haixia"https://zbmath.org/authors/?q=ai:li.haixia"Yang, Wenbin"https://zbmath.org/authors/?q=ai:yang.wenbin"Wei, Meihua"https://zbmath.org/authors/?q=ai:wei.meihua"Wang, Aili"https://zbmath.org/authors/?q=ai:wang.ailiThe effect of diffusion on the dynamics of a predator-prey chemostat modelhttps://zbmath.org/1491.350252022-09-13T20:28:31.338867Z"Nie, Hua"https://zbmath.org/authors/?q=ai:nie.hua"Shi, Yao"https://zbmath.org/authors/?q=ai:shi.yao"Wu, Jianhua"https://zbmath.org/authors/?q=ai:wu.jianhuaComputing the sound of the sea in a seashellhttps://zbmath.org/1491.350262022-09-13T20:28:31.338867Z"Ben-Artzi, Jonathan"https://zbmath.org/authors/?q=ai:ben-artzi.jonathan"Marletta, Marco"https://zbmath.org/authors/?q=ai:marletta.marco"Rösler, Frank"https://zbmath.org/authors/?q=ai:rosler.frankSummary: The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is \(\mathcal{C}^2\). The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.Stability of entire solutions emanating from bistable planar traveling waves in exterior domainshttps://zbmath.org/1491.350272022-09-13T20:28:31.338867Z"Jia, Fu-Jie"https://zbmath.org/authors/?q=ai:jia.fu-jie"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2|wang.zhi-cheng.1Summary: This paper is devoted to the Lyapunov stability of entire solutions emanating from a planar front for bistable (bistable-type or multistable-type) reaction-diffusion equations in exterior domains \(\Omega = \mathbb{R}^N \backslash K\). Applying the super- and sub-solution method, we prove that the entire solution is Lyapunov stable under the small initial perturbations. In addition, the Lyapunov stability is independent of the geometrical conditions of the obstacle \(K\).Global boundedness and stability in a density-suppressed motility model with generalized logistic source and nonlinear signal productionhttps://zbmath.org/1491.350282022-09-13T20:28:31.338867Z"Tao, Xueyan"https://zbmath.org/authors/?q=ai:tao.xueyan"Fang, Zhong Bo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: This work studies the density-suppressed motility model
\[
\begin{aligned}
&u_t= \Delta (\gamma (v)u)+r u -\mu u^{\alpha},&& {x \in \Omega, \,t>0,}\\
&v_t= \Delta v-v+u^{\beta},&& x \in \Omega ,\,t>0,
\end{aligned}
\] under no-flux boundary conditions in a bounded domain \(\Omega \subset{\mathbb{R}}^N \, (N\ge 2)\) with smooth boundary, where \(r, \mu, \beta >0\) and \(\alpha >1\). The positive motility function \(\gamma \in C^3([0,\infty))\) is density-suppressed by assuming \(\gamma '(s)\le 0\) for all \(s>0\). This system has been proposed as a model for experimentally observable stripe pattern formation phenomena in bacterial populations. It is proved that if \(\beta <\frac{2}{N+2}\alpha \), then the system has a classical solution in \(\Omega \times (0,\infty)\) which is globally bounded. Under the assumption that additionally \(\mu\) is sufficiently big, this solution is moreover shown to stabilize toward the constant equilibrium \(\big (\big (\frac{r}{\mu}\big)^{\frac{1}{\alpha -1}}, \big (\frac{r}{\mu}\big)^{\frac{\beta}{\alpha -1}}\big)\) in \((L^{\infty}(\Omega))^2\).Remarks on thin quasilinear plates with mixed boundary conditionshttps://zbmath.org/1491.350292022-09-13T20:28:31.338867Z"Batista, A. A."https://zbmath.org/authors/?q=ai:batista.adriano-a"Clark, H. R."https://zbmath.org/authors/?q=ai:clark.haroldo-rodrigues"Guardia, R. R."https://zbmath.org/authors/?q=ai:guardia.ronald-ramos"Lourêdo, A. T."https://zbmath.org/authors/?q=ai:louredo.aldo-trajano"Milla-Miranda, M."https://zbmath.org/authors/?q=ai:milla-miranda.manuelSummary: This paper deals with initial-boundary value problems for a damped thin quasilinear plate. With restriction on the norms of the initial data, it will be established global weak and global strong solutions. It will also be shown that the strong solution is uniformly stable and unique. Furthermore, using a weak internal damping mechanism, an exponential decay estimate for the energy of weak solutions is established.Asymptotic behavior of solution of Whitham-Broer-Kaup type equations with negative dispersionhttps://zbmath.org/1491.350302022-09-13T20:28:31.338867Z"Bedjaoui, Nabil"https://zbmath.org/authors/?q=ai:bedjaoui.nabil"Kumar, Rajesh"https://zbmath.org/authors/?q=ai:kumar.rajesh-s"Mammeri, Youcef"https://zbmath.org/authors/?q=ai:mammeri.youcefSummary: In this work, we discuss the long time behavior of solutions of the Whitham-Broer-Kaup system with Lipschitz nonlinearity and negative dispersion term. We prove the global well-posedness when \(\alpha+\beta^2<0\) as well as the convergence to 0 of small solutions at rate \(\mathcal{O}(t^{-1/2})\).Optimal relaxation of bump-like solutions of the one-dimensional Cahn-Hilliard equationhttps://zbmath.org/1491.350312022-09-13T20:28:31.338867Z"Biesenbach, Sarah"https://zbmath.org/authors/?q=ai:biesenbach.sarah"Schubert, Richard"https://zbmath.org/authors/?q=ai:schubert.richard"Westdickenberg, Maria G."https://zbmath.org/authors/?q=ai:westdickenberg.maria-gThe authors use a nonlinear energy-based method to derive the optimal relaxation rates for the Cahn-Hilliard equation, with a double-well potential, on the one-dimensional torus and the line. The structure of the initial conditions includes an order-one \(L^1\) disturbance occurring at an arbitrary location in the system. The result extends that previously obtained by the relaxation method developed for a single transition layer (the ``kink'') to the case of two transition layers (the ``bump''). The techniques employed rely on the Nash-type inequalities, duality arguments, and Schauder estimates. It is found that both in the case of the kink and the bump, the energy gap is translation invariant and its decay alone cannot specify to which member of the family of minimizers the solution converges. In the case of the kink, the conserved quantity singles out the longtime limit, while in the case of a bump, a new argument is needed. On the torus, it is quantified the convergence to the bump that is the longtime limit; on the line, the bump-like states are merely metastable and it is found the initial algebraic relaxation behavior.
Reviewer: Gabriela Marinoschi (Bucureşti)Exponential stability for a thermo-viscoelastic Timoshenko system with fading memoryhttps://zbmath.org/1491.350322022-09-13T20:28:31.338867Z"Calsavara, B. M. R."https://zbmath.org/authors/?q=ai:calsavara.bianca-morelli-rodolfo"Gomes Tavares, E. H."https://zbmath.org/authors/?q=ai:gomes-tavares.eduardo-h"Jorge Silva, M. A."https://zbmath.org/authors/?q=ai:jorge-silva.marcio-antonioThe authors study the thermo-viscoelastic Timoshenko beam system in the autonomous form after setting a relative displacement history \(\eta^t(s):=\psi(t)-\psi(t-s),\ t,s>0\)
\begin{align*}
&\rho_1\phi_{tt}-\kappa(\phi_x+\psi)_x+\sigma\theta_x=0, \\
&\rho_2\psi_{tt} - b\psi_{xx}+\kappa(\phi_x+\psi)-\int_0^\infty g(s)\eta_{xx}-\sigma\theta=0,\\
&\rho_3\theta_t-\beta\theta_{xx}+\sigma(\phi_x+\psi)_t=0,\ \eta_t+\eta_s=\psi_t,\ x\in (0,L),\ t>0,\ s>0\\
\end{align*}
with boundary and initial conditions
\begin{align*}
&\phi_x(0,t)=\phi_x(L,t)=\psi(0,t)=\psi(L,t)= \theta(0,t)=\theta(L,t)=0,\ t\ge 0,\\
&\eta^t(0,s)=\eta^t(L,s)=0,\ t,s>0;\\
&(\phi(x,0),\phi_t(x,0), \psi(x,0),\psi_t(x,0),\theta(x,0))=(\phi_0,\phi_1,\psi_0,\psi_1,\theta_0),\ x\in (0,L),\\
&\eta^0(x,s)=\psi_0(x,0)-\psi_0(x,-s),\ (x,s)\in (0,L)\times (0,\infty).
\end{align*}
The problem is rewritten as a first order Cauchy problem, and a semigroup approach is applied. The existence and uniqueness of a mild solution and its exponential stability are verified.
Reviewer: Igor Bock (Bratislava)Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean spacehttps://zbmath.org/1491.350332022-09-13T20:28:31.338867Z"Cannarsa, Piermarco"https://zbmath.org/authors/?q=ai:cannarsa.piermarco"Mendico, Cristian"https://zbmath.org/authors/?q=ai:mendico.cristianSummary: The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant (Mañé critical value) for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a solution to the critical HJB equation on the whole space which coincides with its Lax-Oleinik evolution.Asymptotic behavior of non-autonomous Lamé systems with subcritical and critical mixed nonlinearitieshttps://zbmath.org/1491.350342022-09-13T20:28:31.338867Z"Costa, Alberto L. C."https://zbmath.org/authors/?q=ai:costa.alberto-l-c"Freitas, Mirelson M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Wang, Renhai"https://zbmath.org/authors/?q=ai:wang.renhaiSummary: This paper deals with the asymptotic behavior of solutions to a class of non-autonomous Lamé systems modeling the physical phenomenon of isotropic elasticity. The main feature of this model is that the nonlinearity can be decomposed into a subcritical part and a critical one. We first show that the system generates a non-autonomous dynamical system, and then prove that the system has a minimal universe pullback attractor. The upper-semicontinuity of these pullback attractors is also established as the perturbation parameter of the external force tends to zero. The \texttt{quasi-stability} ideas developed by \textit{I. Chueshov} and \textit{I. Lasiecka} [Von Karman evolution equations. Well-posedness and long-time dynamics. New York, NY: Springer (2010; Zbl 1298.35001); Long-time behavior of second order evolution equations with nonlinear damping. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1151.37059)] and \textit{I. Chueshov} [Dynamics of quasi-stable dissipative systems. Cham: Springer (2015; Zbl 1362.37001)] are used to prove the pullback asymptotic compactness of the solutions in order to overcome the difficulty caused by the critical growthness of the nonlinearity.Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearityhttps://zbmath.org/1491.350352022-09-13T20:28:31.338867Z"Ding, Hang"https://zbmath.org/authors/?q=ai:ding.hang"Zhou, Jun"https://zbmath.org/authors/?q=ai:zhou.jun.1Summary: In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo-Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when \(I (u_0) > 0\), where \(I (u_0)\) denotes the Nehari functional with the initial value \(u_0\). Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when \(I (u_0) < 0\). Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy.On a system of coupled Cahn-Hilliard equationshttps://zbmath.org/1491.350362022-09-13T20:28:31.338867Z"Di Primio, Andrea"https://zbmath.org/authors/?q=ai:di-primio.andrea"Grasselli, Maurizio"https://zbmath.org/authors/?q=ai:grasselli.maurizioSummary: We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono type equation in a bounded domain of \(\mathbb{R}^d\), \(d = 2, 3\). This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters \(u\) and \(v\). The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for \(u, v\) and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for \(u\) and \(v\), namely, they both stay uniformly away from the pure phases \(\pm 1\) in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state.Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional dampinghttps://zbmath.org/1491.350372022-09-13T20:28:31.338867Z"Fareh, Abdelfeteh"https://zbmath.org/authors/?q=ai:fareh.abdelfetehSummary: In this paper we consider a linear thermoelastic system of Timoshenko type where the heat conduction is given by the linearized law of Gurtin-Pipkin. An existence and uniqueness result is proved by the use of a semigroup approach. We establish an exponential stability result without any assumption on the wave speeds once here we have a fully damped system.Stabilization of the wave equation with a nonlinear delay term in the boundary conditionshttps://zbmath.org/1491.350382022-09-13T20:28:31.338867Z"Ghecham, Wassila"https://zbmath.org/authors/?q=ai:ghecham.wassila"Rebiai, Salah-Eddine"https://zbmath.org/authors/?q=ai:rebiai.salah-eddine"Sidiali, Fatima Zohra"https://zbmath.org/authors/?q=ai:sidiali.fatima-zohraSummary: A wave equation in a bounded and smooth domain of \(\mathbb{R}^n\) with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.General decay rate for an abstract weakly dissipative Moore-Gibson-Thompson equationhttps://zbmath.org/1491.350392022-09-13T20:28:31.338867Z"Hassan, Jamilu Hashim"https://zbmath.org/authors/?q=ai:hassan.jamilu-hashim"Messaoudi, Salim A."https://zbmath.org/authors/?q=ai:messaoudi.salim-aSummary: In this paper we study an abstract class of weakly dissipative Moore-Gibson-Thompson equation with finite memory. We establish a general decay rate for the solution of the system under some appropriate conditions on the relaxation function.Global existence and decay of solutions for a higher-order Kirchhoff-type systems with logarithmic nonlinearitieshttps://zbmath.org/1491.350402022-09-13T20:28:31.338867Z"Irkıl, Nazlı"https://zbmath.org/authors/?q=ai:irkil.nazli"Pişkin, Erhan"https://zbmath.org/authors/?q=ai:piskin.erhanSummary: In this paper, we aim to understand the characteristics of dynamical behaviour for a higher order Kirchhoff type systems with logarithmic nonlinearities. Based on the potential well method, the main ingredient of this study is to construct several conditions for initial data leading to the solution global existence in case of \(E (0) < d \). On the other hand, we proved that if the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of energy.Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaceshttps://zbmath.org/1491.350412022-09-13T20:28:31.338867Z"Ishige, Kazuhiro"https://zbmath.org/authors/?q=ai:ishige.kazuhiro"Tateishi, Yujiro"https://zbmath.org/authors/?q=ai:tateishi.yujiroSummary: Let \(H:=-\Delta +V\) be a nonnegative Schrödinger operator on \(L^2(\mathbf{R}^N)\), where \(N\ge 2\) and \(V\) is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of \(e^{-tH}\) and \(\nabla e^{-tH}\) in Lorentz spaces.Dynamical behavior of the indirectly and locally memory-damped Timoshenko systemhttps://zbmath.org/1491.350422022-09-13T20:28:31.338867Z"Jin, Kun-Peng"https://zbmath.org/authors/?q=ai:jin.kunpeng"Liang, Jin"https://zbmath.org/authors/?q=ai:liang.jin"Xiao, Ti-Jun"https://zbmath.org/authors/?q=ai:xiao.ti-junSummary: We are concerned with dynamical behavior of the indirectly and locally memory-damped Timoshenko system. The polynomial/exponential decay results for the long-term dynamical behavior of the Timoshenko system are established when the memory kernels decay polynomially/exponentially. To obtain ideal asymptotic decay rates under basic conditions, we take some analysis processes specially designed for our issues. The obtained long-term dynamical behavior theorems, with the exact uniform decay rates for the solutions to the system, indicate that for the Timoshenko system with indirect memory damping, a local memory effect is enough to produce an entire dissipation mechanism and to ensure the same decay rates as in the case of global memory effect. Moreover, although our conditions on the memory kernels are weaker compared with the ones in the literature, we still derive stronger conclusions. Finally, we present some results of numerical simulation to illustrate quantitatively the behavior of the solution energies of our system, which agree with our theoretical results well.Dynamics of a chain of logistic equations with delay and antidiffusive couplinghttps://zbmath.org/1491.350432022-09-13T20:28:31.338867Z"Kashchenko, S. A."https://zbmath.org/authors/?q=ai:kashchenko.sergey-aleksandrovichSummary: The dynamics of chains of coupled logistic equations with delay are studied using methods of local analysis. It is shown that the critical cases have infinite dimension. As the main results, special nonlinear boundary value problems of the parabolic type describing the evolution of solutions to the initial equation that slowly oscillate at the equilibrium state are constructed.Stability results for laminated beam with thermo-visco-elastic effects and localized nonlinear dampinghttps://zbmath.org/1491.350442022-09-13T20:28:31.338867Z"Khalili, Zineb"https://zbmath.org/authors/?q=ai:khalili.zineb"Ouchenane, Djamel"https://zbmath.org/authors/?q=ai:ouchenane.djamel"El Hamidi, Abdallah"https://zbmath.org/authors/?q=ai:el-hamidi.abdallahSummary: In this article, we investigate a one-dimensional thermoelastic laminated beam system with nonlinear damping and viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under minimal conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the first two equations, we show that the solution energy has an explicit and optimal decay rate from which the exponential and polynomial stability are just particular cases. Moreover, we establish a weaker decay result in the case of non-equal wave of speed propagation and give some examples illustrate our results. This work extends and improves the earlier results in the literature, particularly the result of \textit{S. E. Mukiawa} et al. [AIMS Math. 6, No. 1, 333--361 (2021; Zbl 1484.35149)].Global existence and blowup of solutions for a semilinear Klein-Gordon equation with the product of logarithmic and power-type nonlinearityhttps://zbmath.org/1491.350452022-09-13T20:28:31.338867Z"Khuddush, Mahammad"https://zbmath.org/authors/?q=ai:khuddush.mahammad"Prasad, K. Rajendra"https://zbmath.org/authors/?q=ai:prasad.kapula-rajendra"Bharathi, B."https://zbmath.org/authors/?q=ai:bharathi.bSummary: In this paper we study the initial boundary value problem of a semilinear Klein-Gordon equation with the multiplication of logarithmic and polynomial nonlinearities. By using potential well method and energy method, we obtain the existence of global solutions and finite-time blowup solutions.Existence and decay of global solutions to the three-dimensional Kuramoto-Sivashinsky-Zakharov-Kuznetsov equationhttps://zbmath.org/1491.350462022-09-13T20:28:31.338867Z"Larkin, N. A."https://zbmath.org/authors/?q=ai:larkin.nikolaj-aSummary: In this study, we consider the initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation posed on bounded and unbounded parallelepipeds. The existence and uniqueness of global regular and strong solutions, and their exponential decay are established.Uniform stabilization of a variable coefficient wave equation with nonlinear damping and acoustic boundaryhttps://zbmath.org/1491.350472022-09-13T20:28:31.338867Z"Liu, Yu-Xiang"https://zbmath.org/authors/?q=ai:liu.yuxiangSummary: In this paper we consider uniform stabilization of the wave equation with variable coefficients in a bounded domain. The nonlinear damping is put partly on the interior of the domain and partly on the acoustic boundary. Under some checkable conditions on the coefficients, the energy decay results are established by Riemannian geometry method.On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic mapshttps://zbmath.org/1491.350482022-09-13T20:28:31.338867Z"Li, Ze"https://zbmath.org/authors/?q=ai:li.zeSummary: The results of this paper are twofold. In the first part, we prove that for Schrödinger map flows from hyperbolic planes to Riemannian surfaces with non-positive sectional curvatures, the harmonic maps which are holomorphic or anti-holomorphic of arbitrary size are asymptotically stable. In the second part, we prove that for Schrödinger map flows from hyperbolic planes into Kähler manifolds, the admissible harmonic maps of small size are asymptotically stable. The asymptotic stability results stated here contain two types: one is the convergence in \(L^{\infty}_x\) as the previous works, the other is convergence to harmonic maps plus radiation terms in the energy space, which is new in literature of Schrödinger map flows without symmetric assumptions.Existence and decay of solutions for a higher-order viscoelastic wave equation with logarithmic nonlinearityhttps://zbmath.org/1491.350492022-09-13T20:28:31.338867Z"Pişkin, Erhan"https://zbmath.org/authors/?q=ai:piskin.erhan"Irkil, Nazlı"https://zbmath.org/authors/?q=ai:irkil.nazliSummary: The main goal of this paper is to study for the local existence and decay estimates results for a high-order viscoelastic wave equation with logarithmic nonlinearity. We obtain several results: Firstly, by using Feado-Galerkin method and a logaritmic Sobolev inequality, we proved local existence of solutions. Later, we proved general decay results of solutions.Local indirect stabilization of same coupled evolution systems through resolvent estimateshttps://zbmath.org/1491.350502022-09-13T20:28:31.338867Z"Radhia, Ayechi"https://zbmath.org/authors/?q=ai:radhia.ayechi"Moez, Khenissi"https://zbmath.org/authors/?q=ai:moez.khenissiSummary: In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.The lack of exponential stability for a weakly coupled wave equations through a variable density termhttps://zbmath.org/1491.350512022-09-13T20:28:31.338867Z"Salah, Monia Bel Hadj"https://zbmath.org/authors/?q=ai:bel-hadj-salah.moniaSummary: In this paper, we consider a system of two wave equations coupled through zero order terms. One of these equations has an internal damping, and the other has a boundary damping. We investigate stability properties of the system according to the variable strings densities. Indeed, our main result is to show that the corresponding model is not exponentially stable using a spectral theory which forms the center of this work. Otherwise, we establish a polynomial energy decay rate of type \(\frac{1}{\sqrt{t}}\).Harnack's inequality for singular parabolic equations with generalized Orlicz growth under the non-logarithmic Zhikov's conditionhttps://zbmath.org/1491.350522022-09-13T20:28:31.338867Z"Skrypnik, Igor I."https://zbmath.org/authors/?q=ai:skrypnik.igor-igorievichIn this paper dedicated to Emmanuele Di Benedetto, the author considers general divergence type singular parabolic equations with nonstandard growth conditions that in literature are called Orclitz conditions. He establishes that nonnegative bounded weak solutions satisfy an intrinsic form of the Harnack inequality provided that the equation satisfies suitable condition of singularity. These singularity conditions include new cases of equations with $(p,q)$ nonlinearity and non-logarithmic growth. The results are somewhat sharp as shown by several examples.
Reviewer: Vincenzo Vespri (Firenze)Essential forward weak KAM solution for the convex Hamilton-Jacobi equationhttps://zbmath.org/1491.350532022-09-13T20:28:31.338867Z"Su, Xi Feng"https://zbmath.org/authors/?q=ai:su.xifeng"Zhang, Jian Lu"https://zbmath.org/authors/?q=ai:zhang.jianluSummary: For a convex, coercive continuous Hamiltonian on a closed Riemannian manifold \(M\), we construct a unique forward weak KAM solution of
\[
H(x,{d_x}u) = c(H)
\] by a vanishing discount approach, where \(c(H)\) is the Mañé critical value. We also discuss the dynamical significance of such a special solution.Global existence and stabilization in a forager-exploiter model with general logistic sourceshttps://zbmath.org/1491.350542022-09-13T20:28:31.338867Z"Wang, Jianping"https://zbmath.org/authors/?q=ai:wang.jianping|wang.jianping.1Summary: We study a forager-exploiter model with generalized logistic sources in a smooth bounded domain with homogeneous Neumann boundary conditions. A new boundedness criterion is developed to prove the global existence and boundedness of the solution. Under some conditions on the logistic degradation rates, the classical solution exists globally and remains bounded in the high dimensions. Moreover, the large time behavior of the obtained solution is investigated in the case the nutrient supply is a positive constant or has fast decaying property.Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agarhttps://zbmath.org/1491.350552022-09-13T20:28:31.338867Z"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThe author studies the doubly degenerate parabolic system \[\begin{cases} u_t = \nabla \cdot (uv \nabla u) + l uv, & \quad (x,t) \in \Omega \times (0,\infty),\\
v_t = \Delta v -uv, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases}\] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0, v_0 \in L^\infty(\Omega)\) such that \(u_0 \not\equiv 0\), \(v_0 \not\equiv 0\), and \(\sqrt{v_0} \in W^{1,2}(\Omega)\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^n\) is a bounded convex domain with smooth boundary with \(n \in \mathbb{N}\) as well as \(l \ge 0\). Here, \(u\) denotes the density of a population and \(v\) the food resource distribution, e.g. for Bacillus subtilis in nutrient-poor environments.
The author proves the existence of a global weak solution \((u,v) \in C^0(\overline{\Omega} \times (0,\infty)) \cap C^{2,1} (\overline{\Omega} \times (0,\infty))\) such that \(v \in L^\infty (\Omega \times (0,\infty))\) and \(u \in L^\infty((0,\infty); L^p(\Omega))\) for all \(p \in [1, \frac{n}{(n-2)_+})\). Moreover, it is shown that \((u(\cdot,t), v(\cdot,t))\) converges to \((u_\infty,0)\) as \(t \to \infty\) with respect to a certain topological setting for some \(u_\infty\) belonging to \(L^p(\Omega)\) for all \(p \in [1, \frac{n}{(n-2)_+})\). Finally, in case of \(n \le 5\), a stability property is established which in particular shows that any \((u_0,0)\), with \(u_0\) as above being in addition suitably regular, is a stable steady state of the above problem.
In the proof it is first shown that for suitably regularized initial data there are classical solutions \((u_\varepsilon, v_\varepsilon)\) to a family of regularized approximate problems for \(\varepsilon \in (0,1)\). The existence of a global weak solution to the original problem is based on several a priori estimates for the approximate problems, where particularly global \(L^p\) bounds for \(u_\varepsilon\) and \(u\) are deduced from two novel functional inequalities. The proof of the large time and stability behavior in particular relies on estimates for \((u_\varepsilon)_t\) in dual spaces of appropriate Sobolev spaces.
Reviewer: Christian Stinner (Darmstadt)Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source termshttps://zbmath.org/1491.350562022-09-13T20:28:31.338867Z"Zhang, Hongwei"https://zbmath.org/authors/?q=ai:zhang.hongwei.1"Li, Donghao"https://zbmath.org/authors/?q=ai:li.donghao"Zhang, Wenxiu"https://zbmath.org/authors/?q=ai:zhang.wenxiu"Hu, Qingying"https://zbmath.org/authors/?q=ai:hu.qingyingSummary: This work is devoted to studying a wave equation with degenerate nonlocal nonlinear damping and source terms. By potential well theory, we show the asymptotic stability of energy in the presence of a degenerate damping of polynomial type when the initial energy is small. Also, we firstly derive some sufficient conditions on initial data which lead to finite time blow-up.Large time behavior in a quasilinear chemotaxis model with indirect signal absorptionhttps://zbmath.org/1491.350572022-09-13T20:28:31.338867Z"Zhang, Wenji"https://zbmath.org/authors/?q=ai:zhang.wenji"Liu, Suying"https://zbmath.org/authors/?q=ai:liu.suyingSummary: This paper considers the following chemotaxis system
\[
\begin{cases}
u_t = \nabla \cdot \left( D (u) \nabla u\right) - \nabla \cdot \left(S (u) \nabla v\right), & x \in \Omega, t > 0, \\
v_t = \Delta v - v w, & x \in \Omega, t > 0, \\
w_t = - \delta w + u, & x \in \Omega, t > 0,
\end{cases}
\] under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb{R}^n (n \geqslant 2)\), where the parameter \(\delta > 0\) and \(D, S\) are smooth functions satisfying \(D (u) \geqslant K_0 (u + 1)^\alpha, 0 \leqslant S (u) \leqslant K_1 (u + 1)^{\beta - 1} u\) with \(\alpha, \beta \in \mathbb{R}\) and \(K_0, K_1 > 0\). Suppose that \(\beta - \alpha < \frac{ 1}{ n} + \frac{ 1}{ 2} \), this system admits a global bounded classical solution \((u, v, w)\) fulfilling
\[
\| u \left( \cdot t\right) - \overline{u}_0 \|_{L^\infty \left( \Omega\right)} + \| v \left( \cdot t\right) - 0 \|_{L^\infty \left( \Omega\right)} + \left\| w \left( \cdot t\right) - \frac{ \overline{u}_0}{ \delta} \right\|_{L^\infty \left( \Omega\right)} \to 0
\] as \(t \to \infty \), where \(\overline{u}_0 := \frac{ 1}{ | \Omega |} \int_\Omega u_0\).General decay of a nonlinear viscoelastic wave equation with Balakrishnân-Taylor damping and a delay involving variable exponentshttps://zbmath.org/1491.350582022-09-13T20:28:31.338867Z"Zuo, Jiabin"https://zbmath.org/authors/?q=ai:zuo.jiabin"Rahmoune, Abita"https://zbmath.org/authors/?q=ai:rahmoune.abita"Li, Yanjiao"https://zbmath.org/authors/?q=ai:li.yanjiao(no abstract)Weak mean random attractors for non-local random and stochastic reaction-diffusion equationshttps://zbmath.org/1491.350592022-09-13T20:28:31.338867Z"Caballero, Rubén"https://zbmath.org/authors/?q=ai:caballero.ruben"Marín-Rubio, Pedro"https://zbmath.org/authors/?q=ai:marin-rubio.pedro"Valero, José"https://zbmath.org/authors/?q=ai:valero.joseLong time behavior of a degenerate NPZ model with spatial heterogeneityhttps://zbmath.org/1491.350602022-09-13T20:28:31.338867Z"Cheng, Hongyu"https://zbmath.org/authors/?q=ai:cheng.hongyu"Lv, Yunfei"https://zbmath.org/authors/?q=ai:lv.yunfei"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rongSummary: This work proposes a nutrient-phytoplankton-zooplankton (NPZ) model which is governed by a partially degenerate reaction-diffusion equation. Due to the noncompactness of solution map, the Kuratowski measure is introduced to prove the existence of the global attractor. By an innovative construction of Lyapunov functions, sufficient conditions are given to obtain the global attractivity of steady states of the system with spatial parameters.Weak pullback mean random attractors for stochastic evolution equations and applicationshttps://zbmath.org/1491.350612022-09-13T20:28:31.338867Z"Gu, Anhui"https://zbmath.org/authors/?q=ai:gu.anhuiAttractors and their properties for a class of Kirchhoff models with integro-differential dampinghttps://zbmath.org/1491.350622022-09-13T20:28:31.338867Z"Liu, Gongwei"https://zbmath.org/authors/?q=ai:liu.gongwei"Silva, Marcio A. Jorge"https://zbmath.org/authors/?q=ai:jorge-silva.marcio-antonioSummary: In this paper, we investigate a class of Kirchhoff models with integro-differential damping given by a possibly vanishing memory term in a past history framework and a nonlinear nonlocal strong dissipation
\[
\begin{aligned}
&u_{tt}+ \alpha_\mu \triangle^2 u - \triangle_p u - \int_{-\infty}^t \mu(t-s) \triangle^2 u(s) \mathrm{d}s \\
& \quad - N \left(\int_{\Omega} |\nabla u (t)|^2 \mathrm{d}x \right) \triangle u_t + f(u) = h,
\end{aligned}
\] defined in a bounded \(\Omega\) of \(\mathbb{R}^N\). Our main goal is to show the well-posedness and the long-time behavior through the corresponding autonomous dynamical system by regarding the relative past history. More precisely, under the assumptions that the exponent \(p\) and the growth of \(f(u)\) are up to the critical range, the well-posedness and the existence of a global attractor with its geometrical structure are established. Furthermore, in the subcritical case, such a global attractor has finite fractal dimensions as well as regularity of trajectories. A result on generalized fractal exponential attractor is also proved. These results are presented for a wide class of nonlocal damping coefficient \(N(\cdot)\) and possibly degenerate memory term \((\mu = 0)\), which deepen and extend earlier results on the subject.Limiting behavior of FitzHugh-Nagumo equations driven by colored noise on unbounded thin domainshttps://zbmath.org/1491.350632022-09-13T20:28:31.338867Z"Shi, Lin"https://zbmath.org/authors/?q=ai:shi.lin"Lu, Kening"https://zbmath.org/authors/?q=ai:lu.kening"Wang, Xiaohu"https://zbmath.org/authors/?q=ai:wang.xiaohuBlow up and global solvability for an absorptive porous medium equation with memory at the boundaryhttps://zbmath.org/1491.350642022-09-13T20:28:31.338867Z"Anderson, Jeffrey R."https://zbmath.org/authors/?q=ai:anderson.jeffrey-r"Deng, Keng"https://zbmath.org/authors/?q=ai:deng.kengSummary: We study the characterization of global solvability versus blow up in finite time for a porous medium model including a balance of internal absorption with memory driven flux through the boundary. Such a boundary condition was previously investigated as part of a model for the transmission of tumour-released growth factor from the site of a pre-vascularized tumour up to and across the wall of a nearby capillary, initiating the process of new capillary growth known as angiogenesis. In previous studies of the model without absorption, we have established the characterization of global solvability in a manner that exactly parallels known results for the corresponding model with localized boundary flux conditions. To include models accounting for internal uptake of growth factor, this analysis has recently been extended to a heat equation with absorption, and herein, we consider the case of a porous medium equation with absorption. Conditions for global solvability emerge naturally out of integral estimates and again provide close parallels with results for localized boundary flux models. It is noted that the results provide a complete characterization in a wide range of models considered.Finite time blow-up for a semilinear generalized Tricomi system with mixed nonlinearityhttps://zbmath.org/1491.350652022-09-13T20:28:31.338867Z"Fan, Mengting"https://zbmath.org/authors/?q=ai:fan.mengting"Geng, Jinbo"https://zbmath.org/authors/?q=ai:geng.jinbo"Lai, Ning-An"https://zbmath.org/authors/?q=ai:lai.ningan"Lin, Jiayun"https://zbmath.org/authors/?q=ai:lin.jiayunSummary: There is a large literature about blow-up and lifespan estimate for semilinear generalized Tricomi equation with power type nonlinear terms \(|u|^p\) or \(|u_t|^p\). In this work, we are devoted to studying the blow-up phenomenon for the coupled semilinear generalized Tricomi system including both of these two nonlinearities. To this end, asymptotic behavior of the solutions to two second order ordinary differential equations are investigated.Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensionshttps://zbmath.org/1491.350662022-09-13T20:28:31.338867Z"Fujie, Kentaro"https://zbmath.org/authors/?q=ai:fujie.kentaro"Senba, Takasi"https://zbmath.org/authors/?q=ai:senba.takasiSummary: This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller-Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order to prove our theorem, we deal with some auxiliary identity as an evolution equation with a time dependent operator. In view of this new perspective, the direct consequence of the abstract theory is rich enough to establish global existence of the system.Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent masshttps://zbmath.org/1491.350672022-09-13T20:28:31.338867Z"Ikeda, Masahiro"https://zbmath.org/authors/?q=ai:ikeda.masahiro"Tu, Ziheng"https://zbmath.org/authors/?q=ai:tu.ziheng"Wakasa, Kyouhei"https://zbmath.org/authors/?q=ai:wakasa.kyouheiThe authors study the Cauchy problem of
\[
u_{tt} - \Delta_g u + a(t) u_t + b(t) u = |u|^p
\]
in \(\mathbb{R}^n \times [0,\infty)\) with the intial conditions \(u(x,0) = \varepsilon f(x)\) and \(u_t(x,0) = \varepsilon g(x)\). Here, \(\Delta_g\) is a uniform elliptic operator which is a perturbation of the Laplacian, and \(a(t)\) and \(b(t)\) satisfy \(a(t), tb(t) \in L^1([0,\infty))\). Under these assumptions, one can expect that the damping and the mass terms are perturbations and the behavior of the solution is similar to that of the nonlinear wave equation \(\square u = |u|^p\). Indeed, the authors prove the blow-up of the energy solution as well as the upper bound of the lifespan when \(p\) is less than or equal to the so-called Strauss exponent. In particular, they removed the restriction that \(b(t)\) is nonnegative in the result by [\textit{N.-A. Lai} et al. [in: New tools for nonlinear PDEs and application. Proceedings of the 11th ISAAC congress, Växjö, Sweden, August 14--18, 2017. Cham: Birkhäuser. 217--240 (2019; Zbl 1428.35227)]. The proof is based on the iteration argument and the multiplier method. To apply the multiplier method for the equation with two variable coefficients, the authors introduce ``the double multiplier method'', which employs two suitable functions as a multiplier defined from the coefficients \(a(t)\) and \(b(t)\).
Reviewer: Yuta Wakasugi (Hiroshima)The Cauchy problem for a parabolic \(p\)-Laplacian equation with combined nonlinearitieshttps://zbmath.org/1491.350682022-09-13T20:28:31.338867Z"Lu, Heqian"https://zbmath.org/authors/?q=ai:lu.heqian"Zhang, Zhengce"https://zbmath.org/authors/?q=ai:zhang.zhengceSummary: This article studies the Cauchy problem for the evolution \(p\)-Laplacian equation \(u_t - {\Delta}_p u = \lambda u^m + \mu | {\nabla} u |^q u^r\) in \(\mathbb{R}^N \times(0, T)\). The local existence, global existence and nonexistence of solutions are investigated. In particular, for the case \(\lambda > 0\) and \(\mu > 0\), we obtain an optimal Fujita-type result, which demonstrates the positive gradient term brings about the discontinuity phenomenon of the critical exponent. For the case \(\lambda > 0\) and \(\mu < 0\), the existence and nonexistence of global solutions are also discussed.The Cauchy problem and wave-breaking phenomenon for a generalized sine-type FORQ/mCH equationhttps://zbmath.org/1491.350692022-09-13T20:28:31.338867Z"Qin, Guoquan"https://zbmath.org/authors/?q=ai:qin.guoquan"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.bolingSummary: In this paper, we are concerned with the Cauchy problem and wave-breaking phenomenon for a sine-type modified Camassa-Holm (alias sine-FORQ/mCH) equation. Employing the transport equations theory and the Littlewood-Paley theory, we first establish the local well-posedness for the strong solutions of the sine-FORQ/mCH equation in Besov spaces. In light of the Moser-type estimates, we are able to derive the blow-up criterion and the precise blow-up quantity of this equation in Sobolev spaces. We then give a sufficient condition with respect to the initial data to ensure the occurance of the wave-breaking phenomenon by trace the precise blow-up quantity along the characteristics associated with this equation.Blow-up problems for a parabolic equation coupled with superlinear source and local linear boundary dissipationhttps://zbmath.org/1491.350702022-09-13T20:28:31.338867Z"Sun, Fenglong"https://zbmath.org/authors/?q=ai:sun.fenglong"Wang, Yutai"https://zbmath.org/authors/?q=ai:wang.yutai"Yin, Hongjian"https://zbmath.org/authors/?q=ai:yin.hongjianSummary: In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the solutions to blow up in finite time. In particular, we obtain the existence of finite time blow-up solutions with arbitrary high initial energy. We also derive the upper bound and lower bound of the blow up time.Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficienthttps://zbmath.org/1491.350712022-09-13T20:28:31.338867Z"Tello, J. Ignacio"https://zbmath.org/authors/?q=ai:tello.jose-ignacio|ignacio-tello.jThis paper studies a parabolic-elliptic system of partial differential equations with a chemotactic term in a finite dimensional unit ball describing the behavior of the density of a biological species and a chemical stimulus. The system includes a nonlinear chemotactic coefficient depending on the gradient of chemical stimulus that is in the form of \(p\)-Laplacian equation type. Using a priori estimates of the solution and prove the local existence of weak solutions and uniqueness of solutions under suitable assumptions in the initial data for the proposed model. This paper then studies the radially symmetric solutions under the assumption in the initial mass. Further, for large chemotactic coefficient, it presents conditions in the initial data, such that any regular solution of the problem blows up at finite time.
Reviewer: Lingeshwaran Shangerganesh (Ponda)On effects of the nonlinear signal production to the boundedness and finite-time blow-up in a flux-limited chemotaxis modelhttps://zbmath.org/1491.350722022-09-13T20:28:31.338867Z"Tu, Xinyu"https://zbmath.org/authors/?q=ai:tu.xinyu"Mu, Chunlai"https://zbmath.org/authors/?q=ai:mu.chunlai"Zheng, Pan"https://zbmath.org/authors/?q=ai:zheng.panBlow-up and global solvability of the Cauchy problem for a pseudohyperbolic equation related to the generalized Boussinesq equationhttps://zbmath.org/1491.350732022-09-13T20:28:31.338867Z"Umarov, Kh. G."https://zbmath.org/authors/?q=ai:umarov.khasan-galsanovichSummary: Under consideration is the Cauchy problem in the space of continuous functions for a nonlinear strictly hyperbolic equation related to the generalized Boussinesq equation. We discuss the conditions for existence of a global classical solution and blow-up of a solution to the Cauchy problem on a finite time segment.Blow-up rate of sign-changing solutions to nonlinear parabolic systems in domainshttps://zbmath.org/1491.350742022-09-13T20:28:31.338867Z"Zhanpeisov, Erbol"https://zbmath.org/authors/?q=ai:zhanpeisov.erbolSummary: We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on a convex domain with Sobolev subcritical nonlinearity. We also obtain a blow-up rate estimate on a nonconvex domain under additional assumptions.On a priori estimates of solutions of the Tricomi problem for a certain mixed-type second-order equationhttps://zbmath.org/1491.350752022-09-13T20:28:31.338867Z"Balkizov, G. A."https://zbmath.org/authors/?q=ai:balkizov.g-aSummary: In this paper, a theorem on a priori estimates of solutions of the Tricomi problem for a second-order, mixed-type equation with the Gellerstedt operator in the hyperbolicity domain is proved. The a priori estimate obtained implies the uniqueness of the regular solution of the problem considered.Viscous Hamilton-Jacobi equations in exponential Orlicz heartshttps://zbmath.org/1491.350762022-09-13T20:28:31.338867Z"Blessing, Jonas"https://zbmath.org/authors/?q=ai:blessing.jonas"Kupper, Michael"https://zbmath.org/authors/?q=ai:kupper.michaelSummary: We provide a semigroup approach to viscous Hamilton-Jacobi equations. It turns out that exponential Orlicz hearts are suitable spaces to handle the (quadratic) non-linearity of the Hamiltonian. Based on an abstract extension result for nonlinear semigroups on spaces of continuous functions, we represent the solution of the viscous Hamilton-Jacobi equation as a strongly continuous convex semigroup on an exponential Orlicz heart. As a result, the solution depends continuously on the initial data. Furthermore, we determine the so-called symmetric Lipschitz set which is invariant under the semigroup. This automatically yields a priori estimates and regularity in Sobolev spaces. In particular, on the domain restricted to the symmetric Lipschitz set, the generator can be explicitly determined and linked with the viscous Hamilton-Jacobi equation.The Bernstein technique for integro-differential equationshttps://zbmath.org/1491.350772022-09-13T20:28:31.338867Z"Cabré, Xavier"https://zbmath.org/authors/?q=ai:cabre.xavier"Dipierro, Serena"https://zbmath.org/authors/?q=ai:dipierro.serena"Valdinoci, Enrico"https://zbmath.org/authors/?q=ai:valdinoci.enricoSummary: We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two -- for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the ``pure'' linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.Boundedness criteria for a class of indirect (and direct) chemotaxis-consumption models in high dimensionshttps://zbmath.org/1491.350782022-09-13T20:28:31.338867Z"Frassu, Silvia"https://zbmath.org/authors/?q=ai:frassu.silvia"Viglialoro, Giuseppe"https://zbmath.org/authors/?q=ai:viglialoro.giuseppeSummary: In a bounded and smooth domain \(\Omega\) of \(\mathbb{R}^n, n \geq 5\), we, mainly, consider for some \(\xi, \chi, \delta\) positive and \(T_{max} \in (0, \infty]\) the zero-flux chemotaxis model with indirect signal absorption
\[
u_t = \xi \Delta u - \chi \nabla \cdot (u \nabla v), v_t = \Delta v - w v, w_t = - \delta w + u, \quad \text{in } \Omega \times (0, T_{max}),
\] equipped with sufficiently regular initial data \(u (x, 0) = u_0 (x) \geq 0\), \(v (x, 0) = v_0 (x) \geq 0\) and \(w (x, 0) = w_0 (x) \geq 0\). We establish the existence of \(\xi^\ast = \xi^\ast (n) > 1\) such that whenever \(\chi \| v_0 \|_{L^\infty (\Omega)}\) obeys certain constraints, functions of \(n\) and \(\xi (0 < \xi < \xi^\ast)\), the initial-boundary value problem has a unique classical solution in \(\Omega \times (0, \infty)\), which is bounded. In the frame of both direct and indirect chemotaxis models, our work (partially) improves and generalizes known results.A note on local smoothing estimates for fractional Schrödinger equationshttps://zbmath.org/1491.350792022-09-13T20:28:31.338867Z"Gan, Shengwen"https://zbmath.org/authors/?q=ai:gan.shengwen"Oh, Changkeun"https://zbmath.org/authors/?q=ai:oh.changkeun"Wu, Shukun"https://zbmath.org/authors/?q=ai:wu.shukunSummary: We improve local smoothing estimates for fractional Schrödinger equations for \(\alpha \in(0, 1) \cup(1, \infty)\).A boundary estimate for singular sub-critical parabolic equationshttps://zbmath.org/1491.350802022-09-13T20:28:31.338867Z"Gianazza, Ugo"https://zbmath.org/authors/?q=ai:gianazza.ugo-pietro"Liao, Naian"https://zbmath.org/authors/?q=ai:liao.naianSummary: We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of \(p\)-Laplacian type, with \(p\) in the sub-critical range \(\big(1,\frac{2N}{N+1}\big]\). The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic \(p\)-capacity.Global regularity estimates for the Boltzmann equation without cut-offhttps://zbmath.org/1491.350812022-09-13T20:28:31.338867Z"Imbert, Cyril"https://zbmath.org/authors/?q=ai:imbert.cyril"Silvestre, Luis Enrique"https://zbmath.org/authors/?q=ai:silvestre.luis-enriqueSummary: We derive \(C^\infty\) a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to pointwise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.On the asymptotic behavior of solutions to the Vlasov-Poisson systemhttps://zbmath.org/1491.350822022-09-13T20:28:31.338867Z"Ionescu, Alexandru D."https://zbmath.org/authors/?q=ai:ionescu.alexandru-d"Pausader, Benoit"https://zbmath.org/authors/?q=ai:pausader.benoit"Wang, Xuecheng"https://zbmath.org/authors/?q=ai:wang.xuecheng"Widmayer, Klaus"https://zbmath.org/authors/?q=ai:widmayer.klausSummary: We prove small data modified scattering for the Vlasov-Poisson system in dimension \(d=3\), using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamics related to the scattering mass.Weighted Sobolev regularity of viscosity solutions for fully nonlinear parabolic equationshttps://zbmath.org/1491.350832022-09-13T20:28:31.338867Z"Lee, Mikyoung"https://zbmath.org/authors/?q=ai:lee.mikyoungSummary: We obtain interior regularity estimates in the weighted Orlicz spaces for viscosity solutions of fully nonlinear uniformly parabolic equations \[ u_t - F(D^2 u, x, t) = f(x, t) \text{ in } Q_1\] under relaxed structure conditions on the nonlinear operator \(F\).Estimates of solutions of elliptic differential-difference equations with degenerationhttps://zbmath.org/1491.350842022-09-13T20:28:31.338867Z"Popov, V. A."https://zbmath.org/authors/?q=ai:popov.vyacheslav-aleksandrovich|popov.vladimir-a|popov.vladimir-a.1Summary: We consider a second-order differential-difference equation in a bounded domain \(Q \subset \mathbb{R}^n\). We assume that the differential-difference operator contains difference operators with degenerations corresponding to the differential operators. Also, it is assumed that the considered differential-difference operator cannot be expressed by a composition of a difference operator and a strongly elliptic differential operator. The presence of degenerate difference operators does not allow us to obtain the Gårding inequality. We prove a priori estimates implying that the considered differential-difference operator is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space \({W}_2^1(Q)\). However, using the obtained estimates, we prove smoothness of solutions at least in subdomains \(Q_r\) generated by translations of the boundary, where \(\underset{r}{\cup}\overline{Q_r}=\overline{Q}\).Hölder gradient estimates on \(L^p\)-viscosity solutions of fully nonlinear parabolic equations with VMO coefficientshttps://zbmath.org/1491.350852022-09-13T20:28:31.338867Z"Tateyama, Shota"https://zbmath.org/authors/?q=ai:tateyama.shotaThe author proves interior \(C^{1, \alpha}\) regularity for \(L^p\)-viscosity solutions of a class of fully nonlinear uniformly parabolic equations with coefficients in VMO class under appropriate structure assumptions on the operator and the condition that \(p>n+2\).
Reviewer: Qing Liu (Okinawa)Boundary regularity estimates in Hölder spaces with variable exponenthttps://zbmath.org/1491.350862022-09-13T20:28:31.338867Z"Vita, Stefano"https://zbmath.org/authors/?q=ai:vita.stefanoSummary: We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in Hölder spaces with variable exponent. The procedure allows to extend the estimates up to a portion of the boundary where Dirichlet or Neumann boundary conditions are prescribed and produces a Schauder theory for partial derivatives of solutions of any order \(k\in\mathbb{N}\). The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The given data of the problem are taken in Hölder and Lebesgue spaces, both with variable exponent.Maximum principle for time-fractional parabolic equations with a reaction coefficient of arbitrary signhttps://zbmath.org/1491.350872022-09-13T20:28:31.338867Z"Kopteva, Natalia"https://zbmath.org/authors/?q=ai:kopteva.natalia|kopteva.natalia.1Summary: We consider time-fractional parabolic equations with a Caputo time derivative of order \(\alpha \in (0, 1)\). For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.Heat equations defined by self-similar measures with overlapshttps://zbmath.org/1491.350882022-09-13T20:28:31.338867Z"Tang, Wei"https://zbmath.org/authors/?q=ai:tang.wei"Ngai, Sze-Man"https://zbmath.org/authors/?q=ai:ngai.sze-manA Liouville-type theorem for the Lane-Emden equation in a half-spacehttps://zbmath.org/1491.350892022-09-13T20:28:31.338867Z"Dupaigne, Louis"https://zbmath.org/authors/?q=ai:dupaigne.louis"Sirakov, Boyan"https://zbmath.org/authors/?q=ai:sirakov.boyan-slavchev"Souplet, Philippe"https://zbmath.org/authors/?q=ai:souplet.philippeSummary: We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution that is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution that is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.Carleman estimates and unique continuation property for \(n\)-dimensional Benjamin-Bona-Mahony equationshttps://zbmath.org/1491.350902022-09-13T20:28:31.338867Z"Esfahani, Amin"https://zbmath.org/authors/?q=ai:esfahani.amin"Mammeri, Youcef"https://zbmath.org/authors/?q=ai:mammeri.youcefSummary: We study the unique continuation property for the \(N\)-dimensional BBM equations using Carleman estimates. We prove that if the solution of this equation vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.Unique continuation from a crack's tip under Neumann boundary conditionshttps://zbmath.org/1491.350912022-09-13T20:28:31.338867Z"Felli, Veronica"https://zbmath.org/authors/?q=ai:felli.veronica"Siclari, Giovanni"https://zbmath.org/authors/?q=ai:siclari.giovanniSummary: We derive local asymptotics of solutions to second order elliptic equations at the edge of a \((N - 1)\)-dimensional crack, with homogeneous Neumann boundary conditions prescribed on both sides of the crack. A combination of blow-up analysis and monotonicity arguments provides a classification of all possible asymptotic homogeneities of solutions at the crack's tip, together with a strong unique continuation principle.A quantitative strong unique continuation property of a diffusive SIS modelhttps://zbmath.org/1491.350922022-09-13T20:28:31.338867Z"Wang, Taige"https://zbmath.org/authors/?q=ai:wang.taige"Xu, Dihong"https://zbmath.org/authors/?q=ai:xu.dihongSummary: This article is concerned with a strong unique continuation property of solutions for a diffusive SIS (Susceptible -- Infected -- Susceptible, or SI) model, which belongs to a type of observability inequalities in a time interval \([0,T]\). That is, if one can observe solution on a convex and connected bounded open set \(\omega\) in a bounded domain \(\Omega\) at time \(t=T\), then the norms of solution on \([0,T)\) on \(\Omega\) are observable. In our discussion, boundary condition is a homogeneous Dirichlet one (hostile boundary condition).Natural second-order regularity for parabolic systems with operators having \((p, \delta)\)-structure and depending only on the symmetric gradienthttps://zbmath.org/1491.350932022-09-13T20:28:31.338867Z"Berselli, Luigi C."https://zbmath.org/authors/?q=ai:berselli.luigi-carlo"Růžička, Michael"https://zbmath.org/authors/?q=ai:ruzicka.michaelSummary: In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with \((p, \delta)\)-structure, for all \(p>1\) and for all \(\delta >0\). In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting in the case \(p>2\). However, the same method also yields regularity in the elliptic case and for \(1<p\le 2\), thus proving in a different way results already known.Regularity criterion for the 3D Boussinesq equations with partial viscous terms in the limiting caseshttps://zbmath.org/1491.350942022-09-13T20:28:31.338867Z"Chen, Qionglei"https://zbmath.org/authors/?q=ai:chen.qionglei"Li, Zhen"https://zbmath.org/authors/?q=ai:li.zhenSummary: By means of Fourier analysis and the structure of the equation, for the 3D Boussinesq system with partial viscosity, we show the blow-up criterion of the smooth solution via a Beale-Kato-Majda type of \(\sup_{j \in \mathbb{Z}} \int_0^T \| \dot{\Delta}_j \nabla \times u \|_\infty d t\) and a Serrin type of \(\| u \|_{C ([0, T), B_{\infty, \infty}^{- 1})}\) or especially its oscillation in the \(B_{\infty, \infty}^{- 1}\) norm with small amplitude.Improved regularity criteria for the MHD equations in terms of pressure using an Orlicz normhttps://zbmath.org/1491.350952022-09-13T20:28:31.338867Z"Choe, Hi Jun"https://zbmath.org/authors/?q=ai:choe.hi-jun"Neustupa, Jiří"https://zbmath.org/authors/?q=ai:neustupa.jiri"Yang, Minsuk"https://zbmath.org/authors/?q=ai:yang.minsukSummary: We present new regularity criteria in terms of the negative part of the pressure \(p\) or the positive part of the extended Bernoulli pressure \(\mathcal{B} := p + \frac{ 1}{ 2} | \mathbf{u} |^2 + \frac{ 1}{ 2} | \mathbf{b} |^2\), where \(\mathbf{u}\) is the velocity, and \(\mathbf{b}\) is the magnetic field. The criteria extend the previously known results, and the extension is enabled by the use of an appropriate Orlicz norm.Large-scale regularity of nearly incompressible elasticity in stochastic homogenizationhttps://zbmath.org/1491.350962022-09-13T20:28:31.338867Z"Gu, Shu"https://zbmath.org/authors/?q=ai:gu.shu"Zhuge, Jinping"https://zbmath.org/authors/?q=ai:zhuge.jinpingThe authors consider a bounded and Lipschitz domain \(D\subset \mathbb{R}^{d}\) occupied by a non-homogeneous, anisotropic and nearly incompressible material, thus leading to the system \(\nabla \cdot (A(x)\nabla u)+\nabla (\lambda (x)\nabla \cdot u)=F\) in \(D\), where \(A\) is a tensor-valued function which satisfies the usual ellipticity, continuity and symmetry properties, and \( \lambda \) is a scalar function which satisfies the compressibility condition \(\lambda _{0}\leq \lambda (x)\leq \lambda _{0}+\Lambda \), with \(\lambda _{0}\geq 0\).\ Dirichlet boundary conditions are imposed \(u=f\) on \(\partial D\) . Introducing \(\lambda (x)=\lambda _{0}+b(x)\), this system may be written as \(\nabla \cdot (A(x)\nabla u_{\lambda })+\lambda \nabla (\nabla \cdot u_{\lambda })=F\) in \(D\), \(u_{\lambda }=f\) on \(\partial D\). The authors will consider the problem in a homogenization context \(\nabla \cdot (A^{\varepsilon }(x)\nabla u_{\lambda }^{\varepsilon })+\nabla (\lambda ^{\varepsilon }\nabla \cdot u_{\lambda }^{\varepsilon })=0\) in \(D\), with \( A^{\varepsilon }(x)=A(x/\varepsilon )\) and \(\lambda ^{\varepsilon }(x)=\lambda (x/\varepsilon )\). They consider the \(\sigma \)-algebra \( \mathcal{F}_{D}\) generated by the random elements \((A,\lambda )\rightarrow (\int_{\mathbb{R}^{d}}a_{ij}^{\alpha \beta }(x)\phi (x),\int_{\mathbb{R} ^{d}}\lambda (x)\psi (x))\) with \(\phi ,\psi \in C_{0}^{\infty }(D)\), and a probability measure \(\mathbb{P}\) which satisfies a stationarity property with respect to \(\mathbb{Z}^{d}\)-translations and a unit range of dependence property, that is \(\mathcal{F}_{D}\) and \(\mathcal{F}_{E}\) are \(\mathbb{P}\)-independent for every Borel subsets \(D,E\subset \mathbb{R}^{d}\) satisfying \( dist(D,E)\geq 1\).
The first main result of the paper proves that for any \( s\in (0,d)\) and \(\lambda _{0}\in \lbrack 0,\infty )\), there exist a constant \( C_{0}\), which only depends on \(s,d,\Lambda \) and a random variable \(\mathcal{X }=\mathcal{X}_{s,\lambda }:\Omega \rightarrow \lbrack 1,\infty )\) satisfying \(\mathbb{E}[\exp ((\mathcal{X}/\theta )^{s})]\leq 2\), for some \(s>0\), such that if \(u_{\lambda }^{\varepsilon }\in H^{1}(B_{2}(0);\mathbb{R}^{d})\) is a weak solution to the last preceding problem in \(B_{2}(0)\) then for every \( r\in \lbrack \varepsilon \mathcal{X},1]\), the interior estimate
\[
\left( \frac{1}{\left\vert B_{r}\right\vert }\int_{B_{r}}\left\vert \nabla u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2}+\left( \frac{1}{ \left\vert B_{r}\right\vert }\int_{B_{r}}\left\vert \lambda ^{\varepsilon }\nabla \cdot u_{\lambda }^{\varepsilon }-\frac{1}{\left\vert B_{2}\right\vert }\int_{B_{2}}\lambda ^{\varepsilon }\nabla \cdot u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2}\leq C\left( \frac{1}{ \left\vert B_{2}\right\vert }\int_{B_{2}}\left\vert \nabla u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2}
\]
holds true. The second main result proves the boundary estimate
\[
\left( \frac{1}{\left\vert D_{r}\right\vert }\int_{D_{r}}\left\vert \nabla u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2}+\left( \frac{1}{\left\vert D_{r}\right\vert } \int_{D_{r}}\left\vert \lambda ^{\varepsilon }\nabla \cdot u_{\lambda }^{\varepsilon }-\frac{1}{\left\vert D_{2}\right\vert }\int_{D_{2}}\lambda ^{\varepsilon }\nabla \cdot u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2}\leq C\left( \frac{1}{\left\vert D_{2}\right\vert } \int_{D_{2}}\left\vert \nabla u_{\lambda }^{\varepsilon }\right\vert ^{2}\right) ^{1/2},
\]
where \(D_{r}=D\cap B_{r}\). Here, the authors assume that the domain \(D\) satisfies a \(\varepsilon \)-scale \(C^{1,\alpha }\) condition at 0.
For the proof, the authors first recall the homogenized problem associated with the last preceding problem, and the rate of convergence of the solutions. They recall the existence of a unique weak solution to this problem and a Caccioppoli inequality for this weak solution. They introduce the Stokes problem and recall Meyers' inequality for a weak solution to this Stokes problem and to the problem under consideration. Finally, they prove energy decay estimates using the above ingredients and an iteration process.
Reviewer: Alain Brillard (Riedisheim)Influence of the free parameters and obtained wave solutions from CBS equationhttps://zbmath.org/1491.350972022-09-13T20:28:31.338867Z"Arafat, S. M. Yiasir"https://zbmath.org/authors/?q=ai:arafat.s-m-yiasir"Islam, S. M. Rayhanul"https://zbmath.org/authors/?q=ai:islam.s-m-rayhanul"Bashar, Md Habibul"https://zbmath.org/authors/?q=ai:bashar.md-habibul(no abstract)Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new resultshttps://zbmath.org/1491.350982022-09-13T20:28:31.338867Z"Cherniha, Roman"https://zbmath.org/authors/?q=ai:cherniha.roman-m"Davydovych, Vasyl'"https://zbmath.org/authors/?q=ai:davydovych.vasylSummary: This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.On fractional Fitzhugh-Nagumo equation as a transmission of nerve impulses designhttps://zbmath.org/1491.350992022-09-13T20:28:31.338867Z"Karaman, Bahar"https://zbmath.org/authors/?q=ai:karaman.bahar(no abstract)Exact time-dependent solutions of a Fisher-KPP-like equation obtained with nonclassical symmetry analysishttps://zbmath.org/1491.351002022-09-13T20:28:31.338867Z"McCue, Scott W."https://zbmath.org/authors/?q=ai:mccue.scott-william"Bradshaw-Hajek, Bronwyn H."https://zbmath.org/authors/?q=ai:bradshaw-hajek.bronwyn-h"Simpson, Matthew J."https://zbmath.org/authors/?q=ai:simpson.matthew-jSummary: We consider a family of exact solutions to a nonlinear reaction-diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher-KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher-KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.On the analytical treatment for the fractional-order coupled partial differential equations via fixed point formulation and generalized fractional derivative operatorshttps://zbmath.org/1491.351012022-09-13T20:28:31.338867Z"Rashid, Saima"https://zbmath.org/authors/?q=ai:rashid.saima"Sultana, Sobia"https://zbmath.org/authors/?q=ai:sultana.sobia"Idrees, Nazeran"https://zbmath.org/authors/?q=ai:idrees.nazeran"Bonyah, Ebenezer"https://zbmath.org/authors/?q=ai:bonyah.ebenezer(no abstract)Extended homogeneous balance conditions in the sub-equation methodhttps://zbmath.org/1491.351022022-09-13T20:28:31.338867Z"Song, Chenwei"https://zbmath.org/authors/?q=ai:song.chenwei"Liu, Yinping"https://zbmath.org/authors/?q=ai:liu.yinpingSummary: The sub-equation method is a kind of straightforward algebraic method to construct exact solutions of nonlinear evolution equations. In this paper, the sub-equation method is improved by proposing some extended homogeneous balance conditions. By applying them to several examples, it can be seen that new solutions could indeed be obtained.Solving fractional-order diffusion equations in a plasma and fluids via a novel transformhttps://zbmath.org/1491.351032022-09-13T20:28:31.338867Z"Sunthrayuth, Pongsakorn"https://zbmath.org/authors/?q=ai:sunthrayuth.pongsakorn"Alyousef, Haifa A."https://zbmath.org/authors/?q=ai:alyousef.haifa-a"El-Tantawy, S. A."https://zbmath.org/authors/?q=ai:el-tantawy.s-a"Khan, Adnan"https://zbmath.org/authors/?q=ai:khan.adnan-qadir|khan.adnan-a"Wyal, Noorolhuda"https://zbmath.org/authors/?q=ai:wyal.noorolhuda(no abstract)Exact solutions of a nonlinear diffusion equation on polynomial invariant subspace of maximal dimensionhttps://zbmath.org/1491.351042022-09-13T20:28:31.338867Z"Svirshchevskii, S. R."https://zbmath.org/authors/?q=ai:svirshchevskii.sergey-rSummary: The nonlinear diffusion equation \(u_t=(u^{-4/3}u_x)_x\) is reduced by the substitution \(u=v^{-3/4}\) to an equation with quadratic nonlinearities possessing a polynomial invariant linear subspace of the maximal possible dimension equal to five. The dynamics of the solutions on this subspace is described by a fifth-order nonlinear dynamical system (V.A. Galaktionov).
We found that, on differentiation, this system reduces to a single linear equation of the second order, which is a special case of the Lamé equation, and that the general solution of this linear equation is expressed in terms of the Weierstrass \(\wp\)-function and its derivative. As a result, all exact solutions \(v(x,t)\) on a five-dimensional polynomial invariant subspace, as well as the corresponding solutions \(u(x,t)\) of the original equation, are constructed explicitly.
Using invariance condition, two families of non-invariant solutions are singled out. For one of these families, all types of solutions are considered in detail. Some of them describe peculiar blow-up regimes, while others fade out in finite time.New exact and solitary wave solutions of nonlinear Schamel-KdV equationhttps://zbmath.org/1491.351052022-09-13T20:28:31.338867Z"Tariq, Kalim U."https://zbmath.org/authors/?q=ai:tariq.kalim-u"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Zubair, Muhammad"https://zbmath.org/authors/?q=ai:zubair.muhammad"Osman, Mohamed S."https://zbmath.org/authors/?q=ai:osman.mohamed-sayed-ali"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre(no abstract)Abundant exact traveling wave solutions to the local fractional \((3+1)\)-dimensional Boiti-Leon-Manna-Pempinelli equationhttps://zbmath.org/1491.351062022-09-13T20:28:31.338867Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Wang, Guo-Dong"https://zbmath.org/authors/?q=ai:wang.guodong"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.fengOptimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaceshttps://zbmath.org/1491.351072022-09-13T20:28:31.338867Z"Chikami, Noboru"https://zbmath.org/authors/?q=ai:chikami.noboru"Ikeda, Masahiro"https://zbmath.org/authors/?q=ai:ikeda.masahiro"Taniguchi, Koichi"https://zbmath.org/authors/?q=ai:taniguchi.koichiSummary: The Cauchy problem for the Hardy-Hénon parabolic equation is studied in the critical and subcritical weighted Lebesgue spaces on the Euclidean space \(\mathbb{R}^d\). In earlier works, the well-posedness of singular initial data and the existence of non-radial forward self-similar solutions to the problem were shown for the Hardy and Fujita cases \(( \gamma \leq 0)\). The weighted spaces are used to treat the potential \(| x |^\gamma\) as an increase or decrease in the weight, which enables us to prove the well-posedness of the problem for all \(\gamma \), with \(- \min \{2, d\} < \gamma \), including the Hénon case \(( \gamma > 0)\). As a by-product of global existence, self-similar solutions to the problem are established for all \(\gamma\) without restrictions. Furthermore, the non-existence of a local solution for supercritical data is also shown. Therefore, our critical exponent, \( s_c\) is optimal with regard to solvability.Reliable analysis for the Drinfeld-Sokolov-Wilson equation in mathematical physicshttps://zbmath.org/1491.351082022-09-13T20:28:31.338867Z"Alam, Md Nur"https://zbmath.org/authors/?q=ai:alam.md-nur"Bonyah, Ebenezer"https://zbmath.org/authors/?q=ai:bonyah.ebenezer"Fayz-Al-Asad, Md."https://zbmath.org/authors/?q=ai:fayz-al-asad.mdSummary: The present paper studies the Drinfeld-Sokolov-Wilson (DSW) equation. We per-form the \(S(\xi)\)-expansion method to take some exact solutions and create different solitary wave aspects for each equation. The received perspectives provide the firm mathematical foundation as well as describe the wave generation in soliton physics. As a result, we get some new soliton solutions. Finally, the exact solution and its geometrical properties are constructed, considering Mean curvature and Gaussian curvature as for the DSW equation. The \(S(\xi)\)-expansion method analyzes the solution follow through instantaneously with this equation.Diffusion-convection reaction equations with sign-changing diffusivity and bistable reaction termhttps://zbmath.org/1491.351092022-09-13T20:28:31.338867Z"Berti, Diego"https://zbmath.org/authors/?q=ai:berti.diego"Corli, Andrea"https://zbmath.org/authors/?q=ai:corli.andrea"Malaguti, Luisa"https://zbmath.org/authors/?q=ai:malaguti.luisaSummary: We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.Wavefront solutions for a class of nonlinear highly degenerate parabolic equationshttps://zbmath.org/1491.351102022-09-13T20:28:31.338867Z"Cantarini, Marco"https://zbmath.org/authors/?q=ai:cantarini.marco"Marcelli, Cristina"https://zbmath.org/authors/?q=ai:marcelli.cristina"Papalini, Francesca"https://zbmath.org/authors/?q=ai:papalini.francescaSummary: We consider the following nonlinear parabolic equation
\[
(\mathcal{F} (v))_x + (\mathcal{G} (v))_\tau = (\mathcal{D} (v))_{x x} + \rho(v), \quad v \in [\alpha, \beta]
\]
where \(\mathcal{F}\), \(\mathcal{G}\) are generic \(C^1\)-functions in \([\alpha, \beta]\), \(\mathcal{D} \in C^1 [\alpha, \beta] \cap C^2(\alpha, \beta)\) is positive inside \((\alpha, \beta)\) (possibly vanishing at the extreme points), and finally \(\rho\) is a monostable reaction term.
We investigate the existence and the properties of traveling wave solutions for such an equation and provide their classification between classical and sharp solutions, together with an estimate of the minimal wave speed.Application of the rational \((G^\prime/G)\)-expansion method for solving some coupled and combined wave equationshttps://zbmath.org/1491.351112022-09-13T20:28:31.338867Z"Ekici, Mustafa"https://zbmath.org/authors/?q=ai:ekici.mustafa"Ünal, Metin"https://zbmath.org/authors/?q=ai:unal.metinSummary: In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using the recently established rational \((G^\prime/G)\)-expansion method. We apply this method to the combined KdV-mKdV equation, the reaction-diffusion equation and the coupled Hirota-Satsuma KdV equations. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are also derived from the travelling waves. We have also given some figures for the solutions.Monostable pulled fronts and logarithmic driftshttps://zbmath.org/1491.351122022-09-13T20:28:31.338867Z"Giletti, Thomas"https://zbmath.org/authors/?q=ai:giletti.thomasThis paper is concerned with the issue of logarithmic drifts in the position of the level sets of solutions of monostable reaction-diffusion equations, with respect to the traveling front with minimal speed. It is known that the drift phenomenon disappears when the minimal front speed is nonlinearly determined, and a logarithmic drift occurs when the minimal front speed is linearly determined and the reaction is of KPP type. It is an open problem that whether logarithmic drifts exist when the minimal front speed is linearly determined and the reaction is not of KPP type. The author of this paper proved that a logarithmic drift still appears, but may involve a different factor.
The author first considered a special case when the front has fast decay under the additional assumption that the reaction is linear around zero. A lower estimate on the position of level sets is provided. The result shows that the drift cannot be the same as in the KPP case. Then the author studied the general pulled case. The super- and sub-solutions are constructed in the two cases that the fronts decay fast and slow. Based on the super- and sub-solutions, by the elementary comparison argument, the author obtained that a logarithmic drift still appears.
Reviewer: Guobao Zhang (Lanzhou)Traveling curved fronts in the buffered bistable systems in \(\mathbb{R}^2\)https://zbmath.org/1491.351132022-09-13T20:28:31.338867Z"Jia, Fu-Jie"https://zbmath.org/authors/?q=ai:jia.fu-jie"Wang, Xiaohui"https://zbmath.org/authors/?q=ai:wang.xiaohui"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2|wang.zhi-cheng.1Summary: We consider traveling curved fronts in the buffered bistable systems in \(\mathbb{R}^2\) and show that multiple stationary buffers (where buffers do not diffuse) cannot prevent the existence of V-shaped calcium concentration waves. In other words, for the buffered bistable systems we prove that there exist V-shaped traveling fronts in \(\mathbb{R}^2\) by constructing the proper supersolution and subsolution, applying the comparison principle and the fixed point theory.Traveling wave solutions in predator-prey models with competitionhttps://zbmath.org/1491.351142022-09-13T20:28:31.338867Z"Lin, Guo"https://zbmath.org/authors/?q=ai:lin.guo"Xing, Yibing"https://zbmath.org/authors/?q=ai:xing.yibingTraveling waves for a nonlocal dispersal predator-prey model with two preys and one predatorhttps://zbmath.org/1491.351152022-09-13T20:28:31.338867Z"Zhao, Xu-Dong"https://zbmath.org/authors/?q=ai:zhao.xudong"Yang, Fei-Ying"https://zbmath.org/authors/?q=ai:yang.feiying"Li, Wan-Tong"https://zbmath.org/authors/?q=ai:li.wan-tongSummary: This paper is concerned with a nonlocal dispersal predator-prey model with two preys and one predator. The invasion process of a predator into the habitat of preys is considered, which is characterized by the spreading speed of one predator as well as the minimal wave speed of traveling waves connecting the predator-free state to the coexistence state. It should be pointed out that we need to overcome the difficulties brought by nonlocal dispersive operator and the noncooperation of system itself to get the final state of traveling waves.Soliton solution of some nonlinear PDEs and its applicationshttps://zbmath.org/1491.351162022-09-13T20:28:31.338867Z"Akuamoah, Saviour Worlanyo"https://zbmath.org/authors/?q=ai:akuamoah.saviour-worlanyo"Ayimah, John Coker"https://zbmath.org/authors/?q=ai:ayimah.john-coker"Mahama, Francois"https://zbmath.org/authors/?q=ai:mahama.francois"Bonsi, Prosper Obed"https://zbmath.org/authors/?q=ai:bonsi.prosper-obed(no abstract)On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equationshttps://zbmath.org/1491.351172022-09-13T20:28:31.338867Z"Côte, Raphaël"https://zbmath.org/authors/?q=ai:cote.raphael"Friederich, Xavier"https://zbmath.org/authors/?q=ai:friederich.xavierThis article considers the so-called multi-soliton solutions of nonlinear Schrödinger equations of the form
\[
\partial_{t} u=i(\Delta u+f(|u|^{2}) u)
\]
posed for \((t,x)\in I \times \mathbb{R}^d \), with \(I\subset \mathbb{R}\) being a time interval, and for nonlinearities \(f\colon [0,+\infty) \to \mathbb{R}\) that are \(H^1\)-subcritical. The soliton solutions in this work refer to traveling solitary wave solutions \(R(t,x)\) (not expected to possess the collision properties of solitons admitted by the integrable case \(f(r)=r\), \(d=1\)) constructed from ground-states
\[
R(t,x) = Q_\omega(x) e^{i w t},
\]
where \(\omega >0\) is given and \(Q_{\omega} \in H^{1}(\mathbb{R}^{d})\) is a non-vanishing positive radial solution of the elliptic problem
\[
\Delta Q_{\omega}+f(Q_{\omega}^{2}) Q_{\omega}=\omega Q_{\omega}.
\]
A multi-soliton is a solution \(u\) of (NLS) defined for \(t\in [T_{0},+\infty)\) for some \(T_{0} \in \mathbb{R}\) satisfying
\[
\lim _{t \rightarrow+\infty}\|u(t)-R(t)\|_{H^{1}}=0,
\]
where \(R(t,x)\) is the superposition
\[
R(t,x):=\sum_{k=1}^{K} R_{k}(t,x),\qquad K\in\mathbb{N}\setminus \{0,1\},
\]
of solitons
\[
R_{k}(t, x)=Q_{\omega_{k}}(x-x_{k}^{0}-v_{k} t) e^{i\big(\frac{1}{2} v_{k} \cdot x+(\omega_{k}-\frac{|v_k|^{2}}{4}) t+\gamma_{k}\big)}
\]
moving along the lines \(x=x_{k}^{0}+v_{k} t\) with distinct speeds \(v_k\).
Existence and study of multi-soliton solutions for (NLS) in the pure-power-nonlinearity case, that is, when \(f(r) = r^{\frac{p-1}{2}}\), \(r\geq 0\), where \(1<p<1+\frac{4}{(d-2)}\) if \(d\geq 3\) and \(p>1\) if \(d=1,2\),
date back to the works [\textit{F. Merle}, Commun. Math. Phys. 129, No. 2, 223--240 (1990; Zbl 0707.35021)], [\textit{Y. Martel} and \textit{F. Merle}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 849--864 (2006; Zbl 1133.35093)
], and [\textit{R. Côte} et al., Rev. Mat. Iberoam. 27, No. 1, 273--302 (2011; Zbl 1273.35234)
]. It is known that (see Theorem 1.1 of this article) for \(K\in\mathbb{N}_{>1}\) there exist a constant \(\theta=\theta(v_{k}, \omega_{k})>0\), \(1\leq k \leq K\), a time \(T_0\geq 0\), and a solution \(u\in \mathscr{C}([T_{0},+\infty), H^{1}(\mathbb{R}^{d}))\) of (NLS) such that for all \(t\geq T_0\),
\[
\|u(t)-R(t)\|_{H^{1}} \leq e^{-2 \theta t}.
\]
The main focus of the article is to investigate whether the multi-soliton solutions are in fact smoother, i.e., whether \(u\in \mathscr{C}([T_{0},+\infty), H^{s}(\mathbb{R}^{d}))\) for \(s>1\) and whether \(\|u(t)-R(t)\|_{H^{s}} \rightarrow 0\) as \(t \rightarrow+\infty\) for \(s>1\).
One of the main results (Theorem 1.2) of this work establishes the regularity property mentioned above in the pure-power-nonlinearity case.
Theorem 1.2 completes the foundational result stated above in the following way: Fix \(p\geq 3\), let \(\theta=\theta(v_{k}, \omega_{k})>0\) as in the result above, and let \(s_{0}:=\lfloor p-1\rfloor \geq 2\), or \(s_{0}:=+\infty\) if \(p\) is an odd integer. The authors prove that there exist a time \(T_{1}>0\) and a solution \(u \in \mathscr{C}\left([T_{1},+\infty), H^{s_{0}}(\mathbb{R}^{d})\right)\) of (NLS) such that for all non-negative integer \(s \leq s_{0}\), there exists \(C_{s} \geq 1\) such that for all \(t\geq T_1\), the solution \(u\) satisfies
\[
\|u(t)-R(t)\|_{H^{s}} \leq C_{s} e^{-\frac{2 \theta}{s+1} t}.
\]
Note that the exponential decay rate depends on the regularity parameter \(s>s_0\) and it becomes smaller as \(s\) becomes larger. The authors show that dependence is gone if \(p\) is an odd integer, in which case
\[
\|u(t)-R(t)\|_{H^{s}} \leq \sqrt{C_{s}} e^{-\theta t}
\]
holds for all integers \(s\geq 0\). It is also important to note that this result is limited to the dimensions \(d\leq 3\) due to the assumption that the nonlinearity is both \(H^1\)-subcritical and is of pure-power-type.
The authors also establish a uniqueness result for multi-solitons for dimensions \(d\leq 2\) in the pure-power-nonlinearity case with \(3 \leq p \leq 1+\frac{4}{d}\) (see Theorem 1.4). Under these assumptions, they show that there exists a unique solution \(u \in \mathscr{C}\left([T_{1},+\infty), H^{1}(\mathbb{R}^{d})\right)\) of (NLS) satisfying
\[
\|u(t)-R(t)\|_{H^{1}}=\mathrm{O}\left(\frac{1}{t^{N}}\right),\quad t \rightarrow+\infty
\]
for some \(N\in\mathbb{N}\) sufficiently large. This result in particular implies that the multi-solitons established in Theorem 1.2 are the same solutions as those in the earlier \(H^1\)-result mentioned above in appropriate dimensions.
Extensions of these results to general nonlinearities are also obtained. Doing that of course requires assumptions on the nonlinearity \(f\). The underlying assumptions include those that imply Cauchy problem to be well-posed in \(H^1(\mathbb{R}^d)\) to begin with, and two mutually exclusive coercivity-type assumptions on the linearization of the energy around a ground state. The latter set of assumptions are related to the stability of the ground state.
Reviewer: Deniz Bilman (Cincinnati)Dark soliton solutions for a variable coefficient higher-order Schrödinger equation in the dispersion decreasing fibershttps://zbmath.org/1491.351182022-09-13T20:28:31.338867Z"Zhao, Xue-Hui"https://zbmath.org/authors/?q=ai:zhao.xue-hui"Li, Shuxia"https://zbmath.org/authors/?q=ai:li.shuxiaSummary: Under investigation in this paper is a variable coefficient higher-order Schrödinger equation with the effects of third-order dispersion, self-steepening and stimulated Raman scattering, which describes the propagation of ultrashort optical pulses in the dispersion decreasing fibers. Via the symbolic computation and Hirota method, the bilinear forms, dark one- and two-soliton solutions are derived. Propagation and interaction for the dark solitons are illustrated graphically: the shape of the dark solitons is affected by the third-order dispersion, self-steepening and stimulated Raman scattering, while the amplitude and velocity of those are influenced by the gain coefficient \(\Gamma \). With the increase of the parameter \(\Gamma \), the amplitude of the dark soliton pulse increases, and the velocity of that changes exponentially in the same propagation distance. For the elastic interaction, the anti-bell- and two parabolic-shape dark solitons are discussed respectively, where the amplitudes and shapes remain unchanged after interaction except for certain phase shifts.Automatic pre- and postconditions for partial differential equationshttps://zbmath.org/1491.351192022-09-13T20:28:31.338867Z"Boreale, Michele"https://zbmath.org/authors/?q=ai:boreale.micheleSummary: Based on an automata-theoretic and algebraic framework, we study equational reasoning for Initial Value Problems (\textsc{ivps}) of polynomial Partial Differential Equations (\textsc{pdes}). We first characterize the solutions of a \textsc{pde} system \(\Sigma\) in terms of the final morphism from a coalgebra induced by \(\Sigma\) to the coalgebra of formal power series (\textsc{fps}). \textsc{fps} solutions conservatively extend the classical analytic ones. To express \textsc{ivps} in their general form, we then introduce \textit{stratified} systems, where the specification of a function can be decomposed into distinct subsystems of \textsc{pdes}. We lift the existence and uniqueness result of \textsc{fps} solutions to stratified systems. We then give a relatively complete algorithm to compute weakest preconditions and strongest postconditions for such systems. To some extent, this result reduces equational reasoning on \textsc{pde} initial value problems to algebraic reasoning. We illustrate some experiments conducted with a proof-of-concept implementation of the method.Dirichlet problem of Poisson equations on a type of higher-dimensional fractal setshttps://zbmath.org/1491.351202022-09-13T20:28:31.338867Z"Zhu, Le"https://zbmath.org/authors/?q=ai:zhu.le"Wu, Yipeng"https://zbmath.org/authors/?q=ai:wu.yipeng"Chen, Zhilong"https://zbmath.org/authors/?q=ai:chen.zhi-long"Yao, Kui"https://zbmath.org/authors/?q=ai:yao.kui"Huang, Shuai"https://zbmath.org/authors/?q=ai:huang.shuai"Wang, Yuan"https://zbmath.org/authors/?q=ai:wang.yuan.2|wang.yuan.1|wang.yuan|wang.yuan.3Optimal control for self-organizing target detection model in the 1d casehttps://zbmath.org/1491.351212022-09-13T20:28:31.338867Z"Ryu, Sang-Uk"https://zbmath.org/authors/?q=ai:ryu.sang-ukSummary: This paper is concerned with the optimal control problem associated to the self-organizing target detection model in 1D domains. That is, we show the global existence of weak solution and the existence of optimal control.Rigidity and trace properties of divergence-measure vector fieldshttps://zbmath.org/1491.351222022-09-13T20:28:31.338867Z"Leonardi, Gian Paolo"https://zbmath.org/authors/?q=ai:leonardi.gian-paolo"Saracco, Giorgio"https://zbmath.org/authors/?q=ai:saracco.giorgioSummary: We consider a \(\varphi\)-rigidity property for divergence-free vector fields in the Euclidean \(n\)-space, where \(\varphi(t)\) is a non-negative convex function vanishing only at \(t=0\). We show that this property is always satisfied in dimension \(n=2\), while in higher dimension it requires some further restriction on \(\varphi\). In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e. when \(\varphi(t)=ct^2 )\) in dimension \(n\geq 4\). The validity of the quadratic rigidity, which we prove in dimension \(n=2\), implies the existence of the trace of a divergence-measure vector field \(\xi\) on an \(\mathcal{H}^1 \)-rectifiable set \(S\), as soon as its weak normal trace \([\xi\cdot\nu_S]\) is maximal on \(S\). As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.Time-dependent Hamilton-Jacobi equations on networkshttps://zbmath.org/1491.351232022-09-13T20:28:31.338867Z"Siconolfi, Antonio"https://zbmath.org/authors/?q=ai:siconolfi.antonioSummary: We study well posedness of time-dependent Hamilton-Jacobi equations on a network, coupled with a continuous initial datum and a flux limiter. We show existence and uniqueness of solutions as well as stability properties. The novelty of our approach is that comparison results are proved linking the equation to a suitable semidiscrete problem, bypassing doubling variable method. Further, we do not need special test functions, and perform tests relative to the equations on different arcs separately.On existence and uniqueness of the solution to initial boundary value problem for the first order partial linear systemhttps://zbmath.org/1491.351242022-09-13T20:28:31.338867Z"Gasanov, K. K."https://zbmath.org/authors/?q=ai:hasanov.kazim-k"Guseynova, Kh. T."https://zbmath.org/authors/?q=ai:guseynova.kh-t(no abstract)Variable separation solution for an extended (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equationhttps://zbmath.org/1491.351252022-09-13T20:28:31.338867Z"Li, Lingfei"https://zbmath.org/authors/?q=ai:li.lingfei"Yan, Yongsheng"https://zbmath.org/authors/?q=ai:yan.yongsheng"Xie, Yingying"https://zbmath.org/authors/?q=ai:xie.yingyingSummary: This paper proposes a new variable separation solution for the (3+1)-dimensional nonlinear evolution equation. The new variable separation solution directly gives the analytical form of the solution \(u\) instead of its potential \(u_y\) and renders a distinct way to construct localized excitation. Taking the extended (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation as an example, we test its integrability at first. Then, we analyze the elastic, inelastic head-on collision of two and three folded solitary waves by introducing several suitable multi-valued functions. Specifically, the superimposed structure of folded waves is studied, and plenty of novel patterns have been obtained.Cauchy problem for the BBM equation in \(l^q L^2\)https://zbmath.org/1491.351262022-09-13T20:28:31.338867Z"Wang, Ming"https://zbmath.org/authors/?q=ai:wang.mingSummary: It is shown that the BBM equation is globally well-posed in \(l^q L^2 (\mathbb{R})\) for all \(2 \leq q < \infty\), and locally well-posed in \(l^\infty L^2 (\mathbb{R})\). These imply that the BBM equation is well-posed in larger spaces, and thus improve the results in references.The sharp Gevrey Kotake-Narasimhan theorem with an elementary proofhttps://zbmath.org/1491.351272022-09-13T20:28:31.338867Z"Tartakoff, David S."https://zbmath.org/authors/?q=ai:tartakoff.david-sIn this article, the author is interested in the Gevrey regularity of the Hörmander operators \[P=\sum_{j=1}^{m}X_j^2+X_0+c,\] where the \(X_j\) are real and smooth vector fiels, and where \(c(x)\) is a smooth function, all in the Gevrey class \(G^s\). The operator \(P\) is also assumed to satisfy a subelliptic estimate in an open set \(\Omega_0\): for some \(\varepsilon>0\), there exists a constant \(C\) such that \[\forall v\in C_0^{\infty}(\Omega_0):\Vert v\Vert_{\varepsilon}^2\leq C\left(\vert (Pv,v)\vert+\Vert v\Vert_0^2\right).\]
Under all these assumptions, the author proves that \(G^s(P,\Omega_0)\subset G^{s/\varepsilon}(\Omega_0)\) for all \(s\geq1\), that is the Gevrey growth of derivatives of a vector \(u\) for \(P\) as measured by iterates of \(P\) yields Gevrey regularity for \(u\) in a larger Gevrey class dictated by the size of \(\varepsilon\) in the \textit{a priori} estimate.
Reviewer: Pascal Remy (Carrières-sur-Seine)A remark on Kohn's theorem on sums of squares of complex vector fieldshttps://zbmath.org/1491.351282022-09-13T20:28:31.338867Z"Parmeggiani, Alberto"https://zbmath.org/authors/?q=ai:parmeggiani.albertoSummary: The plan of this paper is to give an alternate proof of Kohn's subelliptic estimate for systems of \(N\) smooth complex vector fields on an open set of \(\mathbb{R}^n\), and to improve it in extending the result to perturbations by a first-order term. A pseudodifferential generalization will also be given.Multiple solutions for semilinear discontinuous variational problems with lack of compactnesshttps://zbmath.org/1491.351292022-09-13T20:28:31.338867Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"dos Santos, Jefferson A."https://zbmath.org/authors/?q=ai:dos-santos.jefferson-aSummary: This article establishes the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity with critical growth by using the Lusternik-Schnirelmann category.Time-harmonic acoustic scattering from locally perturbed half-planeshttps://zbmath.org/1491.351302022-09-13T20:28:31.338867Z"Bao, Gang"https://zbmath.org/authors/?q=ai:bao.gang"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Yin, Tao"https://zbmath.org/authors/?q=ai:yin.taoAn efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equationhttps://zbmath.org/1491.351312022-09-13T20:28:31.338867Z"Berdawood, Karzan"https://zbmath.org/authors/?q=ai:berdawood.karzan"Nachaoui, Abdeljalil"https://zbmath.org/authors/?q=ai:nachaoui.abdeljalil"Saeed, Rostam"https://zbmath.org/authors/?q=ai:saeed.rostam-k"Nachaoui, Mourad"https://zbmath.org/authors/?q=ai:nachaoui.mourad"Aboud, Fatima"https://zbmath.org/authors/?q=ai:aboud.fatimaSummary: Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [\textit{V. A. Kozlov} et al., Comput. Math. Math. Phys. 31, No. 1, 45--2 (1991); translation from Zh. Vychisl. Mat. Mat. Fiz. 31, No. 1, 64--74 (1991; Zbl 0774.65069)]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number \(k\), when the convergence of KMF algorithm's [loc. cit.] is ensured, our algorithm can be used as an acceleration of convergence.
In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.
We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbationhttps://zbmath.org/1491.351322022-09-13T20:28:31.338867Z"Cardone, Giuseppe"https://zbmath.org/authors/?q=ai:cardone.giuseppe"Durante, T."https://zbmath.org/authors/?q=ai:durante.tiziana"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide \(\Pi_l^\varepsilon\) formed by the union of an infinite strip and a narrow box-shaped perturbation of size \(2l\times\varepsilon\), where \(\varepsilon>0\) is a small parameter. We prove the existence of the length parameter \(l_k^\varepsilon=\pi k+O(\varepsilon)\) with any \(k=1,2,3,\dots\) such that the waveguide \(\Pi_{l_k^\varepsilon}^\varepsilon\) supports a trapped mode with an eigenvalue \(\lambda_k^\varepsilon=\pi^2-4\pi^4 l^2\varepsilon^2+O(\varepsilon^3)\) embedded into the continuous spectrum. This eigenvalue is unique in the segment \([0,\pi^2]\), and it is absent in the case \(l\neq l_k^\varepsilon\). The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall \(\partial\Pi_l^\varepsilon\), namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.Nonexistence of positive solutions to \(n\)-th order equations in \(\mathbb{R}^n\)https://zbmath.org/1491.351332022-09-13T20:28:31.338867Z"Dai, Wei"https://zbmath.org/authors/?q=ai:dai.wei.4Summary: In this paper, we are mainly concerned with the following integral equations:
\[
u(x)=C_n\int\limits_{\mathbb{R}^n}\ln\left(\frac{1}{|x-y|}\right)f(y, u(y))d y+\gamma,\quad x\in\mathbb{R}^n,\tag{0.1}
\]
where \(n\geq 2\), \(\gamma\in\mathbb{R}\), \(u\in C(\mathbb{R}^n)\) and \(f(x,u)\) may change signs and satisfies some assumptions. By using the method of scaling spheres developed by the author and \textit{G. Qin} in [``Liouville type theorems for fractional and higher order Hénon-Hardy type equations
via the method of scaling spheres'', Preprint, \url{arXiv:1810.02752}], we first derive nonexistence of positive solutions to the above IEs under some assumptions. Then, based on the equivalence between the above IEs and the following 2D PDEs:
\[
-\Delta u(x)=f(x,u),\quad x\in\mathbb{R}^2,\tag{0.2}
\]
we also obtain nonexistence of positive solutions to the 2D PDEs under some assumptions. One should note that there are no growth conditions on \(u\) and hence \(f(x, u)\) can grow exponentially (or even faster) on \(u\).Inhomogeneous global minimizers to the one-phase free boundary problemhttps://zbmath.org/1491.351342022-09-13T20:28:31.338867Z"De Silva, Daniela"https://zbmath.org/authors/?q=ai:de-silva.daniela"Jerison, David"https://zbmath.org/authors/?q=ai:jerison.david-s"Shahgholian, Henrik"https://zbmath.org/authors/?q=ai:shahgholian.henrikSummary: Given a global 1-homogeneous minimizer \(U_0\) to the Alt-Caffarelli energy functional, with \(sing (F(U_0)) = \{0\}\), we provide a foliation of the half-space \(\mathbb{R}^n \times [0, +\infty)\) with dilations of graphs of global minimizers \(\underline{U} \leq U_0\leq \bar{U}\) with analytic free boundaries at distance 1 from the origin.Spectral stability of the Steklov problemhttps://zbmath.org/1491.351352022-09-13T20:28:31.338867Z"Ferrero, Alberto"https://zbmath.org/authors/?q=ai:ferrero.alberto"Lamberti, Pier Domenico"https://zbmath.org/authors/?q=ai:lamberti.pier-domenicoSummary: This paper investigates the stability properties of the spectrum of the classical Steklov problem under domain perturbation. We find conditions which guarantee the spectral stability and we show their optimality. We emphasize the fact that our spectral stability results also involve convergence of eigenfunctions in a suitable sense according with the definition of connecting system by \textit{G. M. Vainikko} [J. Sov. Math. 15, 675--705 (1981); translation from Itogi Nauki Tekh., Ser. Mat. Anal. 16, 5--53 (1979; Zbl 0582.65046)]. The convergence of eigenfunctions can be expressed in terms of the \(H^1\) strong convergence. The arguments used in our proofs are based on an appropriate definition of compact convergence of the resolvent operators associated with the Steklov problems on varying domains.
In order to show the optimality of our conditions we present alternative assumptions which give rise to a degeneration of the spectrum or to a discontinuity of the spectrum in the sense that the eigenvalues converge to the eigenvalues of a limit problem which does not coincide with the Steklov problem on the limiting domain.New existence results for prescribed mean curvature problem on balls under pinching conditionshttps://zbmath.org/1491.351362022-09-13T20:28:31.338867Z"Fourti, Habib"https://zbmath.org/authors/?q=ai:fourti.habibSummary: We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball \(\mathbb{B}^n\) and on the boundary \(\mathbb{S}^{n - 1}\) the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case \(n \geq 5\), under suitable pinching conditions on the mean curvature function.A fluctuation result for the displacement in the optimal matching problemhttps://zbmath.org/1491.351372022-09-13T20:28:31.338867Z"Goldman, Michael"https://zbmath.org/authors/?q=ai:goldman.michael"Huesmann, Martin"https://zbmath.org/authors/?q=ai:huesmann.martinThe authors investigate the displacement in the optimal matching problem in dimensions two and three, which is esentially given by the solution of the linearized (Poisson) equation. They show that at all mesoscopic scales, this displacement is close to a curl-free Gaussian free field, in some suitable negative Sobolev spaces.
Reviewer: Rodica Luca (Iaşi)Convergence of asymptotic costs for random Euclidean matching problemshttps://zbmath.org/1491.351382022-09-13T20:28:31.338867Z"Goldman, Michael"https://zbmath.org/authors/?q=ai:goldman.michael"Trevisan, Dario"https://zbmath.org/authors/?q=ai:trevisan.darioThe authors prove the existence of the thermodynamic limit for the matching problem of a Poisson point process to reference measure. Then they give the analog theorem for the average minimum cost of a bipartite matching between two Poisson point processes on the unit cube in \(d\) dimensions (\(d\ge 3\)). Stronger convergence results for the corresponding deterministic problem are finally presented.
Reviewer: Rodica Luca (Iaşi)Analysis and computation of the transmission eigenvalues with a conductive boundary conditionhttps://zbmath.org/1491.351392022-09-13T20:28:31.338867Z"Harris, I."https://zbmath.org/authors/?q=ai:harris.isaac"Kleefeld, A."https://zbmath.org/authors/?q=ai:kleefeld.andreasThe authors consider the transmission eigenvalue problem with conductive boundary conditions, where we look for \(k \in \mathbb{C}\) and non-trivial \((w,v) \in L^2(D) \times L^2(D)\) such that
\[\begin{split}& \Delta w +k^2 n w=0, \quad \Delta v + k^2 v=0 \quad\text{in}\quad D, \\
&w-v=0, \quad {\partial_{\nu} w}-{\partial_\nu v}= \eta v \quad \text{on} \quad \partial D,
\end{split}\]
with \(w-v \in H^2(D) \cap H^1_0(D)\) (endowed with \(\| \cdot \| = \|\Delta \cdot \|_{L^2(D)}\)), considering \(D \subset \mathbb{R}^d\) (\(d = 2, 3\)) to be a bounded simply connected open set, with \(\nu\) being the unit outward normal to the boundary \(\partial D\) which is smooth enough so that the well-posedness estimate for the Poisson problem and the \(H^2\) elliptic regularity estimate hold. Here, the refractive index \(n(x)\) is considered as a scalar bounded real-valued function defined in \(D\) and the conductivity parameter \(\eta (x)\) is a scalar bounded real-valued function defined on the boundary \(\partial D\).
This is studied both analytically and numerically. The authors use the variational method to deduce the theoretical results. In particular, it is exhibited that there is a lower bound on the real transmission eigenvalues provided \(\eta\) is strictly positive on \(\partial D\) and \(n-1\) is either uniformly positive or negative in \(D\). In addition, the convergence as \(\eta\) tends to infinity is analysed. Boundary integral equations for various interior transmission eigenvalue problems are derived and used in the included numerical experiments, being possible to compute multiple interior transmission eigenvalue problems with a constant refractive index. Moreover, taking profit from the limiting behavior, numerical estimates of the refractive index are presented.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)A new approach to the complex Helmholtz equation with applications to diffusion wave fields, impedance spectroscopy and unsteady Stokes flowhttps://zbmath.org/1491.351402022-09-13T20:28:31.338867Z"Hauge, Jordan C."https://zbmath.org/authors/?q=ai:hauge.jordan-c"Crowdy, Darren"https://zbmath.org/authors/?q=ai:crowdy.darren-gSummary: A new transform pair representing solutions to the complex Helmholtz equation in a convex 2D polygon is derived using the theory of Bessel's functions and Green's second identity. The derivation is a direct extension of that given by the second author [IMA J. Appl. Math. 80, No. 6, 1902--1931 (2015; Zbl 1338.35357)]
for `Fourier-Mellin transform' pairs associated with Laplace's equation in various domain geometries. It is shown how the new transform pair fits into the collection of ideas known as the Fokas transform where the key step in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of `reciprocal theorems', which are familiar tools in the study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculation of effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methods in diffusion wave field theory, application to the \(3\omega\) method for measuring thermal conductivity and to unsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentz reciprocity in mathematical physics is also pointed out.Correctness of the definition of the Laplace operator with delta-like potentialshttps://zbmath.org/1491.351412022-09-13T20:28:31.338867Z"Kanguzhin, B. E."https://zbmath.org/authors/?q=ai:kanguzhin.baltabek-esmatovich"Tulenov, K. S."https://zbmath.org/authors/?q=ai:tulenov.kanat-serikovichThe goal of this paper is to contribute to the rigorous analysis of the Schrödinger operator with distributional potentials. Specifically, the authors consider a bounded domain \(\Omega \subset {\mathbb R}^d\) with smooth boundary in dimensions \(d \geq 3\), and their analysis is motivated by the problem \(\bigl(\Delta + k\delta(x-x_0)\bigr)u = f \in L^2(\Omega)\) with homogeneous Dirichlet conditions \(u\big|_{\partial\Omega} = 0\), where \(x_0\) is a point in the interior of \(\Omega\). The Laplacian with \(\delta\)-potential at a point is a classical problem in singular perturbation theory, see [\textit{S. Albeverio} and \textit{P. Kurasov}, Singular perturbations of differential operators. Solvable Schrödinger type operators. Cambridge: Cambridge University Press (1999; Zbl 0945.47015)]. The authors also consider generalizations of this problem to allow for more general interactions at the interior point \(x_0\), and aim for conducting a spectral analysis for such problems.
There are some issues with the line of reasoning in this paper that impact the results, some of which do not hold up in the form in which they are stated.
Let \(G(x,\xi)\) be the Green function for the Dirichlet Laplacian in \(\Omega\), and write \(\Omega_0 = \Omega \setminus \{x_0\}\). Let \(\phi_0(x) = G(x,x_0)\) and \(\phi_j(x) = (\partial_{\xi_j}G)(x,x_0)\), \(j=1,\ldots,d\). The authors introduce the function space
\[ W^2_{2,\gamma}(\Omega_0) = \Big\{w = w_0 + \sum\limits_{j=0}^d \gamma_j \phi_j;\; w_0 \in W_2^2(\Omega),\; w_0\big|_{\partial\Omega} = 0,\; \gamma_j \in {\mathbb C}\Big\}.
\] Formally one has \(\Delta w = \Delta w_0 + \gamma_0\delta(x-x_0) - \sum\limits_{j=1}^d\gamma_j(\partial_{x_j}\delta)(x-x_0)\) with \(w\big|_{\partial\Omega} = 0\), so choosing \(w\) such that \(\gamma_0 = kw_0(x_0)\) and \(\gamma_j = 0\) for \(j \geq 1\) yields \(\Delta w = \Delta w_0 + kw_0(x_0)\delta(x-x_0)\). Rigorously the analysis takes place on \(\Omega_0\) rather than \(\Omega\), and the action of the Laplace operator is to be considered in distributions over \(\Omega_0\); the space \(W^2_{2,\gamma}(\Omega_0)\) is a subspace of \({\mathcal D}'(\Omega_0)\), and domains for the Laplace operator are prescribed by choosing subspaces \({\mathcal D} \subset W^2_{2,\gamma}(\Omega_0)\) by imposing linear relations between the function \(w_0\) and the numbers \(\gamma_j\) associated with the elements \(w \in W^2_{2,\gamma}(\Omega_0)\), in this paper specifically by requiring that \(\gamma_j = \gamma_j(w_0)\) are continuous linear functionals depending on \(w_0 \in W_2^2(\Omega)\). These functionals can be expressed in the form \(\gamma_j(w_0) = \langle \Delta w_0,c_j \rangle_{L^2(\Omega_0)}\) for \(c_j \in L^2(\Omega_0)\) by taking advantage of the invertibility of \(\Delta : \{w_0 \in W_2^2(\Omega);\; w_0|_{\partial\Omega} = 0\} \to L^2(\Omega_0)\). One then sees that \(\Delta : {\mathcal D} \subset W^2_{2,\gamma}(\Omega_0) \to L^2(\Omega_0)\) is invertible for any domain \({\mathcal D}\) that is prescribed in this manner. For example, for the regularization of the action of \(\Delta + k\delta(x-x_0)\), the choice of domain is \[ \mathcal D= \Big\{w = w_0 + \sum\limits_{j=0}^d \gamma_j \phi_j\in W^2_{2,\gamma}(\Omega_0);\; \gamma_0(w_0) = kw_0(x_0),\; \gamma_j(w_0) \equiv 0,\; j \geq 1\Big\}. \] This definition makes sense only in dimension \(d=3\) as for higher dimensions the Sobolev regularity is not sufficient for the point evaluation functional \(w_0(x_0)\) to be defined for \(w_0 \in W^2_2(\Omega)\).
The authors then aim to conduct a spectral analysis for the Laplacian with such domains and investigate notions of selfadjointness based on a formal Green formula for the boundary pairing. In this paper they conduct this analysis based on \(L^2(\Omega_0)\), and their claims and arguments here are problematic. In the representation of functions \(w \in W^2_{2,\gamma}(\Omega_0)\) above the function \(\phi_0\) belongs to \(L^2(\Omega_0)\) only in dimension \(d=3\), while none of the \(\phi_j\) for \(j \geq 1\) is in \(L^2(\Omega_0)\) for any dimension \(d \geq 3\); consequently, a domain \({\mathcal D} \subset W^2_{2,\gamma}(\Omega_0)\) for the Laplace operator as described above will satisfy \({\mathcal D} \subset L^2(\Omega_0)\) only if all \(\gamma_j \equiv 0\) for \(j \geq 1\), and \(\gamma_0\) can be nonzero only in dimension \(d=3\). The \(L^2(\Omega_0)\)-based spectral problem \((\Delta - \lambda)u = f\) for \(u \in {\mathcal D}\) and \(f \in L^2(\Omega_0)\) in the form considered by the authors in this paper, as well as their resolvent analysis, only applies to those cases.
The authors derive a Green formula by a formal computation and simplification of \(\langle \Delta u,v \rangle_{L^2(\Omega_0)} - \langle u,\Delta v \rangle_{L^2(\Omega_0)}\) for \(u,v \in W_{2,\gamma}^2(\Omega_0)\); note that generally \(u,v \notin L^2(\Omega_0)\). The problematic pairings are formally regularized as \(\langle \phi_0,\Delta w_0 \rangle_{L^2} = \overline{w_0}(x_0)\) and \(\langle \phi_j,\Delta w_0 \rangle_{L^2} = \overline{\partial_{x_j}w_0}(x_0)\), \(j \geq 1\), and correspondingly \(\langle \Delta w_0,\phi_j \rangle_{L^2} = \overline{\langle \phi_j,\Delta w_0 \rangle_{L^2}}\) for all \(j \geq 0\). Boundary data for \(u = u_0 + \sum\limits_{j=0}^d\gamma_j\phi_j \in W_{2,\gamma}^2(\Omega_0)\) then formally consists of two \((d+1)\)-vectors \(\Gamma_1(u) = [\gamma_0,\ldots,\gamma_d]\) and \(\Gamma_2(u) = [u_0(x_0),\nabla u_0(x_0)]\) (keeping in mind that the Dirichlet condition on \(\partial\Omega\) is imposed), and the formal Green formula then is expressed as a pairing between the \(\Gamma_j(u)\) and \(\Gamma_j(v)\), \(j=1,2\). For the point evaluations to make sense higher Sobolev regularity for \(u_0\) and \(v_0\) is needed than \(W^2_2(\Omega)\) required here, so the Green formula does not extend to \(W_{2,\gamma}^2(\Omega_0)\), which impacts some conclusions drawn in the paper that are based on that formula.
The observations that the functions \(\phi_j\) are generally not \(L^2\)-functions in \(\Omega_0\) in higher dimensions, and that the point evaluation functionals \(w_0(x_0)\) and \(\partial_{x_j}w_0(x_0)\) generally do not make sense on \(W_2^2(\Omega)\), are intimately related and need to be considered when reading this paper.
Reviewer: Thomas Krainer (Altoona)A note on the validity of the Schrödinger approximation for the Helmholtz equationhttps://zbmath.org/1491.351422022-09-13T20:28:31.338867Z"Klumpp, Maximilian"https://zbmath.org/authors/?q=ai:klumpp.maximilian"Schneider, Guido"https://zbmath.org/authors/?q=ai:schneider.guidoSummary: Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation
\[
\Delta u+\omega^2u=0\quad\text{for }(x,y,z)\in{\mathbb{R}}^3.
\] For the evolution of such waves along the \(z\)-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the \(z\)-axis.Existence of solutions for a nonlocal reaction-diffusion equation in biomedical applicationshttps://zbmath.org/1491.351432022-09-13T20:28:31.338867Z"Leon, Cristina"https://zbmath.org/authors/?q=ai:leon.cristina"Kutsenko, Irina"https://zbmath.org/authors/?q=ai:kutsenko.irina"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: The paper is devoted to a nonlocal semi-linear elliptic equation in \(\mathbb{R}^n\) arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray-Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.Directional \(\mathcal{H}^2\) Compression algorithm: optimisations and application to a discontinuous Galerkin BEM for the Helmholtz equationhttps://zbmath.org/1491.351442022-09-13T20:28:31.338867Z"Messaï, Nadir-Alexandre"https://zbmath.org/authors/?q=ai:messai.nadir-alexandre"Pernet, Sebastien"https://zbmath.org/authors/?q=ai:pernet.sebastien"Bouguerra, Abdesselam"https://zbmath.org/authors/?q=ai:bouguerra.abdesselamSummary: This study aimed to specialise a directional \(\mathcal{H}^2(\mathcal{DH}^2)\) compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal \(\mathcal{DH}^2\) approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a \(\mathcal{DH}^2\) gives better computational efficiency than a classical BEM in the case of high-order finite elements and \(hp\) heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splineshttps://zbmath.org/1491.351452022-09-13T20:28:31.338867Z"Nanfuka, Mary"https://zbmath.org/authors/?q=ai:nanfuka.mary"Berntsson, Fredrik"https://zbmath.org/authors/?q=ai:berntsson.fredrik"Mango, John"https://zbmath.org/authors/?q=ai:mango.john-magero(no abstract)Conformal flat metrics with prescribed mean curvature on the boundaryhttps://zbmath.org/1491.351462022-09-13T20:28:31.338867Z"Sharaf, Khadijah Abdullah"https://zbmath.org/authors/?q=ai:sharaf.khadijah-abdullah"Bensouf, Aymen"https://zbmath.org/authors/?q=ai:bensouf.aymen"Chtioui, Hichem"https://zbmath.org/authors/?q=ai:chtioui.hichem"Soumaré, Abdellahi"https://zbmath.org/authors/?q=ai:soumare.abdellahiSummary: We consider the problem of finding conformal metrics on the unit ball \({\mathbb{B}}^n\) of \({\mathbb{R}}^n\), \(n\ge 3\), with zero scalar curvature and prescribed mean curvature on the boundary. We study the lack of compactness of the problem and we prove an existence result based on an index-counting formula.Mathematics of magic angles in a model of twisted bilayer graphenehttps://zbmath.org/1491.351472022-09-13T20:28:31.338867Z"Becker, Simon"https://zbmath.org/authors/?q=ai:becker.simon"Embree, Mark"https://zbmath.org/authors/?q=ai:embree.mark"Wittsten, Jens"https://zbmath.org/authors/?q=ai:wittsten.jens"Zworski, Maciej"https://zbmath.org/authors/?q=ai:zworski.maciejSummary: We provide a mathematical account of the recent letter by \textit{G. Tarnopolsky, A. J. Kruchkov} and \textit{A. Vishwanath} [``Origin of magic angles in twisted bilayer graphene'', Phys. Rev. Lett.
122:10, art. id. 106405 (2019)]. The new contributions are a spectral characterization of magic angles, its accurate numerical implementation and an exponential estimate on the squeezing of all bands as the angle decreases. Pseudospectral phenomena due to the nonhermitian nature of operators appearing in the model considered in the letter of Tarnopolsky et al. play a crucial role in our analysis.Fractional powers of the Schrödinger operator on weigthed Lipschitz spaceshttps://zbmath.org/1491.351482022-09-13T20:28:31.338867Z"Bongioanni, B."https://zbmath.org/authors/?q=ai:bongioanni.bruno"Harboure, E."https://zbmath.org/authors/?q=ai:harboure.eleonor-o"Quijano, P."https://zbmath.org/authors/?q=ai:quijano.pSummary: In the setting of the semigroup generated by the Schrödinger operator \(L= -\Delta +V\) with the potential \(V\) satisfying an appropriate reverse Hölder condition, we consider some non-local fractional differentiation operators. We study their behaviour on suitable weighted smoothness spaces. Actually, we obtain such continuity results for positive powers of \(L\) as well as for the mixed operators \(L^{\alpha /2}V^{\sigma /2}\) and \(L^{-\alpha /2}V^{\sigma /2}\) with \(\sigma >\alpha\), together with their adjoints.Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equationhttps://zbmath.org/1491.351492022-09-13T20:28:31.338867Z"Hong, Younghun"https://zbmath.org/authors/?q=ai:hong.younghun"Jin, Sangdon"https://zbmath.org/authors/?q=ai:jin.sangdonSummary: For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [\textit{J. Bellazzini} et al., Math. Ann. 371, No. 1--2, 707-740 (2018); erratum ibid. 376, No 3--4, 1795--1796 (2020; Zbl 1392.35279)]. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.Fredholm property and essential spectrum of 3-D Dirac operators with regular and singular potentialshttps://zbmath.org/1491.351502022-09-13T20:28:31.338867Z"Rabinovich, Vladimir"https://zbmath.org/authors/?q=ai:rabinovich.vladimir-l|rabinovich.vladimir-sIn the paper, the 3D-Dirac operators with singular potentials supported on both bounded and unbounded surfaces in \(\mathbb{R}^3\) are considered. The approach to the self-adjointness of Dirac operators is based on the study of transmission problems with parameter associated with the Dirac operators. For their invertibility for large values of the parameter, and for the a priori estimates of solutions to associated transmission problems an analogue of Lopatinsky conditions is introduced. Finally, the Fredholm properties and the essential spectrum of transmission problems associated with the Dirac operators with singular potentials with supports on compact surfaces and non-compact surfaces with conical exits to infinity are investigated.
Reviewer: David Kapanadze (Tbilisi)Planar Schrödinger-Choquard equations with potentials vanishing at infinity: the critical casehttps://zbmath.org/1491.351512022-09-13T20:28:31.338867Z"Shen, Liejun"https://zbmath.org/authors/?q=ai:shen.liejun"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Yang, Minbo"https://zbmath.org/authors/?q=ai:yang.minboSummary: We study the following class of stationary Schrödinger equations of Choquard type
\[-\Delta u + V(x) u = [ | x |^{- \mu} \ast ( Q ( x ) F ( u ) ) ] Q(x) f(u), \;\;x \in \mathbb{R}^2,
\] where the potential \(V\) and the weight \(Q\) decay to zero at infinity like \(( 1 + | x |^\gamma )^{- 1}\) and \(( 1 + | x |^\beta )^{- 1}\) for some \((\gamma, \beta)\) in variously different ranges, \( \ast\) denotes the convolution operator with \(\mu \in(0, 2)\), and \(F\) is the primitive of \(f\) that fulfills a critical exponential growth in the Trudinger-Moser sense. By establishing a version of the weighted Trudinger-Moser inequality, we investigate the existence of nontrivial solutions of mountain-pass type for the given problem. Furthermore, we shall establish that the nontrivial solution is a bound state, namely a solution belonging to \(H^1( \mathbb{R}^2)\), for some particular \((\gamma, \beta)\).Fundamental solutions of generalized non-local Schrodinger operatorshttps://zbmath.org/1491.351522022-09-13T20:28:31.338867Z"Tan, Duc Do"https://zbmath.org/authors/?q=ai:tan.duc-doSummary: Let \(d \in \{1, 2, 3, \dots\}\) and \(s \in (0, 1)\) be such that \(d > 2s\). We consider a generalized non-local Schrodinger operator of the form
\[
L=L_K + \nu,
\]
where \(L_K\) is a non-local operator with kernel \(K\) that includes the fractional Laplacian \((-\Delta)^s\) for \(s \in (0, 1)\) as a special case. The potential \(\nu\) is a doubling measure subjected to a certain constraint. We show that the fundamental solution of \(L\) exists, is positive and possesses extra decaying properties.Solvability in the sense of sequences for some non Fredholm operators with drift and superdiffusionhttps://zbmath.org/1491.351532022-09-13T20:28:31.338867Z"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitali"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: We study the solvability of certain linear nonhomogeneous elliptic problems and prove that under some technical assumptions the convergence in \(L^2\) of their right sides implies the existence and the convergence in \(H^1\) of the solutions. The equations contain first order differential operators with or without Fredholm property, in particular the square root of the one dimensional negative Laplacian, on the whole real line or on a finite interval with periodic boundary conditions. We establish that the drift term involved in these problems provides the regularization of solutions.A sharp regularity estimate for the Schrödinger propagator on the spherehttps://zbmath.org/1491.351542022-09-13T20:28:31.338867Z"Chen, Xianghong"https://zbmath.org/authors/?q=ai:chen.xianghong|chen.xianghong.1"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinh"Lee, Sanghyuk"https://zbmath.org/authors/?q=ai:lee.sanghyuk|lee.sang-hyuk"Yan, Lixin"https://zbmath.org/authors/?q=ai:yan.lixinSummary: Let \(\Delta_{\mathbb{S}^n}\) denote the Laplace-Beltrami operator on the \(n\)-dimensional unit sphere \(\mathbb{S}^n\), \(n\geq 2\). In this paper we show that
\[
\|e^{it\Delta_{\mathbb{S}^n}}f\|_{L^4([0,2\pi)\times \mathbb{S}^n)}\leq C\|f\|_{W^{\alpha,4}(\mathbb{S}^n)}
\]
holds if \(\alpha>(n-2)/4\). The range of \(\alpha\) is sharp in the sense that the estimate fails for \(\alpha <(n-2)/4\). As a consequence, we obtain space-time \(L^p\)-estimates for \(e^{it\Delta_{\mathbb{S}^n}}\) for \(2\leq p\leq\infty\). We also prove that the maximal operator \(f\to\sup_{0\leq t<2\pi}|e^{it\Delta_{\mathbb{S}^n}}f|\) is bounded from \(W^{\alpha,2}(\mathbb{S}^n)\) to \(L^{6n/(3n-2)}(\mathbb{S}^n)\) for \(\alpha>1/3\) whenever \(f\) are zonal functions on \(\mathbb{S}^n\).Weighted \(p\)-Laplace approximation of linear and quasi-linear elliptic problems with measure datahttps://zbmath.org/1491.351552022-09-13T20:28:31.338867Z"Eymard, Robert"https://zbmath.org/authors/?q=ai:eymard.robert"Maltese, David"https://zbmath.org/authors/?q=ai:maltese.david"Prignet, Alain"https://zbmath.org/authors/?q=ai:prignet.alainSummary: We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted \(p\)-Laplace term. A well chosen diffeomorphism between \(\mathbb{R}\) and \((- 1, 1)\) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in \(L^1\).Generalized Fountain theorem for locally Lipschitz functionals and applicationhttps://zbmath.org/1491.351562022-09-13T20:28:31.338867Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Batkam, Cyril J."https://zbmath.org/authors/?q=ai:batkam.cyril-joel"Patricio, Geovany F."https://zbmath.org/authors/?q=ai:patricio.geovany-fSummary: In this paper, we obtain a nonsmooth version of the infinite-dimensional Fountain Theorem established by \textit{C. Batkam} and \textit{J. Colin} [J. Math. Anal. Appl. 405, No. 2, 438--452 (2013; Zbl 1312.35096)]. No symmetry condition on the energy functional is needed in our formulation. As an application, we prove the existence of multiple solutions for the following class of elliptic system
\[
\tag{\(S\)} \begin{cases} \begin{aligned}
\Delta u - u & \in [\underline{f} (x, u, v), \overline{f} (x, u, v)] \text{ a.e in } \mathbb{R}^N \\
- \Delta v + v & \in [\underline{g} (x, u, v), \overline{g} (x, u, v)] \text{ a.e in } \mathbb{R}^N, \\
& u, v \in H^1 (\mathbb{R}^N),
\end{aligned} \end{cases}
\] where \(f\) and \(g\) are measurable functions that satisfy some technical conditions.Sign-changing solutions for a modified nonlinear Schrödinger equation in \(\mathbb{R}^N\)https://zbmath.org/1491.351572022-09-13T20:28:31.338867Z"Jing, Yongtao"https://zbmath.org/authors/?q=ai:jing.yongtao"Liu, Haidong"https://zbmath.org/authors/?q=ai:liu.haidongSummary: We consider the modified nonlinear Schrödinger equation
\[
-\Delta u + V(x)u - \Delta(u^2)u = |u|^{p-2}u \text{ in }\mathbb{R}^N,
\]
where \(N\ge 3\), \(V\) is a given potential and \(p>2\). The equation appears in modeling of superfluid film theory in plasma physics. While most existing works in the literature are only for \(p\in [4, 4N/(N-2))\), we propose a new variational approach to deal with exponents \(p\in (2, 4N/(N-2))\) in a unified way and obtain infinitely many sign-changing solutions.Nodal solutions for a weighted \((p,q)\)-equationhttps://zbmath.org/1491.351582022-09-13T20:28:31.338867Z"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhai"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-sLet \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega\). The authors consider a nonlinear Dirichlet problem of the form \[ \begin{cases} -\Delta_{p}^{a_1}u(z)- \Delta_{q}^{a_2}u(z)=f(z,u(z))+|u(z)|^{p^*-2}u(z) \mbox{ in } \Omega,\\
u|_{\partial \Omega}=0, \, 1<q<p<\infty, \end{cases}\] where \(\Delta_p^{a_1} u= \operatorname{div}(a(z)|D u|^{p-2}D u)\) for all \(u \in W^{1,p}_0(\Omega)\) is the weighted \(p\)-Laplace operator, with \(a_1 \in C^{0,1}(\overline{\Omega})\). In the right-hand side, the first term is a locally defined Caratheodory perturbation and the second is a critical term (as \(p^*\) means the critical Sobolev exponent corresponding to \(p\)). Using variational tools and cut-off techniques, the authors show that the problem admits a sequence of arbitrarily small nodal solutions.
Reviewer: Calogero Vetro (Palermo)Upper-bound error estimates for double phase obstacle problems with Clarke's subdifferentialhttps://zbmath.org/1491.351592022-09-13T20:28:31.338867Z"Tam, Vo Minh"https://zbmath.org/authors/?q=ai:tam.vo-minhSummary: The main goal of this article is to investigate upper-bound error estimates (also called error bounds) for a class of double phase obstacle problems. We first recall double phase implicit obstacle problems involving Clarke's subdifferential given by \textit{S. Zeng} et al. [Calc. Var. Partial Differ. Equ. 59, No. 5, Paper No. 176, 17 p. (2020; Zbl 1453.35070)]. Then, based on regularized gap functions introduced by \textit{N. Yamashita} and \textit{M. Fukushima} [SIAM J. Control Optimization 35, No. 1, 273--284 (1997; Zbl 0873.49006)], we establish some regularized gap functions for the double phase obstacle problem. Finally, using the properties of Clarke's subdifferential and double phase operators, the upper-bound error estimates for such double phase obstacle problems in terms of regularized gap functions are provided.On well-defined solvability of the Dirichlet problem for a second order elliptic partial operator-differential equation in Hilbert spacehttps://zbmath.org/1491.351602022-09-13T20:28:31.338867Z"Aslanov, Hamidulla I."https://zbmath.org/authors/?q=ai:aslanov.hamidulla-israfil"Hatamova, Roya F."https://zbmath.org/authors/?q=ai:hatamova.roya-fSummary: In the paper we investigate well-defined solvability of the Dirichlet problem for a second order partial operator-differential equation in a Hilbert space. First we prove a theorem on the isomorphism of the principle part of the given equation. Then it is proved that operator coefficients of the disturbed part of the equation can be chosen from a wider class so that theorems on solvability of a boundary value problem for complete equations hold and are easily verifiable in practical problems. This feature strongly distinguishes our research from the works, where the solvability conditions entail arbitrary smallness of the disturbed part of equations and they are expressed by means of restrictions on the resolvent growth of the appropriate operator beam.Comparison principle for elliptic equations with mixed singular nonlinearitieshttps://zbmath.org/1491.351612022-09-13T20:28:31.338867Z"Durastanti, Riccardo"https://zbmath.org/authors/?q=ai:durastanti.riccardo"Oliva, Francescantonio"https://zbmath.org/authors/?q=ai:oliva.francescantonioSummary: We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by
\[
\begin{aligned}
\begin{cases}
\displaystyle
-{\Delta}_p u= \frac{f}{u^{\gamma}} + g u^q \quad &\text{in } {\Omega},\\
u = 0 \quad &\text{on } \partial{\Omega},
\end{cases}
\end{aligned}
\]
where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\), \(\Delta_p u\) := \(\operatorname{div}(|\nabla u|^{p - 2} \nabla u)\) is the usual \(p\)-Laplacian operator, \(\gamma \geq 0\) and \(0 \leq q \leq p - 1\); \(f\) and \(g\) are nonnegative functions belonging to suitable Lebesgue spaces.On the asymptotic behavior of a solution to an equation with a small parameter in a neighborhood of a boundary inflection pointhttps://zbmath.org/1491.351622022-09-13T20:28:31.338867Z"Lelikova, E. F."https://zbmath.org/authors/?q=ai:lelikova.e-f(no abstract)Non-homogeneous Gagliardo-Nirenberg inequalities in \(\mathbb{R}^N\) and application to a biharmonic non-linear Schrödinger equationhttps://zbmath.org/1491.351632022-09-13T20:28:31.338867Z"Fernández, Antonio J."https://zbmath.org/authors/?q=ai:fernandez.antonio-j"Jeanjean, Louis"https://zbmath.org/authors/?q=ai:jeanjean.louis"Mandel, Rainer"https://zbmath.org/authors/?q=ai:mandel.rainer"Mariş, Mihai"https://zbmath.org/authors/?q=ai:maris.mihaiSummary: We develop a new method, based on the Tomas-Stein inequality, to establish non-homogeneous Gagliardo-Nirenberg-type inequalities in \(\mathbb{R}^N\). Then we use these inequalities to study standing waves minimizing the \textit{energy} when the \(L^2\)-norm (the \textit{mass}) is kept fixed for a fourth-order Schrödinger equation with mixed dispersion. We prove optimal results on the existence of minimizers in the \textit{mass-subcritical} and \textit{mass-critical} cases. In the \textit{mass-supercritical} case global minimizers do not exist. However, if the Laplacian and the bi-Laplacian in the equation have the same sign, we are able to show the existence of local minimizers. The existence of those local minimizers is significantly more difficult than the study of global minimizers in the \textit{mass-subcritical} and \textit{mass-critical} cases. They are global in time solutions with small \(H^2\)-norm that do not scatter. Such special solutions do not exist if the Laplacian and the bi-Laplacian have opposite sign. If the mass does not exceed some threshold \(\mu_0 \in(0, + \infty)\), our results on ``best'' local minimizers are optimal.Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalueshttps://zbmath.org/1491.351642022-09-13T20:28:31.338867Z"Harris, Isaac"https://zbmath.org/authors/?q=ai:harris.isaacSummary: In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl's law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem.The ground state solution for biharmonic Kirchhoff-Schrödinger equations with singular exponential nonlinearities in \(\mathbb{R}^4\)https://zbmath.org/1491.351652022-09-13T20:28:31.338867Z"Liu, Yanjun"https://zbmath.org/authors/?q=ai:liu.yanjun.1"Qi, Shijie"https://zbmath.org/authors/?q=ai:qi.shijieSummary: In this paper, we consider the following singular biharmonic Kirchhoff-Schrödinger problem
\[
\begin{aligned} M\left(\int_{{\mathbb{R}}^4}|\Delta u|^2+V(x)u^2 {\mathrm{d}}x\right) (\Delta^2u+V(x)u) =\frac{f(x, u)}{|x|^{\eta}}, \quad x\in{\mathbb{R}}^4,
\end{aligned}
\]
where \(0<\eta <4\), \(M\) is a Kirchhoff-type function and \(V(x)\) is a continuous function with positive lower bound, \(f(x, t)\) has a critical exponential growth behavior at infinity. Using singular Adams inequality and variational techniques, we get the existence of ground state solution. Moreover, under general assumptions on the nonlinear term, through Cerami sequence for the energy functional, we also obtain the similar result.Axisymmetric residual stresses in a solid cylinder of finite lengthhttps://zbmath.org/1491.351662022-09-13T20:28:31.338867Z"Postolaki, Lesya"https://zbmath.org/authors/?q=ai:postolaki.lesya"Tokovyy, Yuriy"https://zbmath.org/authors/?q=ai:tokovyy.yuriy-vSummary: A solution technique is presented for the determination of residual stresses in a finite-length solid cylinder subject to non-uniform axisymmetric distributions of incompatible residual strains. The problem is reduced to the sequential solving of three individual problems: a problem on the determination of residual stresses in an infinitely long cylinder (the basic state) and two auxiliary problems for evaluating the stresses induced by the end-face effects (the disturbed states). The variational method of homogeneous solutions is implemented in order to determine the disturbed states within the framework of two latter problems. The solution technique is verified numerically for typical distribution profiles of incompatible strains depending on both the radial and axial coordinates. The approach can be used to evaluate residual stresses in a solid cylinder of finite length due to the intense thermal treatment.Ground state solutions for the generalized extensible beam equationshttps://zbmath.org/1491.351672022-09-13T20:28:31.338867Z"Wu, Tsung-fang"https://zbmath.org/authors/?q=ai:wu.tsungfangSummary: In this paper, we study the generalized extensible beam equations with steep potential
\[
\begin{cases}
\Delta^2 u - M \left(\|\nabla u\|_{L^2}^2\right) \Delta u + \lambda V (x) u = f (x, u) \quad \text{in } \mathbb{R}^N, \\
u \in H^2 (\mathbb{R}^N),
\end{cases}
\] where \(N \geq 4\), \(M (t) = a t + b\) with \(a > 0\), \(\lambda > 0\) and \(b \in \mathbb{R}\) are parameters, \(V \in C (\mathbb{R}^N, \mathbb{R})\) and \(f \in C (\mathbb{R}^N \times \mathbb{R}, \mathbb{R})\). Unlike most other papers on this problem, we allow the parameter \(b\) to be any negative number, which has the physical significance. Under some suitable assumptions on \(V(x)\) and \(f (x, u)\), when \(\lambda\) is large enough, we prove the existence of ground state solutions.On a boundary problem for a fourth-order elliptic equation on a planehttps://zbmath.org/1491.351682022-09-13T20:28:31.338867Z"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: The paper consideres a boundary value problem for a fourth-order elliptic equation with constant real coefficients in a multiply connected domain, in which the function and its normal third-order derivative on the boundary of this domain are specified. A convenient Fredholmity criterion is given, and a formula for the index of this problem is presented. Classes of equations for which the Fredholmity criterion is especially simple are indicated, and the exact values of the index are calculated.Generation results for vector-valued elliptic operators with unbounded coefficients in \(L^p\) spaceshttps://zbmath.org/1491.351692022-09-13T20:28:31.338867Z"Angiuli, Luciana"https://zbmath.org/authors/?q=ai:angiuli.luciana"Lorenzi, Luca"https://zbmath.org/authors/?q=ai:lorenzi.luca"Mangino, Elisabetta M."https://zbmath.org/authors/?q=ai:mangino.elisabetta-m"Rhandi, Abdelaziz"https://zbmath.org/authors/?q=ai:rhandi.abdelazizSummary: We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first order, in the Lebesgue space \(L^p({\mathbb{R}}^d;{\mathbb{R}}^m)\) with \(p \in (1,\infty )\). Sufficient conditions to prove generation results of an analytic \(C_0\)-semigroup \(\boldsymbol{T}(t)\), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger systemhttps://zbmath.org/1491.351702022-09-13T20:28:31.338867Z"An, Xiaoming"https://zbmath.org/authors/?q=ai:an.xiaoming"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jing.4|yang.jing.1|yang.jingSummary: This paper deals with the following weakly coupled nonlinear Schrödinger system
\[
\begin{cases}-\Delta{u}_1+a_1(x)u_1=|u_1|^{2p-2}u_1+b|u_1| ^{p-2}|u_2|^pu_1,& x\in\mathbb{R}^N,\\ -\Delta u_2+a_2(x)u_2=|u_2|^{2p-2}u_2+b|u_2|^{p-2}|u_1|^p{u}_2,& x\in\mathbb{R}^N,\end{cases}
\]
where \(N\ge 1\), \(b\in\mathbb{R}\) is a coupling constant, \(2p\in(2,2^{\ast})\), \(2^{\ast}=2N / (N-2)\) if \(N\ge 3\) and \(+\infty\) if \(N=1,2\), \(a_1(x)\) and \(a_2(x)\) are two positive functions. Assuming that \(a_i(x)\) (\(i=1,2\)) satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for \(|b| > 0\) small and a positive segregated solution for \(b< 0\), respectively. We also show that when \(1< p< \min\{2,2^{\ast} / 2\}\), the positive solutions are not unique if \(b> 0\) is small. The asymptotic behavior of the solutions when \(b\to 0\) and \(b\to -\infty\) is also studied.Quasilinear elliptic systems with nonlinear physical datahttps://zbmath.org/1491.351712022-09-13T20:28:31.338867Z"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farah"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussineSummary: Using the theory of Young measures, we prove the existence of weak solutions to the following quasilinear elliptic system:
\[
A(u) = f(x) + \operatorname{div}\sigma_0(x,u),
\]
where \(A(u) = - \operatorname{div}\sigma (x,u,Du)\) and \(f \in{W^{- 1}}{L_{\bar M}}(\Omega;\mathbb{R}^m)\). This problem corresponds to a diffusion phenomenon with a source \(f\) in a moving and dissolving substance, where the motion is described by \(\sigma_0\).Asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and electro-magneto-elastic structureshttps://zbmath.org/1491.351722022-09-13T20:28:31.338867Z"Chkadua, George"https://zbmath.org/authors/?q=ai:chkadua.georgeSummary: In the paper, we consider a three-dimensional model of fluid-solid acoustic interaction when an electro-magneto-elastic body occupying a bounded region \(\Omega^+\) is embedded in an unbounded fluid domain \(\Omega^- =\mathbb{R}^3 \setminus \overline{\Omega^+}\). In this case, we have a five-dimensional electro-magneto-elastic field (the displacement vector with three components, electric potential and magnetic potential) in the domain \(\Omega^+\), while we have a scalar acoustic pressure field in the unbounded domain \(\Omega^-\). The physical kinematic and dynamic relations are mathematically described by the appropriate boundary and transmission conditions. We consider less restrictions on a matrix differential operator of electro-magneto-elasticity by introducing asymptotic classes, in particular, we allow the corresponding characteristic polynomial of the matrix operator to have multiple real zeros.
In the paper, we consider mixed type interaction problem. In particular, except transmission conditions, electric and magnetic potentials are given on one part of the boundary of \(\Omega^+\) (the Dirichlet type condition), while on the other part, normal components of electric displacement and magnetic induction are given (the Neumann type condition).
We derive asymptotic expansion of solutions near the line where different boundary conditions change, and on the basis of asymptotic analysis, we establish optimal Hölder's smoothness results for solutions of the problem.Singular solutions of Toda system in high dimensionshttps://zbmath.org/1491.351732022-09-13T20:28:31.338867Z"Dou, Linlin"https://zbmath.org/authors/?q=ai:dou.linlinSummary: We construct some singular solutions to the four component Toda system. These solutions are almost split into two groups, each one modelled on an explicit solution to the two component Toda system (i.e. Liouivlle equation). These solutions are shown to be stable in high dimensions. This gives a sharp example on the partial regularity of stable solutions to Toda system.Existence of positive solutions of a critical system in \(\mathbb{R}^N\)https://zbmath.org/1491.351742022-09-13T20:28:31.338867Z"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher"Silva, Leticia S."https://zbmath.org/authors/?q=ai:silva.leticia-sSummary: In this paper we show existence of positive solution to the system
\[
\begin{cases} -\Delta u+a(x)u= \frac{1}{2^\ast}K_u(u,\upsilon) &\text{ in } \mathbb{R}^N, \\
-\Delta \upsilon+b(x)\upsilon=\frac{1}{2^\ast}K_\upsilon (u,\upsilon) &\text{ in } \mathbb{R}^N, \\
\qquad \qquad \qquad \qquad u, \upsilon > 0 &\text{ in } \mathbb{R}^N, \\
\qquad \qquad \;\; u, \upsilon \in D^{1,2}(\mathbb{R}^N), &\;N\geq 3.
\end{cases}
\]
We also prove a global compactness result for the associated energy functional similar to that
due to \textit{M. Struwe} [Math. Z. 187, 511--517 (1984; Zbl 0535.35025)]. The basic tool employed here is some information on a limit system of \((S)\) with \(a=b=0\), the concentration compactness due to \textit{P. L. Lions} [Rev. Mat. Iberoam. 1, No. 1, 145--201 (1985; Zbl 0704.49005)] and Brouwer degree theory.Another look at planar Schrödinger-Newton systemshttps://zbmath.org/1491.351752022-09-13T20:28:31.338867Z"Liu, Zhisu"https://zbmath.org/authors/?q=ai:liu.zhisu"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Tang, Chunlei"https://zbmath.org/authors/?q=ai:tang.chunlei"Zhang, Jianjun"https://zbmath.org/authors/?q=ai:zhang.jianjunSummary: In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth
\[
\begin{aligned}
\begin{cases}
- \Delta u + u + \phi u = f (u) \quad &\text{in } \mathbb{R}^2,\\
\Delta \phi = u^2 \quad &\text{in } \mathbb{R}^2,
\end{cases}
\end{aligned}
\]
where \(f\) is a smooth reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space \(H^1(\mathbb{R}^2)\). The analysis developed in this paper also allows to investigate the relationship between a Schrödinger-Newton system of Riesz-type and a Schrödinger-Newton system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrödinger-Newton system and may it have some potential applications in various related problems.Combined effects of homogenization and singular perturbations: quantitative estimateshttps://zbmath.org/1491.351762022-09-13T20:28:31.338867Z"Niu, Weisheng"https://zbmath.org/authors/?q=ai:niu.weisheng"Shen, Zhongwei"https://zbmath.org/authors/?q=ai:shen.zhongwei|shen.zhongwei.1Summary: We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale \(\kappa\) that represents the strength of the singular perturbation and on the length scale \(\epsilon\) of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale \(\epsilon\) and independent of \(\kappa \). This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both \(\epsilon\) and \(\kappa \).Existence and asymptotic behaviour of solutions for a quasilinear Schrödinger-Poisson system in \(\mathbb{R}^3\)https://zbmath.org/1491.351772022-09-13T20:28:31.338867Z"Wei, Chongqing"https://zbmath.org/authors/?q=ai:wei.chongqing"Li, Anran"https://zbmath.org/authors/?q=ai:li.anran"Zhao, Leiga"https://zbmath.org/authors/?q=ai:zhao.leigaSummary: In this paper, we study the existence and asymptotic behaviour of solutions for the following quasilinear Schrödinger-Poisson system in \(\mathbb{R}^3\)
\[
\begin{cases}
-\Delta u + V(x)u + \lambda\phi u = f(x, u),& x\in\mathbb{R}^3,\\
-\Delta\phi - \varepsilon^4\Delta_4\phi = \lambda u^2,& x\in\mathbb{R}^3,
\end{cases}
\]
where \(\lambda\) and \(\varepsilon\) are positive parameters, \(\Delta_4 = \operatorname{div}(|\nabla u|^2\nabla u)\), \(V\) is a continuous and coercive potential function with positive infimum, \(f\) is a Carathéodory function defined on \(\mathbb{R}^3\times\mathbb{R}\) satisfying the classic Ambrosetti-Rabinowitz condition. First, a nontrivial solution is obtained for \(\lambda\) small enough and \(\varepsilon\) fixed by variational methods and truncation technique. Later, the asymptotic behaviour of these solutions is studied whenever \(\varepsilon\) and \(\lambda\) tend to zero respectively. We prove that they converge to a nontrivial solution of a classic Schrödinger-Poisson system and a class of Schrödinger equation associated respectively.Quasilinear logarithmic Choquard equations with exponential growth in \(\mathbb{R}^N\)https://zbmath.org/1491.351782022-09-13T20:28:31.338867Z"Bucur, Claudia"https://zbmath.org/authors/?q=ai:bucur.claudia"Cassani, Daniele"https://zbmath.org/authors/?q=ai:cassani.daniele"Tarsi, Cristina"https://zbmath.org/authors/?q=ai:tarsi.cristinaSummary: We consider the \(N\)-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential \(\alpha\) is equal to the Euclidean dimension \(N\), and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space \(W^{1,N}(\mathbb{R}^N)\), where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable \textit{log}-weighted variant of the Pohozaev-Trudinger inequality which provides a suitable functional framework where we use variational methods.Classification of solutions for some elliptic systemhttps://zbmath.org/1491.351792022-09-13T20:28:31.338867Z"Yu, Xiaohui"https://zbmath.org/authors/?q=ai:yu.xiaohuiSummary: In this paper, we classify the solution of the following elliptic system
\[
\begin{cases}
-\Delta u(x) = e^{3v(x)}, & x\in\mathbb{R}^4, \\
(-\Delta)^2v(x) = u(x)^4, & x\in\mathbb{R}^4.
\end{cases}
\]
Under some assumptions, we will show that the solution has the following form
\[
u(x)=\frac{C_1(\varepsilon)}{\varepsilon^2 + |x - x_0|^2},\quad v(x) = \ln\frac{C_2(\varepsilon)}{\varepsilon^2+|x-x_0|^2},
\]
where \(C_1\), \(C_2\) are two positive constants depending only on \(\varepsilon\) and \(x_0\) is a fixed point in \(\mathbb{R}^4\).On a class of Schrödinger system problem in Orlicz-Sobolev spaceshttps://zbmath.org/1491.351802022-09-13T20:28:31.338867Z"El-Houari, H."https://zbmath.org/authors/?q=ai:el-houari.hassan"Chadli, L. S."https://zbmath.org/authors/?q=ai:chadli.lalla-saadia"Moussa, H."https://zbmath.org/authors/?q=ai:moussa.hassan(no abstract)On a boundary-domain integral equation system for the Robin problem for the diffusion equation in non-homogeneous mediahttps://zbmath.org/1491.351812022-09-13T20:28:31.338867Z"Fresneda-Portillo, Carlos"https://zbmath.org/authors/?q=ai:fresneda-portillo.carlosSummary: The Robin problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one applied by Chkadua, Mikhailov, Natroshvili. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.An elliptic nonlinear system of multiple functions with applicationhttps://zbmath.org/1491.351822022-09-13T20:28:31.338867Z"Kang, Joon Hyuk"https://zbmath.org/authors/?q=ai:kang.joonhyuk"Robertson, Timothy"https://zbmath.org/authors/?q=ai:robertson.timothySummary: The purpose of this paper is to give a sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^n\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of competing species of animals in many biological models.Positive solutions for semilinear elliptic systems with boundary measure datahttps://zbmath.org/1491.351832022-09-13T20:28:31.338867Z"Li, Yimei"https://zbmath.org/authors/?q=ai:li.yimei"Xie, Guangheng"https://zbmath.org/authors/?q=ai:xie.guanghengSummary: In this paper, we study the Dirichlet problem of elliptic systems
\[
\begin{cases}
-\boldsymbol{\Delta} \mathbf{u}=\mathbf{g}(\mathbf{u}) & \text{in } \Omega, \\
\mathbf{u} = \boldsymbol{\varrho\mu} & \text{on } \partial \Omega,
\end{cases}
\]
where \(\varrho \ge 0\), \(\Omega\) is an open bounded \(C^2\) domain in \(\mathbb{R}^N\) with \(N\ge 2\), and \(\mathbf{u}, \mathbf{g}(\mathbf{u}), \boldsymbol{\mu}\) are nonnegative vector-valued functions. We obtain the existence of weak positive solutions for the systems. In the special case \(\mathbf{g}(\mathbf{u})=|\mathbf{u}|^{p-1}\mathbf{u}\) with \(p>1\), we shall give a better description about the positive solutions including the priori estimate, regularity, existence and nonexistence.On nonlocal Choquard system with Hardy-Littlewood-Sobolev critical exponentshttps://zbmath.org/1491.351842022-09-13T20:28:31.338867Z"Luo, Xiaorong"https://zbmath.org/authors/?q=ai:luo.xiaorong"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anmin"Mo, Shuai"https://zbmath.org/authors/?q=ai:mo.shuaiSummary: Standing wave solutions of the following Hartree system with nonlocal interaction and critical exponent are considered:
\[
\begin{cases}
&-(a+b\displaystyle \int_{\Omega }|\nabla u|^2)\Delta u=h(x)\left(\displaystyle \int_{\Omega }\frac{|v(y)|^{2^\ast_\mu }}{|x-y|^\mu}\text{{d}}y\right) |u|^{2^\ast_\mu -2}u \\
& \qquad \qquad \qquad \qquad \qquad \quad + f_\lambda (x)|u|^{q-2}u,\quad \text{in } \Omega, \\
&-(a+b\displaystyle \int_{\Omega }|\nabla v|^2)\Delta v=h(x)\left( \displaystyle \int_{\Omega }\frac{|u(y)|^{2^\ast_\mu }}{|x-y|^\mu }\text{{d}}y\right) |v|^{2^\ast_\mu -2}v \\
& \qquad \qquad \qquad \qquad \qquad \quad + g_\sigma (x)|v|^{q-2}v,\quad \text{in } \Omega, \\
& u,v\ge 0, \quad \text{in }\Omega, \\
& u,v=0, \quad \text{on }\partial \Omega,
\end{cases}
\] where \(1<q<2\), \(2^\ast_{\mu }=\frac{2N-\mu }{N-2}\) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We study the effect of nonlocal interaction on the number of solutions in the case of general response function \(\Psi (x)=|x|^{-\mu }\) (\(0<\mu <N\)), which possesses more information on the mutual interaction between the particles. When parameters pair \((\lambda , \sigma )\) belongs to a certain subset of \(\mathbb{R}^2\), we prove the existence, nonexistence and the limit behavior of the nonnegative vector solutions depending on parameters. In the special case of \(q=2\), existence of nonnegative solution is also established. Our work extends and develops some recent results in the literature.Denumerably many positive radial solutions for the iterative system of elliptic equations in an annulushttps://zbmath.org/1491.351852022-09-13T20:28:31.338867Z"Rajendra Prasad, K."https://zbmath.org/authors/?q=ai:prasad.kapula-rajendra"Khuddush, Mahammad"https://zbmath.org/authors/?q=ai:khuddush.mahammad"Bharathi, B."https://zbmath.org/authors/?q=ai:bharathi.bSummary: Sufficient conditions are derived for the existence of denumerably many positive radial solutions to the iterative system of elliptic equations
\[
\begin{aligned}\Delta \mathtt{u}_\mathtt{j} + \mathtt{P}(|\mathtt{x}|)\mathtt{g}_\mathtt{j}(\mathtt{u}_{\mathtt{j}+1})&=0, \mathtt{R}_1 < |\mathtt{x}| < \mathtt{R}_2, \\
\mathtt{u}_{\ell +1}= \mathtt{u}_1, \mathtt{j}&=1,2, \dots, \ell, \end{aligned}
\]
\(\mathtt{x} \in \mathbb{R}^N\), \(N>2\), subject to a linear mixed boundary conditions at \(\mathtt{R}_1\) and \(\mathtt{R}_2\), by an application of Krasnoselskii's fixed point theorem.Least-energy nodal solutions of critical Schrödinger-Poisson system on the Heisenberg grouphttps://zbmath.org/1491.351862022-09-13T20:28:31.338867Z"Sun, Xueqi"https://zbmath.org/authors/?q=ai:sun.xueqi"Song, Yueqiang"https://zbmath.org/authors/?q=ai:song.yueqiangSummary: In this paper, we study the existence of least-energy nodal (sign-changing) solutions for a class of critical Schrödinger-Poisson system on the Heisenberg group given by
\[
\begin{cases}
-\Delta_H u +\mu\phi|u|^{q-2}u = \lambda f(\xi, u)+|u|^2u, &\text{in } \Omega, \\
-\Delta_H \phi =|u|^q, &\text{in } \Omega, \\
u=\phi =0, &\text{on } \partial \Omega,
\end{cases}
\]
where \(\Delta_H\) is the Kohn-Laplacian on the first Heisenberg group \(\mathbb{H}^1\), and \(\Omega\subset\mathbb{H}^1\) is a smooth bounded domain, \(\lambda > 0\) and \(\mu\in\mathbb{R}\) are some real parameters. Under the suitable conditions on \(f\), together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solution. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient as well as critical nonlinearities.On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constrainthttps://zbmath.org/1491.351872022-09-13T20:28:31.338867Z"Taheri, Ali"https://zbmath.org/authors/?q=ai:taheri.ali-karimi|taheri.ali"Vahidifar, Vahideh"https://zbmath.org/authors/?q=ai:vahidifar.vahidehSummary: The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint:
\[
\begin{cases}
\mathrm{div} \{ \mathsf{A} (|x|, |u|^2, |\nabla u|^2) \nabla u\} + \mathsf{B} (|x|, |u|^2, |\nabla u|^2) u = \mathrm{div} \{ \mathscr{P}(x) [\mathrm{cof} \nabla u]\} & \text{in } \varOmega, \\
\det \nabla u = 1 & \text{in } \varOmega, \\
u = \varphi & \text{on } \partial \varOmega,
\end{cases}
\]
where \(\varOmega \subset \mathbb{R}^n\) (\(n \geq 2\)) is a bounded domain, \(u = (u_1, \ldots, u_n)\) is a vector-map and \(\varphi\) is a prescribed boundary condition. Moreover \(\mathscr{P}\) is a hydrostatic pressure associated with the constraint \(\det \nabla u \equiv 1\) and \(\mathsf{A} = \mathsf{A} (|x|, |u|^2, |\nabla u|^2)\), \(\mathsf{B} = \mathsf{B} (|x|, |u|^2, |\nabla u|^2)\) are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draw upon intimate links with the Lie group \(\mathbf{SO} (n)\), its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably, a discriminant type quantity \(\varDelta = \varDelta (\mathsf{A}, \mathsf{B})\) prompting from the system, will be shown to have a decisive role on the structure and multiplicity of these solutions.Multiple nodal and semi-nodal solutions to a nonlinear Choquard-type systemhttps://zbmath.org/1491.351882022-09-13T20:28:31.338867Z"Wu, Huiling"https://zbmath.org/authors/?q=ai:wu.huilingSummary: We study the nonlinearly coupled Choquard-type system
\[
\begin{cases}
- \Delta u_1 + \mu_1 u_1 = a_1 ( I_\alpha \ast | u_1 |^p ) | u_1 |^{p - 2} u_1 + \beta ( I_\alpha \ast | u_2 |^p ) | u_1 |^{p - 2} u_1, x \in \Omega, \\
- \Delta u_2 + \mu_2 u_2 = a_2 ( I_\alpha \ast | u_2 |^p ) | u_2 |^{p - 2} u_2 + \beta ( I_\alpha \ast | u_1 |^p ) | u_2 |^{p - 2} u_2, x \in \Omega, \\
u_1 = u_2 = 0 \text{ on } \partial {\Omega},
\end{cases}\tag{0.1}
\] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) with \(N \geq 3\), \(p \in(\frac{ N + \alpha}{ N}, \frac{ N + \alpha}{ N - 2})\), \(\alpha \in(0, N)\), \(I_\alpha\) is the Riesz potential, and \(\mu_1, \mu_2, a_1, a_2, \beta\) are positive constants. For every \(k \in \mathbb{N} \), we prove that there exists \(\beta_k > 0\) such that system (0.1) possesses \(k\) nodal solutions and \(k\) semi-nodal solutions for \(\beta \in(0, \beta_k)\) and \(p > 2\). Additionally, the existence of least energy nodal solutions is also obtained.Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz conditionhttps://zbmath.org/1491.351892022-09-13T20:28:31.338867Z"Bonaldo, Lauren M. M."https://zbmath.org/authors/?q=ai:bonaldo.lauren-m-m"Hurtado, Elard J."https://zbmath.org/authors/?q=ai:hurtado.elard-juarez"Miyagaki, Olímpio H."https://zbmath.org/authors/?q=ai:miyagaki.olimpio-hiroshiSummary: In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations \((\mathscr{P}_\lambda)\) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \(\mathscr{L}_{\mathcal{A}K}\) with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem \((\mathscr{P}_\lambda)\) and we show that the problem treated has at least one nontrivial solution for any parameter \(\lambda >0\) small enough as well as that the solution blows up, in the fractional Sobolev norm, as \(\lambda \to 0\). Moreover, for the sublinear case, by imposing some additional hypotheses on the nonlinearity \(f(x,\cdot)\), and by using a new version of the symmetric Mountain Pass Theorem due to \textit{R. Kajikiya} [J. Funct. Anal. 225, No. 2, 352--370 (2005; Zbl 1081.49002)], we obtain the existence of infinitely many weak solutions which tend to zero, in the fractional Sobolev norm, for any parameter \(\lambda >0\). As far as we know, the results of this paper are new in the literature.Generalized Keller-Osserman conditions for fully nonlinear degenerate elliptic equationshttps://zbmath.org/1491.351902022-09-13T20:28:31.338867Z"Capuzzo Dolcetta, I."https://zbmath.org/authors/?q=ai:capuzzo-dolcetta.italo"Leoni, F."https://zbmath.org/authors/?q=ai:leoni.fabiana"Vitolo, A."https://zbmath.org/authors/?q=ai:vitolo.antonioSummary: We discuss the existence of entire (i.e. defined on the whole space) subsolutions of fully nonlinear degenerate elliptic equations, giving necessary and sufficient conditions on the coefficients of the lower order terms which extend the classical Keller-Osserman conditions for semilinear elliptic equations. Our analysis shows that, when the conditions of existence of entire subsolutions fail, a priori upper bounds for local subsolutions can be obtained.Sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the upper half-spaceshttps://zbmath.org/1491.351912022-09-13T20:28:31.338867Z"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Yang, Qiaohua"https://zbmath.org/authors/?q=ai:yang.qiaohua"Zhu, Maochun"https://zbmath.org/authors/?q=ai:zhu.maochunSummary: In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half-spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser (Theorems 1.1 and 1.2) and trace Adams inequalities (Theorems 1.4, 1.5, 1.10 and 1.11) can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders on half-spaces. Furthermore, as an application, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities (Theorem 1.13). It is interesting to note that there are two types of trace Trudinger-Moser and trace Adams inequalities: critical and subcritical trace inequalities under different constraints. Moreover, trace Trudinger-Moser and trace Adams inequalities of exact growth also hold on half-spaces (Theorems 1.6, 1.8 and 1.12).Local Lipschitz continuity for energy integrals with slow growthhttps://zbmath.org/1491.351922022-09-13T20:28:31.338867Z"Eleuteri, Michela"https://zbmath.org/authors/?q=ai:eleuteri.michela"Marcellini, Paolo"https://zbmath.org/authors/?q=ai:marcellini.paolo"Mascolo, Elvira"https://zbmath.org/authors/?q=ai:mascolo.elvira"Perrotta, Stefania"https://zbmath.org/authors/?q=ai:perrotta.stefaniaSummary: We consider some energy integrals under slow growth, and we prove that the local minimizers are locally Lipschitz continuous. Many examples are given, either with subquadratic \(p,q\)-growth and/or anisotropic growth.Exact principal blowup rate near the boundary of boundary blowup solutions to \(k\)-curvature equationhttps://zbmath.org/1491.351932022-09-13T20:28:31.338867Z"Takimoto, Kazuhiro"https://zbmath.org/authors/?q=ai:takimoto.kazuhiroSummary: We consider boundary blowup problems for \(k\)-curvature equations of the form \(H_k[u] = f(u)g(|Du|)\) in a bounded smooth domain \(\Omega \subset \mathbb{R}^n\), where \(f(s)\) behaves like \(s^p\) as \(s \rightarrow \infty\) and \(g(t)\) behaves like \(t^{-q}\) as \(t \rightarrow \infty\). We obtain the exact principal blowup rate of a solution \(u\) near the boundary \(\partial \Omega\) under some conditions.Weighted Lorentz estimates for fully nonlinear elliptic equations with oblique boundary datahttps://zbmath.org/1491.351942022-09-13T20:28:31.338867Z"Zhang, Junjie"https://zbmath.org/authors/?q=ai:zhang.junjie"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhouSummary: We devote this paper to the weighted Lorentz regularity of Hessian for viscosity solution of fully nonlinear elliptic problem with oblique boundary condition \(\beta \cdot Du=0\) under the assumption that the nonlinearity \(F\) has small BMO semi-norms with respect to \(x\)-variable and the boundary of underlying domain belongs to \(C^{2,\alpha}\) for some \(\alpha \in (0,1)\). An optimal global Calderón-Zygmund type estimate in the weighted Lorentz spaces is obtained by constructing the sequence of approximating oblique derivative problems and flattening cover argument on the boundary \(\partial \Omega \). As a direct consequence of main theorem, we also derive global regularity in the variable exponent Morrey spaces to the Hessian of solution.K-ended \(O(m) \times O(n)\) invariant solutions to the Allen-Cahn equation with infinite Morse indexhttps://zbmath.org/1491.351952022-09-13T20:28:31.338867Z"Agudelo, Oscar"https://zbmath.org/authors/?q=ai:agudelo.oscar-mauricio"Rizzi, Matteo"https://zbmath.org/authors/?q=ai:rizzi.matteoSummary: In this work we study existence, asymptotic behaviour and stability properties of \(O(m) \times O(n)\)-invariant solutions of the Allen-Cahn equation \(\Delta u + u(1 - u^2) = 0\) in \(\mathbb{R}^m \times \mathbb{R}^n\) with \(m\), \(n \geq 2\) and \(m + n \geq 8\). We exhibit four families of solutions whose nodal sets are smooth logarithmic corrections of the Lawson cone and with infinite Morse index. This work complements the study started by \textit{F. Pacard} and \textit{J. Wei} in [J. Funct. Anal. 264, No. 5, 1131--1167 (2013; Zbl 1281.35046)] and by the authors and \textit{M. Kowalczyk} in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 216, Article ID 112705, 53 p. (2022; Zbl 1481.35226)].Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equationhttps://zbmath.org/1491.351962022-09-13T20:28:31.338867Z"Allendes, Alejandro"https://zbmath.org/authors/?q=ai:allendes.alejandro"Fuica, Francisco"https://zbmath.org/authors/?q=ai:fuica.francisco"Otárola, Enrique"https://zbmath.org/authors/?q=ai:otarola.enriqueHardy spaces associated to generalized Hardy operators and applicationshttps://zbmath.org/1491.351972022-09-13T20:28:31.338867Z"Bui, The Anh"https://zbmath.org/authors/?q=ai:the-anh-bui."Nader, Georges"https://zbmath.org/authors/?q=ai:nader.georgesSummary: In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator \(L_\alpha = (-\Delta)^{\alpha/2} + a|x|^{-\alpha}\). Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.Appell hypergeometric function and its application to the potential theory for a generalized bi-axially symmetric elliptic equationhttps://zbmath.org/1491.351982022-09-13T20:28:31.338867Z"Ergashev, Tuhtasin"https://zbmath.org/authors/?q=ai:ergashev.tuhtasin-gulamjanovich"Hasanov, Anvar"https://zbmath.org/authors/?q=ai:hasanov.anvar-hSummary: Fundamental solutions of the generalized bi-axially symmetric elliptic equation are expressed in terms of the well-known Appell hypergeometric function in two variables, the properties of which are required to study boundary value problems for the above equation. In this paper, by using some properties of the Appell hypergeometric function \(F_2\), we prove limiting theorems and derive integral equations concerning a denseness of the double- and simple-layer potentials. We apply the results of the constructed potential theory to the study of the Holmgren problem for the generalized bi-axially symmetric elliptic equation in the domain bounded in the first quarter of the plane.Multiplicity of positive solutions to Schrödinger-type positone problemshttps://zbmath.org/1491.351992022-09-13T20:28:31.338867Z"Ko, Eunkyung"https://zbmath.org/authors/?q=ai:ko.eunkyungSummary: We establish multiplicity results for positive solutions to the Schrödinger-type singular positone problem: \(-\Delta u + V (x)u = \lambda f(u)\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N > 2\), \(\lambda\) is a positive parameter, \(V \in L^\infty(\Omega)\) and \(f : [0,\infty) \to (0, \infty)\) is a continuous function. In particular, when \(f\) is sublinear at infinity 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.Existence of positive solutions for a class of fractional Choquard equation in exterior domainhttps://zbmath.org/1491.352002022-09-13T20:28:31.338867Z"Ledesma, César E. Torres"https://zbmath.org/authors/?q=ai:torres-ledesma.cesar-eSummary: In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and Choquard type nonlinearity. We prove the main results using variational method combined with Brouwer theory of degree and Deformation Lemma.Existence results for the mean field equation on a closed symmetric Riemann surfacehttps://zbmath.org/1491.352012022-09-13T20:28:31.338867Z"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjie"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyanSummary: Let \((S, g)\) be a closed Riemann surface, \textbf{G} be a finite isometric group acting on it and \(\sharp \mathbf{G}(x)\) be the number of all distinct points in the set \(\mathbf{G}(x)\) for \(x \in S\). If there exists some \(\ell \in \mathbb{N}^\ast\) satisfying \(\sharp \mathbf{G}(x) \equiv \ell\) for all \(x \in S\), then we show that for any \(\rho \in(8 \pi k \ell, 8 \pi(k + 1) \ell)\) with \(k \in \mathbb{N}^\ast \), the mean field equation
\[
\Delta_g u = \rho \left(\frac{ h e^u}{ \int_S h e^u \mathrm{d} v_g} - \frac{1}{|S|}\right)
\] has a \textbf{G}-invariant solution, where \(h\) is a strictly positive and \textbf{G}-invariant smooth function. Our method is a modification of the min-max scheme due \textit{W. Ding} et al. [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 5, 653--666 (1999; Zbl 0937.35055)], \textit{Z. Djadli} and \textit{A. Malchiodi} [Ann. Math. (2) 168, No. 3, 813--858 (2008; Zbl 1186.53050)] and \textit{Z. Djadli} [Commun. Contemp. Math. 10, No. 2, 205--220 (2008; Zbl 1151.53035)].Existence results for double phase problem in Sobolev-Orlicz spaces with variable exponents in complete manifoldhttps://zbmath.org/1491.352022022-09-13T20:28:31.338867Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Bennouna, Jaouad"https://zbmath.org/authors/?q=ai:bennouna.jaouad"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti-Rabinowitz type condition in the framework of Sobolev-Orlicz spaces with variable exponents in complete manifold. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder inequality, continuous and compact embedding results are proved.Existence and uniqueness results for a class of \(p(x)\)-Kirchhoff-type problems with convection term and Neumann boundary datahttps://zbmath.org/1491.352032022-09-13T20:28:31.338867Z"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"El Ouaarabi, Mohamed"https://zbmath.org/authors/?q=ai:el-ouaarabi.mohamed"Melliani, Said"https://zbmath.org/authors/?q=ai:melliani.saidSummary: We establish an existence and uniqueness results for a homogeneous Neumann boundary value problem involving the \(p(x)\)-Kirchhoff-Laplace operator of the following form
\[
\begin{cases} \begin{aligned}
&-M\Big (\int_{{\Omega}}\frac{1}{p(x)}(\vert \nabla u\vert^{p(x)}+\vert u\vert^{p(x)})\,dx\Big ) \Big (\text{{div}}(\vert \nabla u\vert^{p(x)-2}\nabla u)-\vert u\vert^{p(x)-2}u\Big )\\
&=f(x, u, \nabla u) && \text{in } , \\
&\vert \nabla u\vert^{p(x)-2}\frac{\partial u}{\partial \eta }=0 && \text{on } \partial{{\Omega}}.
\end{aligned} \end{cases}
\] where \({{\Omega}}\) is a smooth bounded domain in \(\mathbb{R}^N\), \(\frac{\partial u}{\partial \eta }\) is the exterior normal derivative, \(p(x)\in C_+({\overline{{\Omega} }})\) with \(p(x)\ge 2\). By means of a topological degree of Berkovits for a class of demicontinuous operators of generalized \((S_+)\) type and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on \(f\) and \(M\), we obtain a results on the existence and uniqueness of weak solution to the considered problem.Multiple solutions for two classes of quasilinear problems defined on a nonreflexive Orlicz-Sobolev spacehttps://zbmath.org/1491.352042022-09-13T20:28:31.338867Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Bahrouni, Sabri"https://zbmath.org/authors/?q=ai:bahrouni.sabri"Carvalho, Marcos L. M."https://zbmath.org/authors/?q=ai:carvalho.marcos-l-mSummary: In this paper we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by \textit{A. Szulkin} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 77--109 (1986; Zbl 0612.58011)] combined with some properties of the weak topology.Solutions of Ginzburg-Landau-type equations involving variable exponenthttps://zbmath.org/1491.352052022-09-13T20:28:31.338867Z"Avci, Mustafa"https://zbmath.org/authors/?q=ai:avci.mustafaSummary: In this article, we are interested in some class of Ginzburg-Landau-type equations involving variable exponent under the homogenous Dirichlet boundary conditions and settled in Musielak-Sobolev spaces. We look for nontrivial weak solutions, that is, critical points of the corresponding Ginzburg-Landau energy functional.Gradient estimates for Orlicz double phase problems with variable exponentshttps://zbmath.org/1491.352062022-09-13T20:28:31.338867Z"Baasandorj, Sumiya"https://zbmath.org/authors/?q=ai:baasandorj.sumiya"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Lee, Ho-Sik"https://zbmath.org/authors/?q=ai:lee.ho-sikSummary: Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón-Zygmund theory.Boundary singular solutions of a class of equations with mixed absorption-reactionhttps://zbmath.org/1491.352072022-09-13T20:28:31.338867Z"Bidaut-Véron, Marie-Françoise"https://zbmath.org/authors/?q=ai:bidaut-veron.marie-francoise"Garcia-Huidobro, Marta"https://zbmath.org/authors/?q=ai:garcia-huidobro.marta"Véron, Laurent"https://zbmath.org/authors/?q=ai:veron.laurentThe paper is devoted to the equation \[-\Delta u+|u|^{p-1}u-M|\nabla u|^q=0,\leqno(1)\] in a bounded domain \(\Omega\) of \(\mathbb{R}^N\) or in the half-space \(\mathbb{R}_+^N\), where \(M>0\), \(p>q\) and \(q\in (1,2)\). The authors study the existence of various types of positive solutions of \((1)\) and their properties, with a particular interest in the analysis of boundary singularities of the solutions. They concentrate on the solutions of equation \((1)\) which vanish on the boundary except one point. Conditions for the removability of compact boundary sets, and the Dirichlet problem associated to \((1)\) with a measure as boundary data are also investigated.
Reviewer: Rodica Luca (Iaşi)Multiple solutions for quasilinear Schrödinger equations involving local nonlinearity termhttps://zbmath.org/1491.352082022-09-13T20:28:31.338867Z"Chen, Chunfang"https://zbmath.org/authors/?q=ai:chen.chunfang"Zhu, Wenjie"https://zbmath.org/authors/?q=ai:zhu.wenjieSummary: In this paper, we deal with the existence and multiplicity of nontrivial solutions for the following quasilinear Schrödinger equation:
\[
-\Delta u+V(x)u+\frac{1}{2}\Delta (u^2)u=\lambda f(u),\quad x\in \mathbb{R}^N,
\]
where \(N\geq 3\), \(\lambda >0\) is a real parameter, \(V(x):\mathbb{R}^N\rightarrow \mathbb{R}\) is a potential function, the nonlinearity \(f(t)\in \mathcal{C}(\mathbb{R},\mathbb{R})\) and just superlinear near the origin. By using a change of variables, truncation argument, the Mountain Pass theorem and Fountain theorem, we prove the existence and multiplicity of nontrivial solutions for the above equation when \(\lambda\) is large enough.Fractional elliptic problems with nonlinear gradient sources and measureshttps://zbmath.org/1491.352092022-09-13T20:28:31.338867Z"da Silva, João Vitor"https://zbmath.org/authors/?q=ai:da-silva.joao-vitor"Ochoa, Pablo"https://zbmath.org/authors/?q=ai:ochoa.pablo-d"Silva, Analía"https://zbmath.org/authors/?q=ai:silva.analiaSummary: In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data.On solutions for a class of fractional Kirchhoff-type problems with Trudinger-Moser nonlinearityhttps://zbmath.org/1491.352102022-09-13T20:28:31.338867Z"de Souza, Manassés"https://zbmath.org/authors/?q=ai:de-souza.manasses-x"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"Luiz do Rêgo, Thiago"https://zbmath.org/authors/?q=ai:luiz-do-rego.thiagoExistence of solution to Kirchhoff type problem with gradient nonlinearity and a perturbation termhttps://zbmath.org/1491.352112022-09-13T20:28:31.338867Z"Dwivedi, Gaurav"https://zbmath.org/authors/?q=ai:dwivedi.gaurav"Gupta, Shilpa"https://zbmath.org/authors/?q=ai:gupta.shilpa-dasSummary: This article deals with the existence of a weak solution to the Kirchhoff problem:
\[
\begin{cases} \begin{aligned}
&-A\left( x,\int_{\Omega } |\nabla u|^2dx \right) \Delta u =f(x,u,\nabla u)+\lambda h(x,u) && \text{in } \Omega, \\
&u = 0 && \text{on } \partial \Omega.
\end{aligned} \end{cases}
\] where \(\Omega\) is a bounded and smooth domain in \(\mathbb{R}^N\) (\(N\ge 2\)). We assume that \(f\), \(h\) and \(A\) are continuous functions and the growth of the non linearity \(f:\overline{\Omega }\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}\) is dependent on \(u\) and \(\nabla u\). We do not assume any growth condition on the perturbation term \(h\). In the case of \(N=2\), we consider the exponential growth in the second variable of \(f\). The proof of our main existence result uses an iterative technique based on the mountain pass theorem.Strongly nonlinear coupled system in Orlicz-Sobolev spaces without \(\Delta_2\)-conditionhttps://zbmath.org/1491.352122022-09-13T20:28:31.338867Z"Elarabi, Rabab"https://zbmath.org/authors/?q=ai:elarabi.rababSummary: The aim of this paper is to prove in Orlicz-Sobolev spaces, the existence of capacity solution to a strongly nonlinear elliptic coupled system without assuming the \(\Delta_2\)-condition on the N-funtion. This system may be regarded as a modified version of the well-known thermistor problem; in this case, the unknowns are the temperature in a conductor and the electrical potential.On a \(p(x)\)-Kirchhoff fourth order problem involving Leray-Lions type operatorshttps://zbmath.org/1491.352132022-09-13T20:28:31.338867Z"Filali, Mohammed"https://zbmath.org/authors/?q=ai:filali.mohammed"Soualhine, Khalid"https://zbmath.org/authors/?q=ai:soualhine.khalid"Talbi, Mohamed"https://zbmath.org/authors/?q=ai:talbi.mohamed"Tsouli, Najib"https://zbmath.org/authors/?q=ai:tsouli.najibSummary: The aim of this work is to study the existence and the multiplicity of nontrivial weak solutions for a class of \(p(x)\)-Kirchhoff type problems involving Leray-Lions type operators and a changing sign weight under no flux boundary condition. By using the mountain pass type theorem and the Ekeland's variational principle, we obtain at least two nontrivial weak solutions; moreover, by following the steps described by the Fountain Theorem, we will find an infinitely many weak solutions.Three weak solutions for a degenerate nonlocal singular sub-linear problemhttps://zbmath.org/1491.352142022-09-13T20:28:31.338867Z"Heidarkhani, Shapour"https://zbmath.org/authors/?q=ai:heidarkhani.shapour"Kou, Kit Ian"https://zbmath.org/authors/?q=ai:kou.kit-ian|kou.kitian"Salari, Amjad"https://zbmath.org/authors/?q=ai:salari.amjadSummary: Based on one recent abstract critical point result for differentiable and parametric functionals which was recently proved by Ricceri, we establish the existence of three weak solutions for a class of degenerate nonlocal singular sub-linear problems when the nonlinear term admits some hypotheses on the behavior at infinitely or perturbation property.Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical caseshttps://zbmath.org/1491.352152022-09-13T20:28:31.338867Z"Kumar, Deepak"https://zbmath.org/authors/?q=ai:kumar.deepak"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Sreenadh, Konijeti"https://zbmath.org/authors/?q=ai:sreenadh.konijetiThe authors produce a qualitative analysis of an equation involving the \((p,q)\)-fractional Laplacian and a special nonlinearity.
Reviewer: Dumitru Motreanu (Perpignan)Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponentshttps://zbmath.org/1491.352162022-09-13T20:28:31.338867Z"Liang, Sihua"https://zbmath.org/authors/?q=ai:liang.sihua"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanni"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlinSummary: In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problem with critical variable exponent. By using constraint variational method and quantitative deformation lemma we show the existence of one least energy solution, which is strictly larger than twice of that of any ground state solution.Multiplicity and concentration results for generalized quasilinear Schrödinger equations with nonlocal termhttps://zbmath.org/1491.352172022-09-13T20:28:31.338867Z"Li, Quanqing"https://zbmath.org/authors/?q=ai:li.quanqing"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.1"Nie, Jianjun"https://zbmath.org/authors/?q=ai:nie.jianjun"Wang, Wenbo"https://zbmath.org/authors/?q=ai:wang.wenboSummary: This paper is concerned with the following generalized quasilinear Schrödinger equation with nonlocal term
\[
-\varepsilon^2\operatorname{div}(g^2(u)\nabla u) + \varepsilon^2g(u)g^\prime(u)|\nabla u|^2 + V(x)u = \varepsilon^{\mu - N}[|x|^{-\mu}\ast|u|^p]|u|^{p-2}u + f(u)
\]
for \(x\in\mathbb{R}^N\), where \(N\ge 3\), \(\varepsilon >0\), \(g:\mathbb{R}\rightarrow\mathbb{R}^+\) is a bounded \(C^1\) even function, \(g(0)=1\), \(g^\prime (s)\ge 0\) for all \(s\ge 0\), \(\lim\nolimits_{|s|\rightarrow +\infty}\frac{g(s)}{|s|^{\alpha -1}}:=\beta >0\) for some \(\alpha\in [1,2]\) and \((\alpha -1)g(s)\ge g^\prime(s)s\) for all \(s\ge 0\), \(2\alpha \le p< \frac{2\alpha(N - \mu )}{N-2}\), \(0<\mu <N\). By using the variational methods and Ljusternik-Schnirelmann category theory, we establish the existence, multiplicity of semi-classical solutions and characterize the concentration behavior and exponential decay property for the above problem.Multi-bump solutions for Kirchhoff equation in \(\mathbb{R}^3 \)https://zbmath.org/1491.352182022-09-13T20:28:31.338867Z"Liu, Weiming"https://zbmath.org/authors/?q=ai:liu.weimingSummary: This paper deals with the existence of multi-bump solutions for the following Kirchhoff equation
\[
-\left( a+b \int \limits_{{\mathbb{R}}^3}|\nabla u|^2dx\right) \Delta u+u=(1-\varepsilon q(x))|u|^{p-2}u, \quad x\in{\mathbb{R}}^3,
\] where \(a,b>0\), \(2<p<6\) and \(q(x)\in C({\mathbb{R}}^3,{\mathbb{R}}^+)\) satisfies some suitable conditions. By using the Lyapunov-Schmidt reduction method, we extend the results in [\textit{L. Lin} and \textit{Z. Liu}, J. Funct. Anal. 257, No. 2, 485--505 (2009; Zbl 1171.35114)] to the Kirchhoff problem.A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearityhttps://zbmath.org/1491.352192022-09-13T20:28:31.338867Z"Liu, Zhisu"https://zbmath.org/authors/?q=ai:liu.zhisu"Lou, Yijun"https://zbmath.org/authors/?q=ai:lou.yijun"Zhang, Jianjun"https://zbmath.org/authors/?q=ai:zhang.jianjunSummary: By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form
\[
-\left( a+ b\int_{\mathbb{R}^3}|\nabla u|^2\right) \Delta{u}+V(x)u=f(u),\,\,x\in \mathbb{R}^3,
\]
where \(a,b>0\) are constants, \(V\in C(\mathbb{R}^3,\mathbb{R}), f\in C(\mathbb{R},\mathbb{R})\). The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on \(f\) nor the coercivity condition on \(V\) is required. Our result improves the study made by \textit{Y. Deng} et al. [J. Funct. Anal. 269, No. 11, 3500--3527 (2015; Zbl 1343.35081)] in the sense that, in the present paper, the nonlinearities include the power-type case \(f(u)=|u|^{p-2}u\) for \(p\in (2,4)\), in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, \textit{energy doubling} is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small \(b>0\).The singular Kirchhoff equation under small tensionhttps://zbmath.org/1491.352202022-09-13T20:28:31.338867Z"Montenegro, Marcelo"https://zbmath.org/authors/?q=ai:montenegro.marceloSummary: We show existence of a positive solution for the Kirchhoff equation with small tension forces and general nonlinearities, including singular terms. We use an approximation scheme of Galerkin.Generalized quasilinear equations with sign-changing unbounded potentialhttps://zbmath.org/1491.352212022-09-13T20:28:31.338867Z"Oliveira, José Carlos jun."https://zbmath.org/authors/?q=ai:oliveira.jose-carlos-jun"Moreira, S. I."https://zbmath.org/authors/?q=ai:moreira.sandra-imaculadaSummary: We are interested in studying the existence of solution for the generalized quasilinear Schrödinger equation:
\[
\begin{aligned}
&-\mathrm{div}(g^2(u) \nabla u)+g(u)g^\prime (u) |\nabla u|^2 + V(x)u = h(x,u), \\
&u \in H^1(\mathbb{R}^N)
\end{aligned} \tag{\text{P}}
\] in \(\mathbb{R}^N\), where \(N \geq 3\), \(g\) and \(h\) are suitable smooth functions and \(V\) is a potential that may change sign and be unbounded. By using a change of variables and variational arguments, we obtain the existence of a nontrivial solution for the given problem.Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces via Leray-Schauder's nonlinear alternativehttps://zbmath.org/1491.352222022-09-13T20:28:31.338867Z"Sbai, Abdelaaziz"https://zbmath.org/authors/?q=ai:sbai.abdelaaziz"El Hadfi, Youssef"https://zbmath.org/authors/?q=ai:el-hadfi.youssef"Srati, Mohammed"https://zbmath.org/authors/?q=ai:srati.mohammed"Aboutabit, Noureddine"https://zbmath.org/authors/?q=ai:aboutabit.noureddineSummary: In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm
\[
\begin{cases}
-M\Big(\int_{\Omega}\varPhi(|\nabla u|)dx\Big)\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u) \quad \text{in }\Omega,\\
u=0 \quad \text{on } \partial\Omega,
\end{cases}
\]
where \(\Omega\) is a bounded subset in \(\mathbb{R}^N\), \(N\geq 1\) with Lipschitz boundary \(\partial\Omega\). The used technical approach is mainly based on Leray-Shauder's non linear alternative.Nehari-Pohožaev-type ground state solutions of Kirchhoff-type equation with singular potential and critical exponenthttps://zbmath.org/1491.352232022-09-13T20:28:31.338867Z"Su, Yu"https://zbmath.org/authors/?q=ai:su.yu.2|su.yu|su.yu.4|su.yu.1|su.yu.3"Liu, Senli"https://zbmath.org/authors/?q=ai:liu.senliSummary: In this paper, we focus on a Kirchhoff-type equation with singular potential and critical exponent. By virtue of the generalized version of Lions-type theorem and the Nehari-Pohožaev manifold, we established the existence of Nehari-Pohožaev-type ground state solutions to the mentioned equation. Some recent results from the literature are generally improved and extended.Three solutions for a new Kirchhoff-type problemhttps://zbmath.org/1491.352242022-09-13T20:28:31.338867Z"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.6"Wei, Qi-Ping"https://zbmath.org/authors/?q=ai:wei.qiping"Suo, Hong-Min"https://zbmath.org/authors/?q=ai:suo.hong-minSummary: This article concerns on the existence of multiple solutions for a Kirchhoff-type problem with positive and negative modulus. By applying the variational methods and algebraic analysis, we prove that there exist the only three solutions when the parameter is absolutely small than a constant, only two solutions when the parameter is absolutely equals with the constant and an unique solution when the parameter is absolutely greater than the constant. Moreover, we use the algebraic analysis to calculating the constant with the help of one of the Mountain Pass Lemma, Ekeland variational principle, and Minimax principle.Ground state solutions for modified quasilinear Schrödinger equations coupled with the Chern-Simons gauge theoryhttps://zbmath.org/1491.352252022-09-13T20:28:31.338867Z"Xiao, Yingying"https://zbmath.org/authors/?q=ai:xiao.yingying"Zhu, Chuanxi"https://zbmath.org/authors/?q=ai:zhu.chuanxi"Chen, Jianhua"https://zbmath.org/authors/?q=ai:chen.jianhuaSummary: In this paper, we study the following modified quasilinear Schrödinger equation
\[
\begin{aligned}
&-\Delta u + V(x)u- \kappa u \Delta (u^2)+q \frac{h^2(|x|)}{|x|^2}(1+\kappa u^2)u \\
& \quad +q \left(\int_{|x|}^{+\infty} \frac{h(s)}{s}(2+\kappa u^2 (s))u^2 (s) \mathrm{d}s \right) u = |u|^{p-2} u \quad \text{in } \mathbb{R}^2.
\end{aligned}
\] where \(\kappa>0\), \(q>0\), \(p>6\) and \(V \in \mathcal{C}(\mathbb{R}^2, \mathbb{R})\). We develop a more direct and simpler approach to prove the existence of ground state solutions. Our method is based on the Pohožaev type identity and monotone trick.On critical variable-order Kirchhoff type problems with variable singular exponenthttps://zbmath.org/1491.352262022-09-13T20:28:31.338867Z"Zuo, Jiabin"https://zbmath.org/authors/?q=ai:zuo.jiabin"Choudhuri, Debajyoti"https://zbmath.org/authors/?q=ai:choudhuri.debajyoti"Repovš, Dušan D."https://zbmath.org/authors/?q=ai:repovs.dusan-dSummary: We establish a continuous embedding \(W^{s (\cdot), 2}(\Omega) \hookrightarrow L^{\alpha (\cdot)}(\Omega)\), where the variable exponent \(\alpha(x)\) can be close to the critical exponent \(2_s^\ast(x) = \frac{2N}{N - 2 \overline{s}(x)}\), with \(\overline{s}(x) = s(x, x)\) for all \(x \in \overline{\Omega}\). Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponent. Moreover, we also obtain the uniform \(L^\infty\)-estimate of these infinite solutions by a bootstrap argument.Existence of solution for nonlinear elliptic inclusion problems with degenerate coercivity and \(L^1\)-datahttps://zbmath.org/1491.352272022-09-13T20:28:31.338867Z"Akdim, Youssef"https://zbmath.org/authors/?q=ai:akdim.youssef"Belayachi, Mohammed"https://zbmath.org/authors/?q=ai:belayachi.mohammed"Ouboufettal, Morad"https://zbmath.org/authors/?q=ai:ouboufettal.moradSummary: This paper is concerned with the study of the existence results to the general class of nonlinear elliptic problem associated with the differential inclusion having degenerate coercivity, whose prototype is giving by:
\[
\begin{cases}
\beta (u)-div \,\left( \dfrac{|\nabla u|^{p-2} \nabla u }{(1+|u|)^{\theta (p-1)}}\right) \ni f &\text{in } \Omega, \\
u=0 & \text{on} \, \partial \Omega,
\end{cases}
\] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^N\) (\(N\ge 2\)) with sufficiently smooth boundary \(\partial \Omega\), \(0 \le \theta < 1\), \(1<p<N\), \(\beta\) is a maximal monotone mapping such that \(0 \in \beta (0)\) and the right hand side \(f\) is assumed to belong to \(L^1(\Omega )\). We show the existence of entropy solutions for this non-coercive differential inclusion and we will conclude some regularity results.Multiple solutions for \(\Delta_\gamma \)-Laplace problems without the ambrosetti-Rabinowitz conditionhttps://zbmath.org/1491.352282022-09-13T20:28:31.338867Z"Luyen, Duong Trong"https://zbmath.org/authors/?q=ai:luyen.duong-trongA variational principle for pulsating standing waves and an Einstein relation in the sharp interface limithttps://zbmath.org/1491.352292022-09-13T20:28:31.338867Z"Morfe, Peter S."https://zbmath.org/authors/?q=ai:morfe.peter-sSummary: This paper investigates the connection between the effective, large scale behavior of Allen-Cahn functionals with periodic coefficients and the sharp interface limit of the associated \(L^2\) gradient flows. By introducing a Percival-type Lagrangian in the cylinder \(\mathbb{R} \times\mathbb{T}^d\), we establish a link between the \(\Gamma\)-convergence results of Anisini, Braides, and Chiadò Piat and the sharp interface limit results of Barles and Souganidis. In laminar media, we prove a sharp interface limit in a graphical setting, making no assumptions other than sufficient smoothness of the coefficients, and we prove that the effective interface velocity and surface tension satisfy an Einstein relation. A number of pathologies are presented to highlight difficulties that do not arise in the spatially homogeneous setting.The Neumann problem for a multidimensional elliptic equation with several singular coefficients in an infinite domainhttps://zbmath.org/1491.352302022-09-13T20:28:31.338867Z"Ergashev, T. G."https://zbmath.org/authors/?q=ai:ergashev.tuhtasin-gulamjanovich|ergashev.tukhtasin-gulamzhanovich"Tulakova, Z. R."https://zbmath.org/authors/?q=ai:tulakova.z-rSummary: In this article the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain is studied. Using the method of the integral energy, the uniqueness of solution is proved. In proof of existence of the explicit solution of the Neumann problem a differentiation formula, some adjacent and limiting relations for the Lauricella hypergeometric functions and the values of some multidimensional improper integrals are used.Fractional Kirchhoff Hardy problems with singular and critical Sobolev nonlinearitieshttps://zbmath.org/1491.352312022-09-13T20:28:31.338867Z"Fiscella, Alessio"https://zbmath.org/authors/?q=ai:fiscella.alessio"Mishra, Pawan Kumar"https://zbmath.org/authors/?q=ai:mishra.pawan-kumarSummary: The paper deals with the following singular fractional problem
\[
\begin{aligned}
\begin{cases}
M\left(\displaystyle \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right) (-\Delta)^s u-\mu \displaystyle \frac{u}{|x|^{2s}}= \lambda f(x)u^{-\gamma}+ g(x){u^{2^*_s-1}}\quad &\text{in } \Omega,\\
u>0 \quad &\text{in } \Omega,\\
u=0 \quad &\text{in } \mathbb{R}^N\setminus \Omega,
\end{cases}
\end{aligned}
\]
where \(\Omega \subset\mathbb{R}^N\) is an open bounded domain, with \(0\in \Omega\), dimension \(N>2s\) with \(s\in (0,1)\), \(2^*_s=2N/(N-2s)\) is the fractional critical Sobolev exponent, \(\lambda\) and \(\mu\) are positive parameters, exponent \(\gamma \in (0,1)\), \(M\) models a Kirchhoff coefficient, \(f\) is a positive weight while \(g\) is a sign-changing function. The main feature and novelty of our problem is the combination of the critical Hardy and Sobolev nonlinearities with the bi-nonlocal framework and a singular nondifferentiable term. By exploiting the Nehari manifold approach, we provide the existence of at least two positive solutions.Extensions of symmetric operators that are invariant under scaling and applications to indicial operatorshttps://zbmath.org/1491.352322022-09-13T20:28:31.338867Z"Krainer, Thomas"https://zbmath.org/authors/?q=ai:krainer.thomasSummary: Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural scaling invariance property with respect to dilations of the radial variable. In this paper we discuss extensions of symmetric indicial operators from a functional analytic point of view. In the first, purely abstract part of this paper, we consider a general unbounded symmetric operator that exhibits invariance with respect to an abstract scaling action on a Hilbert space, and we describe its extensions in terms of generalized eigenspaces of the infinitesimal generator of this action. Among others, we obtain a Green formula for the adjoint pairing, an algebraic formula for the signature, and in the semibounded case explicit descriptions of the Friedrichs and Krein extensions. In the second part we consider differential operators of Fuchs type on the half axis with unbounded operator coefficients that are invariant under dilation, and show that under suitable ellipticity assumptions on the indicial family these operators fit into the framework of the first part, which in this case furnishes a description of extensions in terms of polyhomogeneous asymptotic expansions. We also obtain an analytic formula for the signature of the adjoint pairing in terms of the spectral flow of the indicial family for such operators.Singular anisotropic elliptic problems with variable exponentshttps://zbmath.org/1491.352332022-09-13T20:28:31.338867Z"Naceri, Mokhtar"https://zbmath.org/authors/?q=ai:naceri.mokhtarSummary: In this paper, we prove the existence and regularity results of positive solutions for anisotropic elliptic problems with variable exponents and a singular nonlinearity having also a variable exponent. The functional setting involves anisotropic Sobolev spaces with variable exponents.A metric potential capacity: some qualitative properties of Schrödinger's equations with a non negative potentialhttps://zbmath.org/1491.352342022-09-13T20:28:31.338867Z"Rakotoson, Jean Michel"https://zbmath.org/authors/?q=ai:rakotoson.jean-michelSummary: We introduce a new notion that we call a metric potential capacity associated to a positive potential \(V\) and a weak-star topology of a suitable Banach space. We show among other things that if the metric potential capacity of a closed set \(F\) is zero and the square root of the potential is locally \(q\)-integrable for some \(q\in [1,+\infty[\), then the usual Sobolev capacity in \(W^{1, q}\) of the set \(F\) is zero. Properties of potential capacities are presented and applications to Schrödinger type equations are also made with very singular potentials including the modified Pösh-Teller potential \(V_\alpha(x)=\Bigl|\sin |x|\Bigr|^{-\alpha}\), \(\alpha >0\). We discuss about existence and non existence of a solution.Existence and regularity results for nonlinear anisotropic unilateral elliptic problems with degenerate coercivityhttps://zbmath.org/1491.352352022-09-13T20:28:31.338867Z"Benaichouche, Noureddine"https://zbmath.org/authors/?q=ai:benaichouche.noureddine"Ayadi, Hocine"https://zbmath.org/authors/?q=ai:ayadi.hocine"Mokhtari, Fares"https://zbmath.org/authors/?q=ai:mokhtari.fares"Hakem, Ali"https://zbmath.org/authors/?q=ai:hakem.aliSummary: In this paper we prove some existence and regularity results for a class of nonlinear anisotropic unilateral problems with degenerate coercivity by using the penalty method. Our results are a natural generalization of some existing ones in the context of isotropic exponents.Ground state solution for a class of Choquard equation with indefinite periodic potentialhttps://zbmath.org/1491.352362022-09-13T20:28:31.338867Z"Chen, Fulai"https://zbmath.org/authors/?q=ai:chen.fulai"Liao, Fangfang"https://zbmath.org/authors/?q=ai:liao.fangfang"Geng, Shifeng"https://zbmath.org/authors/?q=ai:geng.shifengSummary: This paper is concerned with a class of nonlinear Choquard equation with potential. We assume the potential satisfies general indefinite periodic condition, so the Schrödinger operator has purely continuous spectrum and the associate energy functional is strongly indefinite. Applying the generalized Nehari manifold method developed by Szulkin and Weth, we prove the existence of ground state solution.Asymptotic expansions for singular solutions of conformal Q-curvature equationshttps://zbmath.org/1491.352372022-09-13T20:28:31.338867Z"Guo, Zongming"https://zbmath.org/authors/?q=ai:guo.zongming"Liu, Zhongyuan"https://zbmath.org/authors/?q=ai:liu.zhongyuan"Wan, Fangshu"https://zbmath.org/authors/?q=ai:wan.fangshuThe authors consider the following conformally invariant equation of fourth order
\[
\begin{cases}
\Delta^2 u=e^u&\text{in }B\backslash\{0\},\\
\int_{B\backslash\{0\}}e^{u(x)}\,dx<\infty \end{cases}\tag{1}
\]
where \(B=\{x\in \mathbb{R}^4: |x|<1\}\).
The authors establish asymptotic expansions at an isolated singular point of solutions to the \(Q\)-curvature equation (1). Furthermore, they derive the asymptotic behavior near an isolated singularity of conformally flat metrics \(g_w = e^{2w}|dx|^2\) with constant positive \(Q\)-curvature, where \(|dx|^2\) is the Euclidean metric.
They also obtain the asymptotic expansions at \(|x|\rightarrow\infty\) of solutions \(u\in C^4(\mathbb{R}^4\backslash{\overline B})\) to the problem
\[
\begin{cases}
\Delta^2 u=e^u\quad\mbox{in}\,\,\mathbb{R}^4\backslash{\overline B},\\
\int_{\mathbb{R}^4\backslash{\overline B}}e^{u(x)}\,dx<\infty. \end{cases}
\]
Reviewer: Said El Manouni (Berlin)Existence of solutions to Chern-Simons-Higgs equations on graphshttps://zbmath.org/1491.352382022-09-13T20:28:31.338867Z"Hou, Songbo"https://zbmath.org/authors/?q=ai:hou.songbo"Sun, Jiamin"https://zbmath.org/authors/?q=ai:sun.jiaminSummary: Let \(G=(V, E)\) be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation
\[
\Delta u = -\lambda e^{g(u)}\left(e^{g(u)} - 1\right)^2 + 4\pi\sum\limits_{j=1}^N\delta_{p_j}
\]
on \(G\), where \(\lambda\) is a positive constant; \(g(u)\) is the inverse function of \(u = f(\upsilon) = 1 + \upsilon - e^\upsilon\) on \((-\infty, 0]\); \(N\) is a positive integer; \(p_1, p_2, \dots, p_N\) are distinct vertices of \(V\) and \(\delta_{p_j}\) is the Dirac delta mass at \(p_j\). We prove that there is critical value \(\lambda_c\) such that the generalized Chern-Simons-Higgs equation has a solution if and only if \(\lambda \ge \lambda_c\). We also prove the existence of solutions to the Chern-Simons-Higgs equation \[
\Delta u = \lambda e^u(e^u-1)+4\pi \sum \limits_{j=1}^N\delta_{p_j}
\]
on \(G\) when \(\lambda\) takes the critical value \(\lambda_c\) and this completes the results of \textit{A. Huang} et al. [Commun. Math. Phys. 377, No. 1, 613--621 (2020; Zbl 1447.35338)].Multiple Delaunay ends solutions of the Cahn-Hilliard equationhttps://zbmath.org/1491.352392022-09-13T20:28:31.338867Z"Kowalczyk, Michał"https://zbmath.org/authors/?q=ai:kowalczyk.michal"Rizzi, Matteo"https://zbmath.org/authors/?q=ai:rizzi.matteoThe authors study the Cahn-Hilliard equation, which arises from the phase transition theory, describing a geometric approach based on Lyapunov-Schmidt reduction for obtaining new solutions.
Reviewer: Dumitru Motreanu (Perpignan)Sign-changing solutions to the critical Choquard equationhttps://zbmath.org/1491.352402022-09-13T20:28:31.338867Z"Luo, Xiaorong"https://zbmath.org/authors/?q=ai:luo.xiaorong"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anminSummary: We consider the following Choquard problem
\[
\begin{cases}
- \Delta u = \lambda u + f (x) (|x|^{- \mu} \ast |u|^{2_\mu^\ast}) |u|^{2_\mu^\ast - 2} u &\text{in } \Omega, \\
u = 0 & \text{on } \partial \Omega,
\end{cases}
\] where \(0 < \lambda < \lambda_1 (\Omega)\), \(|\Omega| < \infty\). \(2_\mu^\ast\) is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We prove the existence of sign-changing solution by combining Nehari manifold method with the Ljusternik-Schnirelman theory. The main results extend and complement the earlier works [\textit{L. Guo} et al., Calc. Var. Partial Differ. Equ. 58, No. 4, Paper No. 128, 34 p. (2019; Zbl 1422.35077); \textit{V. Moroz} and \textit{J. Van Schaftingen}, Commun. Contemp. Math. 17, No. 5, Article ID 1550005, 12 p. (2015; Zbl 1326.35109)].Existence and non-existence results for the higher order Hardy-Hénon equations revisitedhttps://zbmath.org/1491.352412022-09-13T20:28:31.338867Z"Ngô, Qu\^óc Anh"https://zbmath.org/authors/?q=ai:ngo-quoc-anh."Ye, Dong"https://zbmath.org/authors/?q=ai:ye.dong.2|ye.dong.1Summary: This paper is devoted to the study of non-negative, non-trivial (classical, punctured, or distributional) solutions to higher order Hardy-Hénon equations
\[
(-\Delta)^mu=|x|^\sigma u^p
\]
in \(\mathbb{R}^n\) with \(p>1\). We show that the condition
\[
n-2m-\frac{2m+\sigma}{p-1}>0
\]
is necessary for the existence of distributional solution. For \(n\geq 2m\) and \(\sigma>-2m\), we prove that any distributional solution satisfies an integral equation and weak super polyharmonic properties. We establish also some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if \(n\geq 2m\), \(\sigma>-2m\), there is no non-negative, non-trivial classical solution to the equation if
\[
1<p<\frac{n+2m+2\sigma}{n-2m}=:p_{\mathsf{S}}(m,\sigma);
\]
and classical positive radial solutions exist for \(n>2m\), \(\sigma>-2m\) and \(p\geq p_{\mathsf{S}}(m,\sigma)\). Our approach is very different from most previous works on this subject, which enables us to have more understanding of distributional solutions, to get sharp results, hence closes several open questions.Local behavior of solutions to a biharmonic equation with isolated singularityhttps://zbmath.org/1491.352422022-09-13T20:28:31.338867Z"Wu, Ke"https://zbmath.org/authors/?q=ai:wu.ke"Yao, Ruofei"https://zbmath.org/authors/?q=ai:yao.ruofeiSummary: This paper is concerned with positive classical solutions \(u\) of the biharmonic equation \(\Delta^2 u + u^p = 0\) in the punctured ball \(B_2 \setminus \{0\}\) of \(\mathbb{R}^n\), where \(n \geq 5\) and \(p >(n + 1) /(n - 3)\). We classify the isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for \(- \Delta u\).Isolated singularities of solutions to the Yamabe equation in dimension 6https://zbmath.org/1491.352432022-09-13T20:28:31.338867Z"Xiong, J."https://zbmath.org/authors/?q=ai:xiong.jiayuan|xiong.junda|xiong.jinzhi|xiong.jinbo|xiong.juxia|xiong.junjie|xiong.jie.1|xiong.junjing|xiong.jiafeng|xiong.junjiang|xiong.jiecheng|xiong.jing|xiong.jiangen|xiong.jinhua|xiong.juntao|xiong.juhua|xiong.jiangang|xiong.jianxin|xiong.jicheng|xiong.jinquan|xiong.jiaming|xiong.jinghong|xiong.jianyi|xiong.junqing|xiong.jason|xiong.jin|xiong.jiabing|xiong.jun|xiong.jay|xiong.jingang|xiong.jinbiao|xiong.jingjing|xiong.jundi|xiong.jiawen|xiong.jijun|xiong.jianhua|xiong.jiawei|xiong.jintao|xiong.jiang|xiong.jingting|xiong.jie|xiong.jiajun|xiong.jiaojiao|xiong.junlin|xiong.junfeng|xiong.jingqi|xiong.joshua|xiong.jiangtao|xiong.jianfei|xiong.jingwei|xiong.juan|xiong.jiguang|xiong.jinsong|xiong.jiandong|xiong.jia|xiong.jian|xiong.jiechao|xiong.jinjun|xiong.jingyi|xiong.jincheng|xiong.jiping"Zhang, L."https://zbmath.org/authors/?q=ai:zhang.lanzhu|zhang.lingwei|zhang.lianjun|zhang.lupeng|zhang.liancheng|zhang.li.1|zhang.lei|zhang.lingxia|zhang.linfeng|zhang.liyuan|zhang.lipai|zhang.lingjie|zhang.lai|zhang.luyin|zhang.linran|zhang.liwen|zhang.li.3|zhang.lei.9|zhang.lisha|zhang.lianzeng|zhang.lingfu|zhang.lingchen|zhang.letian|zhang.leigang|zhang.liuyang|zhang.li.5|zhang.lanfang|zhang.li.9|zhang.lin.1|zhang.laping|zhang.longting|zhang.lumei|zhang.lizao|zhang.li.12|zhang.lei.17|zhang.liangqi|zhang.li.2|zhang.lishi|zhang.ludan|zhang.limin|zhang.leilei|zhang.lixian|zhang.liguo|zhang.longjun|zhang.lige|zhang.lijie|zhang.lianzhen|zhang.linjie|zhang.liangxiu|zhang.lei.4|zhang.lingqin|zhang.likun|zhang.linzhong|zhang.lansheng|zhang.liwei.1|zhang.liangxin|zhang.linhua|zhang.lei.5|zhang.lieping|zhang.liangjin|zhang.linru|zhang.lei.21|zhang.lianyang|zhang.lian|zhang.liuliu|zhang.lifei|zhang.linqiao|zhang.linyun|zhang.leon|zhang.lianhai|zhang.laihui|zhang.liangyong|zhang.linlan|zhang.lingyi|zhang.lirong|zhang.lihui|zhang.linwen|zhang.lefei|zhang.liang|zhang.linnan|zhang.linzi|zhang.lizhao|zhang.li.7|zhang.lianhong|zhang.lianzhu|zhang.liqiao|zhang.liaojun|zhang.lanlan|zhang.li|zhang.longbing|zhang.liangchi|zhang.linghong|zhang.lianyi|zhang.lingyuan|zhang.linmiao|zhang.luojia|zhang.ledi|zhang.linda|zhang.lichun|zhang.luying|zhang.lingbo|zhang.leiwu|zhang.liao|zhang.lijiang|zhang.lun|zhang.lequan|zhang.lingxin|zhang.luyao|zhang.liantang|zhang.lequn|zhang.lefeng|zhang.lida|zhang.lanhua|zhang.luzou|zhang.liangzhe|zhang.lingye|zhang.laiping|zhang.liren|zhang.lifeng|zhang.lizhu|zhang.li.10|zhang.long|zhang.lanhui|zhang.lei.20|zhang.leike|zhang.lingjuan|zhang.lizou|zhang.lei.25|zhang.lidong|zhang.lide|zhang.lin|zhang.liangyun|zhang.lisheng|zhang.linbo|zhang.luoping|zhang.liangliang|zhang.liting|zhang.luchan|zhang.lianfang|zhang.lyuyuan|zhang.libiao|zhang.lijia|zhang.lilin|zhang.longteng|zhang.li.4|zhang.lixing|zhang.linjing|zhang.lipeng|zhang.liming|zhang.luwan|zhang.lunkai|zhang.liangwei|zhang.lingxi|zhang.lei.2|zhang.li-xin|zhang.louzin|zhang.licen|zhang.louxin|zhang.liqian|zhang.lijing|zhang.lei.11|zhang.lanju|zhang.lingzhong|zhang.lianmin|zhang.liyang|zhang.linyan|zhang.lifan|zhang.lingyu|zhang.lianzheng|zhang.liling|zhang.liquing|zhang.lihe|zhang.li.6|zhang.lingmin|zhang.lianying|zhang.lijuan|zhang.lingling|zhang.lang|zhang.lanxia|zhang.liang.2|zhang.lianwen|zhang.lufang|zhang.longbin|zhang.lei.13|zhang.lili|zhang.litao|zhang.lei.24|zhang.longyao|zhang.lingming|zhang.leishi|zhang.lipan|zhang.lintao|zhang.leyou|zhang.lilun|zhang.libang|zhang.luping|zhang.liyan|zhang.lanhong|zhang.lixiu|zhang.lianhe|zhang.lei-hong|zhang.leiming|zhang.liang.3|zhang.liufeng|zhang.longfei|zhang.lizhong|zhang.linwan|zhang.liyou|zhang.linqing|zhang.linxi|zhang.lichuan|zhang.liu|zhang.lingxian|zhang.liwei|zhang.luning|zhang.liangquan|zhang.liangyue|zhang.linxia|zhang.lufeng|zhang.longchuan|zhang.lixiang|zhang.lipu|zhang.lv|zhang.linfen|zhang.linrang|zhang.lianping|zhang.lingqian|zhang.lichao|zhang.lingying|zhang.lingrui|zhang.libing|zhang.lezhong|zhang.lei.10|zhang.luo|zhang.liangying|zhang.lilong|zhang.liang.1|zhang.lanling|zhang.lingyue|zhang.lukun|zhang.liuhua|zhang.liping|zhang.lianzhong|zhang.liangrui|zhang.longbo|zhang.libao|zhang.liya|zhang.lei.15|zhang.luyu|zhang.lianming|zhang.lejun|zhang.lijun|zhang.lingyun|zhang.liping.1|zhang.liqiong|zhang.longhui|zhang.lin.3|zhang.liuping|zhang.lei.16|zhang.liuyue|zhang.lei.22|zhang.liangdi|zhang.lei.23|zhang.liuwei|zhang.leyan|zhang.longsheng|zhang.lingping|zhang.lele|zhang.lianshui|zhang.linchuang|zhang.landing|zhang.licheng|zhang.longxin|zhang.laixi|zhang.liangjun|zhang.liying|zhang.lianmei|zhang.liangbin|zhang.liqun|zhang.ling|zhang.lihua|zhang.linan|zhang.lixi|zhang.liangzhong|zhang.liehui|zhang.linke|zhang.libo|zhang.linna|zhang.liyu|zhang.lingsong|zhang.liqiang|zhang.ligang|zhang.liqin|zhang.liqi|zhang.lichen|zhang.lvyuan|zhang.longge|zhang.liabiao|zhang.lejie|zhang.liangpei|zhang.lianmeng|zhang.liyong|zhang.liangyin|zhang.lixun|zhang.lihai|zhang.likai|zhang.liye|zhang.lihao|zhang.le|zhang.lei.14|zhang.langwen|zhang.lu|zhang.li.11|zhang.lixuan|zhang.lifu|zhang.lifa|zhang.laiwu|zhang.lingchuan|zhang.liquan|zhang.lei.1|zhang.lisa|zhang.leying|zhang.lixia|zhang.lanyong|zhang.lingfeng|zhang.lingjun|zhang.lingyan|zhang.lunchuan|zhang.linjun|zhang.lingmei|zhang.laicheng|zhang.linghua|zhang.liuqing|zhang.liangchao|zhang.luyang|zhang.limei|zhang.luona|zhang.lingxiang|zhang.lixin|zhang.lihong|zhang.lantian|zhang.liangcheng|zhang.lining|zhang.lichi|zhang.longtian|zhang.linjuan|zhang.linjia|zhang.lianghao|zhang.legui|zhang.luchao|zhang.luyi|zhang.li.8|zhang.liyi|zhang.lingqi|zhang.liansheng|zhang.lianfu|zhang.lianyong|zhang.lina|zhang.lailiang|zhang.lukai|zhang.libin|zhang.lei.18|zhang.lianfeng|zhang.lifang|zhang.linyuan|zhang.liyun|zhang.lufei|zhang.lingmi|zhang.lei.7|zhang.luming|zhang.lizhen|zhang.lianhua|zhang.lidan|zhang.lyuou|zhang.lingli|zhang.lianxing|zhang.leping|zhang.linlin|zhang.lingfan|zhang.linghai|zhang.linyang|zhang.liqing|zhang.linli|zhang.lijian|zhang.longxiang|zhang.liumei|zhang.linmeng|zhang.lanyu|zhang.lilian|zhang.longjie|zhang.lixu|zhang.lie|zhang.liupiao|zhang.letao|zhang.liandi|zhang.laobing|zhang.lan|zhang.liruo|zhang.lin.2|zhang.lulu|zhang.linsen|zhang.liangcai|zhang.lizhiSummary: We study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not conformally flat. We prove that, in dimension 6, any solution is asymptotically close to a Fowler solution, which is an extension of the same result for lower dimensions by \textit{F. C. Marques} [Calc. Var. Partial Differ. Equ. 32, No. 3, 349--371 (2008; Zbl 1143.35323)].On \(p(z)\)-Laplacian system involving critical nonlinearitieshttps://zbmath.org/1491.352442022-09-13T20:28:31.338867Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Bennouna, Jaouad"https://zbmath.org/authors/?q=ai:bennouna.jaouad"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandra(no abstract)Harnack inequality for elliptic \((p, q)\)-Laplacian with partially Muckenhoupt weighthttps://zbmath.org/1491.352452022-09-13T20:28:31.338867Z"Aliyev, M. J."https://zbmath.org/authors/?q=ai:aliyev.mushviq-j"Alkhutov, Yu. A."https://zbmath.org/authors/?q=ai:alkhutov.yuriy-alexandrovich"Tikhomirov, R. N."https://zbmath.org/authors/?q=ai:tikhomirov.r-nSummary: We prove the Harnack inequality for nonnegative solutions to an equation with \(p(x)\)-Laplacian and partially Muckenhoupt weight, where \(p(x)\) takes two constant values \(p\) and \(q\), the phase interface is a hyperplane, and the weight satisfies the Muckenhoupt \(A_p\)-condition in one part and the Muckenhoupt \(A_q\)-condition in another part of the domainExistence of weak solutions for some local and nonlocal \(p\)-Laplacian problemhttps://zbmath.org/1491.352462022-09-13T20:28:31.338867Z"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Hilal, Khalid"https://zbmath.org/authors/?q=ai:hilal.khalid"Ait Temghart, Said"https://zbmath.org/authors/?q=ai:ait-temghart.saidSummary: We study the existence of weak solutions for some elliptic \(p\)-Laplacian problems in the case where \(p\) depends on the solution itself. We consider two situations, when \(p\) is a local and nonlocal quantity. The main aim of this paper is to extend the results established by M. \textit{Chipot} and \textit{H. B. de Oliveira} [Math. Ann. 375, No. 1--2, 283--306 (2019; Zbl 1430.35106); correction ibid. 375, No. 1--2, 307--313 (2019)].Comparison results for solutions to \(p\)-Laplace equations with Robin boundary conditionshttps://zbmath.org/1491.352472022-09-13T20:28:31.338867Z"Amato, Vincenzo"https://zbmath.org/authors/?q=ai:amato.vincenzo.1"Gentile, Andrea"https://zbmath.org/authors/?q=ai:gentile.andrea"Masiello, Alba Lia"https://zbmath.org/authors/?q=ai:masiello.alba-liaSummary: In the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in
[\textit{A. Alvino, C. Nitsch, C. Trombetti} and \textit{ A. Talenti }, ``Comparison result for solutions to elliptic problems with Robin boundary conditions'', to appear in Commun. Pure Appl. Math.]
to the case of \(p\)-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in Alvino et al. [loc. cit.] only in the planar case, holds true in any dimension if \(p\) is sufficiently small.Existence results for fractional \(p\)-Laplacian systems via Young measureshttps://zbmath.org/1491.352482022-09-13T20:28:31.338867Z"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farah"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussineSummary: In this paper, we show the existence result of the following fractional \(p\)-Laplacian system
\[
(-\varDelta)^s_pu=f(x,u)\quad\text{in }\varOmega,\quad u=0\quad\text{in }\mathbb{R}^n\setminus\varOmega
\]
for a given datum \(f\). The existence of weak solutions is obtained by using the theory of Young measures.On the strong comparison principle for degenerate elliptic problems with convectionhttps://zbmath.org/1491.352492022-09-13T20:28:31.338867Z"Benedikt, Jiří"https://zbmath.org/authors/?q=ai:benedikt.jiri"Girg, Petr"https://zbmath.org/authors/?q=ai:girg.petr"Kotrla, Lukáš"https://zbmath.org/authors/?q=ai:kotrla.lukas"Takáč, Peter"https://zbmath.org/authors/?q=ai:takac.peterSummary: The weak and strong comparison principles (\textit{\textbf{WCP}} and \textit{\textbf{SCP}}, respectively) are investigated for quasilinear elliptic boundary value problems with the \(p\)-Laplacian in one space dimension, \(\Delta_p(u) \overset{{def}}{=} \frac{{d}}{{d}x} (| u^\prime |^{p - 2} u^\prime )\). We treat the ``degenerate'' case of \(2 < p < \infty\) and allow also for the nontrivial \(\textbf{convection velocity }b : [- 1, 1] \to \mathbb{R}\) in the underlying domain \(\Omega = (- 1, 1)\). We establish the \textit{\textbf{WCP}} under a rather general, ``natural sufficient condition'' on the convection velocity, \(b(x)\), and the reaction function, \( \varphi(x, u)\). Furthermore, we establish also the \textit{\textbf{SCP}} under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the \textit{\textbf{SCP}} and discuss their applications to an interesting classical problem of fluid flow in porous medium, \textit{``seepage flow of fluids in inclined bed''}. Our methods are based on a mixture of classical and new techniques.Necessary condition in a Brezis-Oswald-type problem for mixed local and nonlocal operatorshttps://zbmath.org/1491.352502022-09-13T20:28:31.338867Z"Biagi, Stefano"https://zbmath.org/authors/?q=ai:biagi.stefano"Mugnai, Dimitri"https://zbmath.org/authors/?q=ai:mugnai.dimitri"Vecchi, Eugenio"https://zbmath.org/authors/?q=ai:vecchi.eugenioSummary: In this note we complete the study started in [the authors, ``A Brezis-Oswald approach for mixed local and nonlocal operators'', Preprint, \url{arXiv:2103.11382}] providing a full characterization of the existence of a unique positive weak solution of a \(p\)-sublinear Dirichlet boundary value problem driven by a mixed local-nonlocal operator.Steklov eigenvalues problems for generalized \((p,q)\)-Laplacian type operatorshttps://zbmath.org/1491.352512022-09-13T20:28:31.338867Z"Boukhsas, Abdelmajid"https://zbmath.org/authors/?q=ai:boukhsas.abdelmajid"Ouhamou, Brahim"https://zbmath.org/authors/?q=ai:ouhamou.brahimSummary: In this paper, we study the following class of \((p,q)\) elliptic problems under Steklov-type boundary conditions
\[
\begin{cases}
-\mathrm{div}\big(a(|\nabla u|^p)|\nabla u|^{p-2}\nabla u\big)+a(|u|^p)|u|^{p-2}u=0 \;\text{in }\Omega, \\
a(|\nabla u|^p)|\nabla u|^{p-2}\nabla u\cdot\nu=\lambda |u|^{m-2}u \;\text{on }\partial\Omega,
\end{cases}
\]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) (\(N \geq 2\)), \(\nu\) is the outward unit normal vector on \(\partial\Omega\), \(2 \leq p < N\), \(m\in\mathbb{R}\) with \(m > 1\) in suitable ranges listed later and \(a\) is a \(C^1\) real function and \(\lambda > 0\) is a real parameter. Using variational methods, we establish the existence of a continuous and unbounded set of positive generalized eigenvalues.Nonexistence of solutions for quasilinear Schrödinger equation in \(\mathbb{R}^N\)https://zbmath.org/1491.352522022-09-13T20:28:31.338867Z"Chen, Lijuan"https://zbmath.org/authors/?q=ai:chen.lijuan"Chen, Caisheng"https://zbmath.org/authors/?q=ai:chen.caisheng"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei"Xiu, Zonghu"https://zbmath.org/authors/?q=ai:xiu.zonghuSummary: In this paper, we study the nonexistence of solutions to the quasilinear Schrödinger equations:
\[
-\Delta_p u + V(x) |u|^{p-2} u - \Delta_p (|u|^{2 \alpha}) |u|^{2 \alpha-2} u = g(x,u), \quad x \in \mathbb{R}^N, \tag{1}
\] where \(p\)-Laplacian operator \(\Delta_p u = \mathrm{div}(|\nabla u|^{p-2} \nabla u)\), \(1<p<N\) and \(\alpha > 1/2\) is a parameter. Under suitable conditions on \(g(x,u)\), the nonexistence of solutions to Equation (1) is investigated.Positive solutions for anisotropic singular Dirichlet problemshttps://zbmath.org/1491.352532022-09-13T20:28:31.338867Z"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Scapellato, Andrea"https://zbmath.org/authors/?q=ai:scapellato.andreaLet \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega\). The authors consider an anisotropic singular Dirichlet problem of the form \[ \begin{cases} -\Delta_{p(z)}u(z)- \Delta_{q(z)}u(z)=\lambda u(z)^{-\eta(z)}+f(z,u(z)) \mbox{ in } \Omega,\\
u|_{\partial \Omega}=0, \, \lambda >0, \, u > 0, \end{cases}\tag{\(\mathrm P_\lambda\)}\] where \(\Delta_{p(z)} u= \operatorname{div}(|D u|^{p(z)-2}D u)\) for all \(u \in W^{1,p(z)}_0(\Omega)\) is the \(p(z)\)-Laplace operator. In the right-hand side, the first (singular) term is parametric with \(\eta \in C(\overline{\Omega})\) and \(0<\eta_-=\min_{\overline{\Omega}} \eta \leq \max_{\overline{\Omega}} \eta =\eta_+<1\) and the second (Caratheodory) term is \((p_+-1)\)-superlinear but does not satisfy the Ambrosetti-Rabinowitz condition.
The authors first study the auxiliary purely singular problem related to \((P_\lambda)\) (that is, they consider \((P_\lambda)\) without the second term in the right-hand side) establishing an existence and uniqueness result. Thus, using this solution, they are able to bypass the singularity in \((P_\lambda)\). Finally, using variational tools from the critical point theory together with truncation and comparison techniques, they establish a bifurcation-type theorem describing the changes in the set of positive solutions to \((P_\lambda)\).
Reviewer: Calogero Vetro (Palermo)On a class of singular anisotropic \((p, q)\)-equationshttps://zbmath.org/1491.352542022-09-13T20:28:31.338867Z"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Winkert, Patrick"https://zbmath.org/authors/?q=ai:winkert.patrickSummary: We consider a Dirichlet problem driven by the anisotropic \((p, q)\)-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.Solvable optimization problems involving a \(p\)-Laplacian type operatorhttps://zbmath.org/1491.352552022-09-13T20:28:31.338867Z"Qiu, Chong"https://zbmath.org/authors/?q=ai:qiu.chong"Yang, Xiaoqi"https://zbmath.org/authors/?q=ai:yang.xiaoqi"Zhou, Yuying"https://zbmath.org/authors/?q=ai:zhou.yuyingSummary: This paper is concerned with maximization and minimization problems related to a boundary value problem involving a \(p\)-Laplacian type operator. These optimization problems are formulated relative to the rearrangement of a fixed function. Firstly, by introducing a truncated function, we establish the existence and uniqueness of the solution of the boundary value problem involving a \(p\)-Laplacian type operator, and then, we show that both optimization problems are solvable under some suitable assumptions. Furthermore, we show that the solution of the minimization problem is unique and has some symmetric property if the domain considered is a ball.Positive solutions for the \(p(x)-\)Laplacian : application of the Nehari methodhttps://zbmath.org/1491.352562022-09-13T20:28:31.338867Z"Taarabti, Said"https://zbmath.org/authors/?q=ai:taarabti.saidSummary: In this paper, we study the existence of positive solutions of the following equation
\[
\begin{cases}-\Delta_{p(x)}u+V(x)|u|^{p(x)-2}u&=\quad\lambda k(x)|u|^{\alpha(x)-2}u\\&+\quad h(x)|u|^{\beta(x)-2}u\quad\text{in}\quad\Omega \\ &u\quad=0\quad\text{on }\partial\Omega.\tag{\(P_\lambda\)}\end{cases}
\]
The study of the problem (\(P_\lambda\)) needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem \((P_\lambda\)) in \(W_0^{1,p(x)}(\Omega)\).Existence of three weak solutions for some singular elliptic problems with Hardy potentialhttps://zbmath.org/1491.352572022-09-13T20:28:31.338867Z"Tavani, Mohammad Reza Heidari"https://zbmath.org/authors/?q=ai:tavani.mohammad-reza-heidari"Khodabakhshi, Mehdi"https://zbmath.org/authors/?q=ai:khodabakhshi.mehdiSummary: In this paper, under growth conditions on the nonlinearity, we obtain the existence of at least three weak solutions for some singular elliptic Dirichlet problems involving the \(p\)-Laplacian, subject to Dirichlet boundary conditions in a smooth bounded domain in \(\mathbb{R}^N\). The approach is based on variational methods and critical point theory.Asymptotic behavior of positive solutions for quasilinear elliptic equationshttps://zbmath.org/1491.352582022-09-13T20:28:31.338867Z"Wang, Biao"https://zbmath.org/authors/?q=ai:wang.biao"Zhang, Zhengce"https://zbmath.org/authors/?q=ai:zhang.zhengceSummary: We systematically inquire positive radial solutions of the quasilinear elliptic equation
\[
-\Delta_p\phi =|x|^\sigma\phi^q
\]
with \(\Delta_p\phi :=\operatorname{div}(|\nabla\phi|^{p-2}\nabla\phi)\) is the \(p\)-Laplace operator, \(q>p-1\), \(\sigma > -p\) and \(1 < p < N\). It is known that the radial solutions of this equation can be classified into three different types: the M-solutions (singular at \(r=0\)), the E-solutions (regular at \(r=0\)) and the F-solutions (whose existence begins away from \(r=0\)). For these radial solutions, we find a substitution which can transfer this equation into an autonomous Lotka-Volterra system, such that the F-, E- and M-solutions can be comprehensively characterized. In particular, the M-solution has extremely plentiful properties, and its asymptotic expansions are more complicated. For the subcritical case of \(q\), by virtue of a priori estimates, we employ an iterative method which can improve the accuracy step by step, to derive their precise asymptotic expansions. For the supercritical case of \(q\), with the help of the autonomous Lotka-Volterra system and an inverse transformation, we obtain the asymptotic expansions of the M-solutions near the origin. In the same manner, we could acquire more accurate asymptotic expansions of the E-solutions via the iterative method whereas the F-solutions can be described by the autonomous Lotka-Volterra system.Nonuniqueness of solutions to the \(L_p\) dual Minkowski problemhttps://zbmath.org/1491.352592022-09-13T20:28:31.338867Z"Li, Qi-Rui"https://zbmath.org/authors/?q=ai:li.qirui"Liu, Jiakun"https://zbmath.org/authors/?q=ai:liu.jiakun"Lu, Jian"https://zbmath.org/authors/?q=ai:lu.jian.1|lu.jian.2|lu.jianSummary: The \(L_p\) dual Minkowski problem with \(p<0<q\) is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the nonuniqueness of solutions to this problem.Trace formulae for the heat-volume potential of the time-fractional heat equationhttps://zbmath.org/1491.352602022-09-13T20:28:31.338867Z"Oralsyn, Gulaiym"https://zbmath.org/authors/?q=ai:oralsyn.gulaiymSummary: In this note we discuss shortly a method for constructing trace formulae for the heat-volume potential of the time-fractional heat equation to lateral surfaces of cylindrical domains and related nonlocal initial boundary value problems for the time-fractional heat equation.
For the entire collection see [Zbl 1436.46003].On the placement of an obstacle so as to optimize the Dirichlet heat contenthttps://zbmath.org/1491.352612022-09-13T20:28:31.338867Z"Li, Liangpan"https://zbmath.org/authors/?q=ai:li.liangpanReachable states for the distributed control of the heat equationhttps://zbmath.org/1491.352622022-09-13T20:28:31.338867Z"Chen, Mo"https://zbmath.org/authors/?q=ai:chen.mo"Rosier, Lionel"https://zbmath.org/authors/?q=ai:rosier.lionelSummary: We are concerned with the determination of the reachable states for the distributed control of the heat equation on an interval. We consider either periodic boundary conditions or homogeneous Dirichlet boundary conditions. We prove that for a \(L^2\) distributed control, the reachable states are in the Sobolev space \(H^1\) and that they have complex analytic extensions on squares whose horizontal diagonals are regions where no control is applied.An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensionshttps://zbmath.org/1491.352632022-09-13T20:28:31.338867Z"Watschinger, Raphael"https://zbmath.org/authors/?q=ai:watschinger.raphael"Of, Günther"https://zbmath.org/authors/?q=ai:of.guntherSummary: While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps, we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.Regularity of IBVP for parabolic equations in polyhedral domainshttps://zbmath.org/1491.352642022-09-13T20:28:31.338867Z"Luong, Vu Trong"https://zbmath.org/authors/?q=ai:vu-trong-luong."Van Loi, Do"https://zbmath.org/authors/?q=ai:van-loi.doSummary: In this paper, we study the initial-boundary value problem with Dirichlet boundary condition for higher order parabolic equations in polyhedral domains. We get assertions of the well-posedness and the regularity of the solution.Quasi-normal forms of two-component singularly perturbed systemshttps://zbmath.org/1491.352652022-09-13T20:28:31.338867Z"Kashchenko, I. S."https://zbmath.org/authors/?q=ai:kashchenko.ilya-s"Kashchenko, S. A."https://zbmath.org/authors/?q=ai:kashchenko.sergey-aleksandrovich(no abstract)An analytic study for a family of nonlinear diffusion equations of the Fisher typehttps://zbmath.org/1491.352662022-09-13T20:28:31.338867Z"Zhang, Yinghui"https://zbmath.org/authors/?q=ai:zhang.yinghui"Li, Huixu"https://zbmath.org/authors/?q=ai:li.huixu"Yu, Demin"https://zbmath.org/authors/?q=ai:yu.demin"Wan, Zhengsu"https://zbmath.org/authors/?q=ai:wan.zhengsu(no abstract)Mixed problems for plane parabolic systems and boundary integral equationshttps://zbmath.org/1491.352672022-09-13T20:28:31.338867Z"Baderko, E. A."https://zbmath.org/authors/?q=ai:baderko.elena-a"Cherepova, M. F."https://zbmath.org/authors/?q=ai:cherepova.m-fSummary: We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of the solution as well.Propagation dynamics of a Lotka-Volterra competition model with stage structure in time-space periodic environmenthttps://zbmath.org/1491.352682022-09-13T20:28:31.338867Z"Zhao, Xiao"https://zbmath.org/authors/?q=ai:zhao.xiao"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rongSummary: We consider a diffusive Lotka-Volterra competition model with stage structure and time-space periodic habitats. Firstly, we investigate that the system has two periodic semi-trivial solutions \((u^\ast (t, x), 0)\) and \((0, v^\ast (t, x))\). Next, the existence of the principal eigenvalue for the eigenvalue problem associated with a linear time-space periodic reaction-diffusion cooperative system with time delay is established. By using eigenvalue functions and semi-trivial solutions, we construct upper and lower solutions. Then by Schauder's fixed point, there is a pulsating wave solution \((U (t, x, x + c t)\), \(V (t, x, x + c t))\) connecting \(( 0, v^\ast (t, x))\) and \((u^\ast (t,x), 0 )\).An elastic flow for nonlinear spline interpolations in \(\mathbb{R}^n\)https://zbmath.org/1491.352692022-09-13T20:28:31.338867Z"Lin, Chun-Chi"https://zbmath.org/authors/?q=ai:lin.chun-chi"Schwetlick, Hartmut R."https://zbmath.org/authors/?q=ai:schwetlick.hartmut-r"Tran, Dung The"https://zbmath.org/authors/?q=ai:tran.dung-theSummary: In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in \(n\)-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.Local and global existence and blow-up of solutions to a polytropic filtration system with nonlinear memory and nonlinear boundary conditionshttps://zbmath.org/1491.352702022-09-13T20:28:31.338867Z"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.9"Su, Meng-Long"https://zbmath.org/authors/?q=ai:su.menglong"Fang, Zhong-Bo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system
\[
\begin{cases}
u_t=(|(u^{m_1})_x|^{p_1-1}(u^{m_1})_x)_x+u^{l_{11}}\int_0^av^{l_{12}}(\xi, t)\mathrm{d}\xi,(x, t)&\text{in}\ [0, a]\times (0, T),\\
v_t=(|(v^{m_2})_x|^{p_2-1}(v^{m_2})_x)_x+v^{l_{22}}\int_0^au^{l_{21}}(\xi, t)\mathrm{d}\xi,(x, t) &\text{in}\ [0,a]\times (0, T)
\end{cases}
\]
with nonlinear boundary conditions \(u_x|_{x=0}=0\), \(u_x|_{x=a}= u^{q_{11}}v^{q_{12}}|_{x=a}\), \(v_x|_{x=0}=0\), \(v_x|_{x=a}= u^{q_{21}}v^{q_{22}}|_{x=a}\) and the initial data \((u_0, v_0)\), where \(m_1,m_2\geq 1\), \(p_1, p_2 > 1\), \(l_{11}, l_{12}, l_{21}, l_{22}, q_{11},q_{12}, q_{21}, q_{22} > 0\). Under appropriate hypotheses, the authors establish local theory of the solutions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt typehttps://zbmath.org/1491.352712022-09-13T20:28:31.338867Z"Bäcker, Jan-Phillip"https://zbmath.org/authors/?q=ai:backer.jan-phillip"Röger, Matthias"https://zbmath.org/authors/?q=ai:roger.matthiasThe authors study the bulk-surface Gierer-Meinhardt system
\[
u_t=\varepsilon^2\Delta u-u+\frac{u^p}{v^q}+\sigma,\ \tau_sv_t=D_s\Delta v-(1+K)v+Kw+\varepsilon^{-1}\frac{u^r}{v^s} \quad \text{on }\Gamma \times (0,T)
\]
with
\[
\tau_sw_t=D_b\Delta w-w \quad \text{in }\Omega\times (0,T), \qquad \left. D_b\frac{\partial w}{\partial \nu}\right\vert_\Gamma=Kv-Kw,
\]
where \(\Omega\subset \mathbb{R}^n\) is an open set with boundary \(\Gamma=\partial\Omega\). Having global-in-time well-posedness, there arises the weak convergence of the solution to that of the nonlocal surface Gierer-Meinhardt system
\[
\begin{aligned}
u_t= \varepsilon^2\Delta u-u+\frac{u^p}{v^q}+\sigma,\ \tau_sv_t=D_s\Delta v-(1+K)v+Kw+\varepsilon^{-1}\frac{u^r}{v^s} \quad &\text{on }\Gamma\times (0,T),\\
\tau_b\frac{dw}{dt}=-(1+K\frac{\vert \Gamma\vert}{\vert\Omega\vert})w+\frac{K}{\vert \Omega\vert}\int_\Gamma v \quad &\text{in }(0,T)
\end{aligned}
\]
as \(D_b\rightarrow \infty\).
Reviewer: Takashi Suzuki (Osaka)Long time evolutionary dynamics of phenotypically structured populations in time-periodic environmentshttps://zbmath.org/1491.352722022-09-13T20:28:31.338867Z"Iglesias, Susely Figueroa"https://zbmath.org/authors/?q=ai:iglesias.susely-figueroa"Mirrahimi, Sepideh"https://zbmath.org/authors/?q=ai:mirrahimi.sepidehAsymptotic profiles of a diffusive mussel-algae system in closed advective environmentshttps://zbmath.org/1491.352732022-09-13T20:28:31.338867Z"Qu, Anqi"https://zbmath.org/authors/?q=ai:qu.anqi"Wang, Jinfeng"https://zbmath.org/authors/?q=ai:wang.jinfengSummary: In this paper, we are concerned with a system modeling the interaction of mussels and algae, where the advection pushes algae in one direction but not out of the domain. We obtain the influence of diffusion and advection on positive steady states. As the advection rate goes to infinity, the individuals concentrate at the downstream end. When the diffusion rate of the algae tends to zero, a part of algae concentrates at the downstream end, the rest part of algae disperses in a non-homogeneous manner; and the density of mussels is positive on the entire habitat. Our results show that the impact of advection on the spatial distribution is virtual.Global solutions for semilinear rough partial differential equationshttps://zbmath.org/1491.352742022-09-13T20:28:31.338867Z"Hesse, Robert"https://zbmath.org/authors/?q=ai:hesse.robert"Neamţu, Alexandra"https://zbmath.org/authors/?q=ai:neamtu.alexandraOn a semilinear parabolic problem with non-local (Bitsadze-Samarskii type) boundary conditions in more dimensionshttps://zbmath.org/1491.352752022-09-13T20:28:31.338867Z"Slodička, Marián"https://zbmath.org/authors/?q=ai:slodicka.marianSummary: This paper studies a semilinear parabolic equation in a bounded domain \(\Omega\subset\mathbb{R}^d\) along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior \((d-1)\)-dimensional manifold lying inside \(\Omega\). Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.Solvability of a boundary value problem of chaotic dynamics of polymer molecule in the case of bounded interaction potentialhttps://zbmath.org/1491.352762022-09-13T20:28:31.338867Z"Starovoĭtov, Victor N."https://zbmath.org/authors/?q=ai:starovoitov.victor-nikolayevichSummary: This paper deals with a boundary value problem for a parabolic differential equation that describes a chaotic motion of a polymer chain in water. The equation is nonlocal in time as well as in space. It includes a so called interaction potential that depends on the integrals of the solution over the entire time interval and over the space domain where the problem is being solved. The time nonlocality appears since the time plays the role of the arc length along the chain and each segment interacts with all others through the surrounding fluid. The weak solvability of the problem is proven for the case of the bounded continuous interaction potential. The proof of the solvability does not use any continuity properties of the solution with respect to the time and is based on the energy estimate only.Partial existence result for homogeneous quasilinear parabolic problems beyond the duality pairinghttps://zbmath.org/1491.352772022-09-13T20:28:31.338867Z"Adimurthi, Karthik"https://zbmath.org/authors/?q=ai:adimurthi.karthik"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Kim, Wontae"https://zbmath.org/authors/?q=ai:kim.wontaeSummary: In this paper, we study the existence of distributional solutions solving (1.3) on a bounded domain \(\Omega\) satisfying a uniform capacity density condition where the nonlinear structure \(\mathcal{A}(x, t, \nabla u)\) is modelled after the standard parabolic \(p\)-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates is fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. \textit{As a consequence, our a priori estimates are new even for the heat equation on bounded domains.} This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.Global solutions and blow-up profiles for a nonlinear degenerate parabolic equation with nonlocal sourcehttps://zbmath.org/1491.352782022-09-13T20:28:31.338867Z"Zhong, Guangsheng"https://zbmath.org/authors/?q=ai:zhong.guangsheng(no abstract)Existence and regularity results for a singular parabolic equations with degenerate coercivityhttps://zbmath.org/1491.352792022-09-13T20:28:31.338867Z"El Ouardy, Mounim"https://zbmath.org/authors/?q=ai:el-ouardy.mounim"El Hadfi, Youssef"https://zbmath.org/authors/?q=ai:el-hadfi.youssef"Ifzarne, Aziz"https://zbmath.org/authors/?q=ai:ifzarne.azizSummary: The aim of this paper is to prove existence and regularity of solutions for the following nonlinear singular parabolic problem
\[
\begin{cases}
\frac{\partial u}{\partial t}-\operatorname{div}\left(\frac{a(x,t,u,\nabla u)}{(1+|u|)^{\theta(p-1)}}\right)+g(x,t,u)=\frac{f}{u^{\gamma}} &\text{in } Q,\\ u(x,0)=0 &\text{on }\Omega,\\ u=0 &\text{on }\Gamma.
\end{cases}
\]
Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\) (\(N > p\geq 2\)), \( T> 0\) and \(f\) is a non-negative function that belong to some Lebesgue space, \(f\in L^m(Q)\), \(Q=\Omega\times (0,T)\), \(\Gamma=\partial\Omega\times (0,T)\), \(g(x,t,u)=|u|^{s-1}u\), \(s\geq 1\), \(0\leq\theta< 1\) and \(0<\gamma<1\).Entropy solutions for unilateral parabolic problems with \(L^1\)-data in Musielak-Orlicz-Sobolev spaceshttps://zbmath.org/1491.352802022-09-13T20:28:31.338867Z"El Amarty, Nourdine"https://zbmath.org/authors/?q=ai:el-amarty.nourdine"El Haji, Badr"https://zbmath.org/authors/?q=ai:el-haji.badr"El Moumni, Mostafa"https://zbmath.org/authors/?q=ai:el-moumni.mostafaSummary: We prove the existence of entropy solution for the obstacle parabolic equations:
\[
\frac{\partial u}{\partial t} - \operatorname{div} \biggl(a(x,t,u,\nabla u)+ \Phi(u)\biggr)+ g(u)\varphi (x, |\nabla u|)=f \text{ in } Q,
\]
where \(-\operatorname{div} \biggl(a(x,t,u, \nabla u)\biggl)\) is a Leray-Lions operator, \(\Phi \in C^0 (\mathbb{R},\mathbb{R}^N)\). The function \(g(u) \varphi(x,|\nabla u|)\) is a nonlinear lower order term with natural growth with respect to \(|\nabla u|\), without satisfying the sign condition and the datum is assumed belongs to \(L^1(Q)\).Solvability of initial boundary value problems for non-autonomous evolution equationshttps://zbmath.org/1491.352812022-09-13T20:28:31.338867Z"Pyatkov, S. G."https://zbmath.org/authors/?q=ai:pyatkov.sergei-gSummary: The initial boundary value problems for linear non-autonomous first-order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with \(t\). We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results, we use only the continuity assumption on the operators in the main part of the equation.The Cauchy problem for singular systems of heat equations with Fredholm operator of time-derivative in Banach spaceshttps://zbmath.org/1491.352822022-09-13T20:28:31.338867Z"Slastnaya, Ol'ga Viktoravna"https://zbmath.org/authors/?q=ai:slastnaya.olga-viktoravna(no abstract)On the behavior of blow-up solutions to a parabolic problem with critical exponenthttps://zbmath.org/1491.352832022-09-13T20:28:31.338867Z"Cheng, Ting"https://zbmath.org/authors/?q=ai:cheng.ting"Lan, Haipeng"https://zbmath.org/authors/?q=ai:lan.haipeng"Yang, Jinmei"https://zbmath.org/authors/?q=ai:yang.jinmei"Zheng, Gao-Feng"https://zbmath.org/authors/?q=ai:zheng.gao-fengSummary: The blow-up behavior of the solution to a semilinear equation with critical exponent
\[
\begin{cases}
u_t=\Delta u+|u|^{p-1}u&\text{in}\ \Omega\times(0,T),\\
u=0&\text{on}\ \partial\Omega\times (0,T),\\
u(\cdot,0)=u_0&\text{on}\ \Omega,
\end{cases}
\]
is established. We obtain that if \(\Omega\) is star-shaped about \(a\in\Omega\) and \(n\geq 3\), \(p=\frac{n+2}{n-2}\), then \(\lim_{t\to T}(T-t)^\beta u(a+y\sqrt{T-t},t)\) exists and equals 0 or \(\pm \kappa\) in \(L_{loc}^2(\mathbb{R}^n)\), where \(\beta=\frac{1}{p-1}\), \(\kappa=\beta^\beta\).Critical curves for a fast diffusive p-Laplacian equation with nonlocal sourcehttps://zbmath.org/1491.352842022-09-13T20:28:31.338867Z"Zheng, Yadong"https://zbmath.org/authors/?q=ai:zheng.yadong"Fang, Zhong Bo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: This paper deals with the critical curves for a fast diffusive p-Laplacian equation with nonlocal inner source under nonlinear boundary flux in half-line. We obtain new critical global existence curve and critical Fujita curve by constructing various self-similar super- and sub-solutions.Semilinear effectively damped wave models with general relaxation functionhttps://zbmath.org/1491.352852022-09-13T20:28:31.338867Z"Aslan, Halit Sevki"https://zbmath.org/authors/?q=ai:aslan.halit-sevki"Reissig, Michael"https://zbmath.org/authors/?q=ai:reissig.michaelSummary: In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for semilinear effective damped wave models with general relaxation function in the source term. Our aim is to generalize some known results for special nonlinear memory terms, where the convolution is given with respect to the time variable. We first present auxiliary estimates for integrals. Then we prove results on global (in time) existence of small data Sobolev solutions for different classes of data. We distinguish between low regular data, data with suitable higher regularity and large regular data.A second-order evolution equation and logarithmic operatorshttps://zbmath.org/1491.352862022-09-13T20:28:31.338867Z"Bezerra, F. D. M."https://zbmath.org/authors/?q=ai:bezerra.flank-david-moraisSummary: In this paper we introduce a matrix representation of the logarithmic wave operator and we study a second-order semilinear evolution equation governed by this operator. We present a result of local well-posedness for this problem and properties of the logarithmic wave operator in terms of the logarithmic negative Dirichlet Laplacian operator.Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticityhttps://zbmath.org/1491.352872022-09-13T20:28:31.338867Z"Bongarti, Marcelo"https://zbmath.org/authors/?q=ai:bongarti.marcelo"Lasiecka, Irena"https://zbmath.org/authors/?q=ai:lasiecka.irena"Rodrigues, José H."https://zbmath.org/authors/?q=ai:rodrigues.jose-hSummary: The Jordan-Moore-Gibson-Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third-order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at \textit{finite} speed. This is due to the heat phenomenon known as \textit{second sound} which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called ``critical'' case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not ``controlled'', and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).The boundary value problem for one class of higher-order nonlinear partial differential equationshttps://zbmath.org/1491.352882022-09-13T20:28:31.338867Z"Kharibegashvili, Sergo"https://zbmath.org/authors/?q=ai:kharibegashvili.sergo"Midodashvili, Bidzina"https://zbmath.org/authors/?q=ai:midodashvili.bidzinaSummary: In this paper, we consider the boundary value problem for one class of higher-order nonlinear partial differential equations. We prove theorems on existence, uniqueness and nonexistence of solutions of this problem.On the wave interactions for the drift-flux equations with the Chaplygin gashttps://zbmath.org/1491.352892022-09-13T20:28:31.338867Z"Li, Shuangrong"https://zbmath.org/authors/?q=ai:li.shuangrong"Shen, Chun"https://zbmath.org/authors/?q=ai:shen.chunThe authors study solutions to the system (modeling two-phase flows)
\[
\left\{ \begin{array}{l} \partial_t\rho_1 +\partial_x(\rho_1 u)=0,\\
\partial_t\rho_2 +\partial_x(\rho_2 u)=0,\\
\partial_t\big((\rho_1+\rho_2) u\big) +\partial_x\Big((\rho_1+\rho_2)u^2- (\frac{1}{\rho_1}+\frac{1}{\rho_2})\Big)=0, \end{array}\right.
\]
emerging from piecewise-constant inital data. The system is strictly hyperbolic with all three characteristic fields being linearly degenerate.
The first part of the work deals with the Riemann problem. The authors exhibit two distinct situations: in one case the solution is described through three contact discontinuities, while the other case produces a less standard ``delta shock wave'' solution.
In the second part, the authors investigate the interactions between contact discontinuities and delta shock waves, contact discontinuities and contact discontinuities, delta shock waves and delta shock waves, emerging from piecewise constant initial data with three states.
Reviewer: Vincent Duchêne (Rennes)The Riemann problem for the nonisentropic Baer-Nunziato model of two-phase flowshttps://zbmath.org/1491.352902022-09-13T20:28:31.338867Z"Thanh, Mai Duc"https://zbmath.org/authors/?q=ai:mai-duc-thanh."Vinh, Duong Xuan"https://zbmath.org/authors/?q=ai:vinh.duong-xuanSummary: The Riemann problem for the well-known Baer-Nunziato model of two-phase flows is solved. The system consists of seven partial differential equations with nonconservative terms. The most challenging problem is that this model possesses a double eigenvalue. Although characteristic speeds coincide, the curves of composite waves associated with different characteristic fields can be still constructed. They will also be incorporated into composite wave curves to form solutions of the Riemann problem. Solutions of the Riemann problem will be constructed when initial data are in supersonic regions, subsonic regions, or in both kinds of regions. A unique solution and solutions with resonance are also obtained.Global small data solutions for semilinear waves with two dissipative termshttps://zbmath.org/1491.352912022-09-13T20:28:31.338867Z"Chen, Wenhui"https://zbmath.org/authors/?q=ai:chen.wenhui"D'Abbicco, Marcello"https://zbmath.org/authors/?q=ai:dabbicco.marcello"Girardi, Giovanni"https://zbmath.org/authors/?q=ai:girardi.giovanniSummary: In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity \(|u|^p\) or nonlinearity of derivative type \(|u_t|^p\), in any space dimension \(n\geqslant 1\), for supercritical powers \(p>{\bar{p}} \). The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive \(L^r-L^q\) long time decay estimates for the solution in the full range \(1\leqslant r\leqslant q\leqslant \infty \). The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers \(p<{\bar{p}} \).Soliton resolution for critical co-rotational wave maps and radial cubic wave equationhttps://zbmath.org/1491.352922022-09-13T20:28:31.338867Z"Duyckaerts, Thomas"https://zbmath.org/authors/?q=ai:duyckaerts.thomas"Kenig, Carlos"https://zbmath.org/authors/?q=ai:kenig.carlos-e"Martel, Yvan"https://zbmath.org/authors/?q=ai:martel.yvan"Merle, Frank"https://zbmath.org/authors/?q=ai:merle.frankIn this seminal paper, for the co-rotational wave map equation, the authors prove the soliton resolution conjecture for all times, for all solutions in the energy space. This is the first such result for all initial data in the energy space for a wave-type equation. They also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.
Reviewer: Dongbing Zha (Shanghai)Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein-Gordon equationhttps://zbmath.org/1491.352932022-09-13T20:28:31.338867Z"Fukuizumi, Reika"https://zbmath.org/authors/?q=ai:fukuizumi.reika"Hoshino, Masato"https://zbmath.org/authors/?q=ai:hoshino.masato"Inui, Takahisa"https://zbmath.org/authors/?q=ai:inui.takahisaBoundary-value problem for a loaded hyperbolic-parabolic equation with degeneration of orderhttps://zbmath.org/1491.352942022-09-13T20:28:31.338867Z"Khubiev, K. U."https://zbmath.org/authors/?q=ai:khubiev.kazbek-uzeirovichSummary: In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains.Boundary-value problems for Sobolev-type equations with irreversible operator coefficient of the highest derivativeshttps://zbmath.org/1491.352952022-09-13T20:28:31.338867Z"Kozhanov, A. I."https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovichSummary: This paper is devoted to the study of the solvability of boundary-value problems for differential equations of the form
\[ \left({\alpha}_0(t)+{\alpha}_1(t)\Delta \right){u}_{tt}-B{u}_t- Cu=f\left(x,t\right),\]
where \(\Delta\) is the Laplace operator acting with respect to spatial variables and \(B\) and \(C\) are also second-order differential acting with respect to spatial variables. A feature of the equations considered is the condition that the functions \(\alpha_0(t)\) and \(\alpha_1(t)\) may not possess the fixed sign property on the range \((0, T)\) of the temporal variable; in particular, the operator \(\alpha_0(t)+ \alpha_1(t) \Delta\) may be irreversible at any point of the interval \((0, T)\), including any strictly inner segments. For problems considered, we prove theorems on the existence and uniqueness of regular solutions (i.e., solutions possessing all generalized derivatives in the Sobolev sense).Quantitative stability estimates for a two-phase Serrin-type overdetermined problemhttps://zbmath.org/1491.352962022-09-13T20:28:31.338867Z"Cavallina, Lorenzo"https://zbmath.org/authors/?q=ai:cavallina.lorenzo"Poggesi, Giorgio"https://zbmath.org/authors/?q=ai:poggesi.giorgio"Yachimura, Toshiaki"https://zbmath.org/authors/?q=ai:yachimura.toshiakiSummary: In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.On properties of the spectrum of an operator pencil arising in viscoelasticity theoryhttps://zbmath.org/1491.352972022-09-13T20:28:31.338867Z"Davydov, A. V."https://zbmath.org/authors/?q=ai:davydov.aleksandr-vadimovich"Tikhonov, Yu. A."https://zbmath.org/authors/?q=ai:tikhonov.yu-aSummary: The main goal of the present paper is the spectral analysis of operator functions that are symbols
of integro-differential equations with unbounded operator coefficients in a separable Hilbert space.
These abstract integro-differential equations can be realized as integro-differential equations with partial derivatives arising in various sections of viscoelasticity theory.Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related resultshttps://zbmath.org/1491.352982022-09-13T20:28:31.338867Z"Colbois, Bruno"https://zbmath.org/authors/?q=ai:colbois.bruno"Provenzano, Luigi"https://zbmath.org/authors/?q=ai:provenzano.luigiSummary: We study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. We discuss the weak formulation and the classical boundary conditions, and we describe a few properties of the eigenvalues. Moreover, we establish upper bounds compatible with the Weyl's law under a given lower bound on the Ricci curvature.On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusionshttps://zbmath.org/1491.352992022-09-13T20:28:31.338867Z"Khelifi, Abdessatar"https://zbmath.org/authors/?q=ai:khelifi.abdessatar"Jaouabi, Ahlem"https://zbmath.org/authors/?q=ai:jaouabi.ahlemSummary: In this paper, we provide a rigorous derivation of an asymptotic formula for the perturbation of eigenvalues associated to the Stokes eigenvalue problem with Dirichlet conditions and in the presence of small deformable inclusions. Taking advantage of the small sizes of the inclusions immersed in an incompressible Newtonian fluid having kinematic viscosity different from the background one, we show that our asymptotic formula can be expressed in terms of the eigenvalue in the absence of the inclusions and in terms of the viscous moment tensor (VMT).Fundamental gaps of spherical triangleshttps://zbmath.org/1491.353002022-09-13T20:28:31.338867Z"Seto, Shoo"https://zbmath.org/authors/?q=ai:seto.shoo"Wei, Guofang"https://zbmath.org/authors/?q=ai:wei.guofang"Zhu, Xuwen"https://zbmath.org/authors/?q=ai:zhu.xuwenGiven a domain \(\Omega\) in \(\mathbb{S}^2\), the fundamental (or mass) gap is the difference between the first two eigenvalues \(\lambda_2 - \lambda_1 > 0\) of the Laplacian on \(\Omega\) with Dirichlet boundary condition. \par The main result of the article under review reads as follows: The equilateral spherical triangle with angle \(\pi/2\) is a strict local minimum for the gap on the space of the spherical triangles with diameter \(\pi/2\). Moreover \[ \lambda_2(T(t)) - \lambda_1 (T(t)) \geq \lambda_2(T(0)) - \lambda_1 (T(0)) + \frac{16}{\pi}t + O(t^2), \] where \(T(t)\) is the triangle with vertices \((0, 0)\), \((\pi/2, 0)\) and \((\pi/2 - bt, \pi/2 - at)\) with \(a^2 + b^2 = 1\), \(a \geq 0\), \(b \geq 0\) under geodesic polar coordinates centered at the north pole. \par This theorem is similar to a result by \textit{Z. Lu} and \textit{J. Rowlett} [Commun. Math. Phys. 319, No. 1, 111--145 (2013; Zbl 1310.58008)] for the gap of triangles on the plane and is obtained by the same method, i.e., by computing and estimating the first derivatives of \(\lambda_1\) and \(\lambda_2\) at \(t=0\).
Reviewer: Victor Alexandrov (Novosibirsk)Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalitieshttps://zbmath.org/1491.353012022-09-13T20:28:31.338867Z"Vikulova, Anastasia V."https://zbmath.org/authors/?q=ai:vikulova.anastasia-vSummary: In this paper we show how the method of parallel coordinates can be extended to three dimensions. As an application, we prove the conjecture of Antunes et al. [Adv. Calc. Var. 10, No. 4, 357--379 (2017; Zbl 1375.35284)] that ``the ball maximises the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area'' under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex. We also provide partial results in arbitrary dimensions.A perturbation problem for transmission eigenvalueshttps://zbmath.org/1491.353022022-09-13T20:28:31.338867Z"Ambrose, David M."https://zbmath.org/authors/?q=ai:ambrose.david-m"Cakoni, Fioralba"https://zbmath.org/authors/?q=ai:cakoni.fioralba"Moskow, Shari"https://zbmath.org/authors/?q=ai:moskow.shariIn this paper, the authors study the following scattering problem. Consider
the Helmholtz equation for the incident wave \(\nu\) of monochromatic radiation
with frequency \(\omega\)
\[
\triangle\nu+k^{2}\nu=0\text{ \ in \ }\mathbb{R}^{d},
\]
with \(d=2,3\), in a bounded region \(D\) with the refractive index \(n\) being a
bounded real-valued function such that \(\sup\left( n-1\right) =\bar{D}\), and
\(k\) is the wave number proportional to \(\omega\). The total field \(u\) is
decomposed as \(u=u^{s}+\nu\), where the scattered field \(u^{s}\in H_{loc}
^{2}\left( \mathbb{R}^{d}\right) \) satisfies
\[
\triangle u^{s}+k^{2}nu^{s}=-k^{2}\left( n-1\right) \nu,\text{ \ in
\ }\mathbb{R}^{d},
\]
with the outgoing Sommerfeld radiation condition
\[
\lim_{r\rightarrow\infty}r^{\frac{d-1}{2}}\left( \frac{\partial u^{s}
}{\partial r}-iku^{s}\right) =0,
\]
uniformly with respect to \(\hat{x}=x/\left\vert x\right\vert \), \(r=\left\vert
x\right\vert \). \(k\) is a non-scattering wave number if \(u^{s}\) is zero outside
\(D\). Non-scattering wave numbers are a subset of real transmission
eigenvalues. From the previous results in the literature, if the media \(D\) is
Lipschitz and \(n\) a bounded function, an infinite discrete set of real
transmission eigenvalues is shown to exist under the assumption that either
\(n-1\geq\alpha>0\), or else, \(1-n\geq\alpha>0\) a.e. in \(D\). Moreover, when \(D\)
is a ball of radius \(a\) centered at the origin and \(n(r)\) is a radial
function, the existence of real transmission eigenvalues is known for \(n\in
C^{2}[0,a]\) under the additional assumption \(\frac{1}{a}\int_{0}^{a}
\sqrt{n\left( \rho\right) }d\rho\neq1\). The aim of the present paper is to
extend the above results and to provide examples of existence of real
transmission eigenvalues for classes of refractive index that do not satisfy
the above conditions. The authors use a perturbation method based on the
implicit function theorem, meaning that the ''irregular'' refractive index is a
perturbation of a refractive index for which a real transmission eigenvalues
is known to exist. Several examples of spherical perturbations of spherically
symmetric media are presented and partial results for general media are obtained.
Reviewer: Ivan Naumkin (Nice)Existence and multiplicity of solutions of \(p(x)\)-triharmonic problemhttps://zbmath.org/1491.353032022-09-13T20:28:31.338867Z"Belakhdar, Adnane"https://zbmath.org/authors/?q=ai:belakhdar.adnane"Belaouidel, Hassan"https://zbmath.org/authors/?q=ai:belaouidel.hassan"Filali, Mohammed"https://zbmath.org/authors/?q=ai:filali.mohammed"Tsouli, Najib"https://zbmath.org/authors/?q=ai:tsouli.najibSummary: We prove the existence and nonexistence of eigenvalues for \(p(x)\)-triharmonic problem with Navier boundary value conditions on a bounded domain in \(\mathbb{R}^N\). Our technique is based on variational approaches and the theory of variable exponent Lebesgue spaces.The diatomic Hartree model at dissociationhttps://zbmath.org/1491.353042022-09-13T20:28:31.338867Z"Cazalis, Jean"https://zbmath.org/authors/?q=ai:cazalis.jeanA Riemann-Hilbert type problem for a singularly perturbed Cauchy-Riemann equation with a singularity in the coefficienthttps://zbmath.org/1491.353052022-09-13T20:28:31.338867Z"Fedorov, Yu. S."https://zbmath.org/authors/?q=ai:fedorov.yury-sergeevichSummary: We consider the Riemann-Hilbert problem for a singularly perturbed system of partial differential equations of the Cauchy-Riemann type. Using the Lomov regularization method, we obtain sufficient conditions under which the asymptotic solutions of the problem converge in the usual sense.Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial datahttps://zbmath.org/1491.353062022-09-13T20:28:31.338867Z"Li, Zhi-Qiang"https://zbmath.org/authors/?q=ai:li.zhiqiang.1"Tian, Shou-Fu"https://zbmath.org/authors/?q=ai:tian.shoufu"Yang, Jin-Jie"https://zbmath.org/authors/?q=ai:yang.jinjieSummary: In this work, we employ the \(\bar{\partial}\)-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space \(\mathcal{H}(\mathbb{R})\). The long time asymptotic behavior of the solution \(q(x, t)\) is derived in a fixed space-time cone \(S(y_1,y_2,v_1,v_2)=\{(y,t)\in\mathbb{R}^2: y=y_0+vt, ~y_0\in [y_1,y_2], v\in [v_1,v_2]\}\). Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by \(N(\mathcal{I})\)-soliton on discrete spectrum and the \(t^{-\frac{1}{2}}\) order term on continuous spectrum with residual error up to \(O(t^{-\frac{3}{4}})\).\(N\)-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithmhttps://zbmath.org/1491.353072022-09-13T20:28:31.338867Z"Peng, Wei-Qi"https://zbmath.org/authors/?q=ai:peng.weiqi"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yongSummary: In this paper, we systematically investigate the nonlocal Hirota equation with nonzero boundary conditions via Riemann-Hilbert method and multi-layer physics-informed neural networks algorithm. Starting from the Lax pair of nonzero nonlocal Hirota equation, we first give out the Jost function, scattering matrix, their symmetry and asymptotic behavior. Then, the Riemann-Hilbert problem with nonzero boundary conditions are constructed and the precise formulae of \(N\)-double poles solutions and \(N\)-simple poles solutions are written by determinants. Different from the local Hirota equation, the symmetry of scattering data for nonlocal Hirota equation is completely different, which results in disparate discrete spectral distribution. In particular, it could be more complicated and difficult to obtain the symmetry of scattering data under the circumstance of double poles. Besides, we also analyze the asymptotic state of one-double poles solution as \(t \to \infty\). Whereafter, the multi-layer physics-informed neural networks algorithm is applied to research the data-driven soliton solutions of the nonzero nonlocal Hirota equation by using the training data obtained from the Riemann-Hilbert method. Most strikingly, the integrable nonlocal equation is firstly solved via multi-layer physics-informed neural networks algorithm. As we all know, the nonlocal equations contain the \(\mathcal{PT}\) symmetry \(\mathcal{P} : x \to - x\), or \(\mathcal{T} : t \to - t\), which are different with local ones. Adding the nonlocal term into the neural network, we can successfully solve the integrable nonlocal Hirota equation by multi-layer physics-informed neural networks algorithm. The numerical results show that the algorithm can recover the data-driven soliton solutions of the integrable nonlocal equation well. Noteworthily, the inverse problems of the integrable nonlocal equation are discussed for the first time through applying the physics-informed neural networks algorithm to discover the parameters of the equation in terms of its soliton solution.On a Riemann-Hilbert boundary value problem for \((\varphi,\psi)\)-harmonic functions in \(\mathbb{R}^m\)https://zbmath.org/1491.353082022-09-13T20:28:31.338867Z"Serrano Ricardo, José Luis"https://zbmath.org/authors/?q=ai:serrano-ricardo.jose-luis"Abreu Blaya, Ricardo"https://zbmath.org/authors/?q=ai:abreu-blaya.ricardo"Bory Reyes, Juan"https://zbmath.org/authors/?q=ai:bory-reyes.juan|moreno-garcia.tania"Sánchez Ortiz, Jorge"https://zbmath.org/authors/?q=ai:sanchez-ortiz.jorgeSummary: The purpose of this paper is to solve a kind of the Riemann-Hilbert boundary value problem for \((\varphi,\psi)\)-harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space \(\mathbb{R}^m\). We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain \(\Omega\subset\mathbb{R}^m\) with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class \(\operatorname{Lip}(1+\alpha,\Gamma)\).On a Riemann-Hilbert problem for the negative-order KdV equationhttps://zbmath.org/1491.353092022-09-13T20:28:31.338867Z"Yuan, Shengyang"https://zbmath.org/authors/?q=ai:yuan.shengyang"Xu, Jian"https://zbmath.org/authors/?q=ai:xu.jianSummary: In this paper, we study the initial value problem for the negative order integrable KdV (NKdV) equation by Riemann-Hilbert problem method. The solutions of the NKdV equation are constructed in terms of the solution of a \(2 \times 2\)-matric Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at one singularity point, \( \lambda = \infty \). And the one-soliton and two-soliton solutions are discussed in detail.Regularization estimates and hydrodynamical limit for the Landau equationhttps://zbmath.org/1491.353102022-09-13T20:28:31.338867Z"Carrapatoso, Kleber"https://zbmath.org/authors/?q=ai:carrapatoso.kleber"Rachid, Mohamad"https://zbmath.org/authors/?q=ai:rachid.mohamad"Tristani, Isabelle"https://zbmath.org/authors/?q=ai:tristani.isabelleSummary: In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and we obtain the instantaneous expected anisotropic gain of regularity (see [\textit{M. Rachid}, ``Hypoelliptic and spectral estimates for the linearized Landau operator'', Preprint, \url{arXiv:2004.09300}] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.A stability result for the identification of a permeability parameter on Navier-Stokes equationshttps://zbmath.org/1491.353112022-09-13T20:28:31.338867Z"Aguayo, Jorge"https://zbmath.org/authors/?q=ai:aguayo.jorge"Osses, Axel"https://zbmath.org/authors/?q=ai:osses.axelSecondary flows from a linear array of vortices perturbed principally by a Fourier modehttps://zbmath.org/1491.353122022-09-13T20:28:31.338867Z"Chen, Zhi-Min"https://zbmath.org/authors/?q=ai:chen.zhiminSummary: In the understanding of primary bifurcating flows of a linear array of electromagnetically forced vortices in an experimental fluid motion, a theoretical study on the nonlinear instability is presented. The existence of the bifurcating flows is obtained from a Fourier mode perturbation. This large-scale perturbation, leading to the primary bifurcation observed in a laboratory experiment, was found to be generated principally from a single vortex mode.An exact solution for the semi-stationary compressible Stokes problemhttps://zbmath.org/1491.353132022-09-13T20:28:31.338867Z"Dong, Jianwei"https://zbmath.org/authors/?q=ai:dong.jianweiSummary: In this note, we present an exact solution for the semi-stationary compressible Stokes problem in \(\mathbb{R}^N\). In the case of radial symmetry, an exact solution with velocity of the form \(c(t)r^s\) is obtained for \(s=\frac{1-N\gamma +\gamma}{\gamma +1}\), where \(\gamma >1\) is the adiabatic index and \(r=|x|\). Some interesting properties of the exact solution are analyzed.Low Mach number limit for the full compressible magnetohydrodynamic equations without thermal conductivityhttps://zbmath.org/1491.353142022-09-13T20:28:31.338867Z"Guo, Liang"https://zbmath.org/authors/?q=ai:guo.liang"Li, Fucai"https://zbmath.org/authors/?q=ai:li.fucaiSummary: In this paper we consider the low Mach number limit of the full compressible magnetohydrodynamic equations for the polytropic ideal gas with zero thermal conductivity coefficient in the whole space \(\mathbb{R}^n\) (\(n=2, 3\)). We focus on the case that the pressure varies near its equilibrium state. It means that the density and the temperature may change around their limit functions, and hence generalize the case on the perturbation of the constant states for the density and the temperature. We establish this limit process rigorously when the initial data is well-prepared. Moreover, we also obtain the convergence rates.Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domainhttps://zbmath.org/1491.353152022-09-13T20:28:31.338867Z"Nersesyan, Vahagn"https://zbmath.org/authors/?q=ai:nersesyan.vahagnSummary: The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers [\textit{S. Kuksin} et al., Geom. Funct. Anal. 30, No. 1, 126--187 (2020; Zbl 1442.35437)] and [\textit{A. Shirikyan}, J. Eur. Math. Soc. (JEMS) 23, No. 4, 1381--1422 (2021; Zbl 1470.37008)] and using the asymptotic compactness of the dynamics.Stability and periodicity of solutions to Navier-Stokes equations on non-compact Riemannian manifolds with negative curvaturehttps://zbmath.org/1491.353162022-09-13T20:28:31.338867Z"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha."Nguyen, Thi Van"https://zbmath.org/authors/?q=ai:van-nguyen.thiSummary: Let \((M, g)\) be a non-compact Riemannian manifold having negative Ricci curvature tensor. Then, we consider the Navier-Stokes Equations (NSE) for vector fields on \((M, g)\) and prove the existence of a bounded solution to NSE on \((M, g)\). Moreover we show the stability on a small neighborhood for such a solution. Then, using such a local stability we show the existence of a time-periodic solution to NSE under the action of a time-periodic external force. Our result can be considered as a Serrin-type theorem for the case of non-compact Riemannian manifolds with negative curvature tensors.Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 datahttps://zbmath.org/1491.353172022-09-13T20:28:31.338867Z"Paicu, Marius"https://zbmath.org/authors/?q=ai:paicu.marius"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.3Summary: We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using the Cattaneo type law instead of the Fourier law, evolving in a thin strip \(\mathbb{R} \times (0, \epsilon)\). The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data. Compared with [\textit{M. Paicu} et al., Adv. Math. 372, Article ID 107293, 41 p. (2020; Zbl 1446.35105)] for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data, here the initial data belongs to the Gevrey class 2, which is very sophisticated even for the well-posedness of the classical Prandtl system (see [\textit{H. Dietert} and \textit{D. Gérard-Varet}, Ann. PDE 5, No. 1, Paper No. 8, 51 p. (2019; Zbl 1428.35355)] and [\textit{C. Wang}, \textit{Y. Wang,} and \textit{P. Zhang}, ``On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class'', Preprint, \url{arXiv:2103.00681}]); furthermore, the estimate of the pressure term in the hyperbolic Prandtl system give rise to additional difficulties.Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channelhttps://zbmath.org/1491.353182022-09-13T20:28:31.338867Z"Patriarca, Clara"https://zbmath.org/authors/?q=ai:patriarca.claraSummary: In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a \textit{Poiseuille flow} profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent the obstacle from going excessively far from the equilibrium position and colliding with the boundary of the channel, we introduce a \textit{strong force} in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.On numerical approximations to fluid-structure interactions involving compressible fluidshttps://zbmath.org/1491.353192022-09-13T20:28:31.338867Z"Schwarzacher, Sebastian"https://zbmath.org/authors/?q=ai:schwarzacher.sebastian"She, Bangwei"https://zbmath.org/authors/?q=ai:she.bangweiSummary: In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.New thought on Matsumura-Nishida theory in the \(L_p\)-\(L_q\) Maximal regularity frameworkhttps://zbmath.org/1491.353202022-09-13T20:28:31.338867Z"Shibata, Yoshihiro"https://zbmath.org/authors/?q=ai:shibata.yoshihiroSummary: This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier-Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to \textit{A. Matsumura} and \textit{T. Nishida} [Commun. Math. Phys. 89, 445--464 (1983; Zbl 0543.76099)]. In [loc. cit.], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to \(L_2\) in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the \(L_\infty\) norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [loc. cit.] in \(L_p\) (\(1 <p \le 2\)) in time and \(L_2\cap L_6\) in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [loc. cit.] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal \(L_p\) (\(1 <p \le 2\)) in time and \(L_2\cap L_6\) in space regularity and \(L_p\)-\(L_q\) decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach's fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the \(L_p\)-\(L_q\) maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal \(L_p\)-\(L_q\) regularity and \(L_p\)-\(L_q\) decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.Time optimal control problem of the 3D Navier-Stokes-\( \alpha\) equationshttps://zbmath.org/1491.353212022-09-13T20:28:31.338867Z"Son, Dang Thanh"https://zbmath.org/authors/?q=ai:son.dang-thanh"Thuy, Le Thi"https://zbmath.org/authors/?q=ai:thuy.le-thi-hong|le-thi-thuy.Summary: In this paper, we study an optimal control problem for the three-dimensional Navier-Stokes-\(\alpha\) equations in bounded domains with Dirichlet boundary conditions, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions. Then we establish the first-order and second-order necessary optimality conditions, and the second-order sufficient optimality conditions. The second-order optimality ones obtained in the paper seem to be optimal in the sense that the gap between them is minimal.Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\)https://zbmath.org/1491.353222022-09-13T20:28:31.338867Z"Wang, Dixi"https://zbmath.org/authors/?q=ai:wang.dixi"Yu, Cheng"https://zbmath.org/authors/?q=ai:yu.cheng"Zhao, Xinhua"https://zbmath.org/authors/?q=ai:zhao.xinhuaSummary: In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\). In particular, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in \(\mathbb{R}^3\), which yields the uniform bounds of \(\alpha^{th}\)-order fractional derivatives of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2_x\) for some \(\alpha > 0\), independent of the viscosity. The uniform bounds can provide strong convergence of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2\) space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusionhttps://zbmath.org/1491.353232022-09-13T20:28:31.338867Z"Wang, Wenjun"https://zbmath.org/authors/?q=ai:wang.wenjun"Wen, Huanyao"https://zbmath.org/authors/?q=ai:wen.huanyaoSummary: We consider the full compressible Navier-Stokes equations with reaction diffusion. A global existence and uniqueness result of the strong solution is established for the Cauchy problem when the initial data is in a neighborhood of a trivially stationary solution. The appearance of the difference between energy gained and energy lost due to the reaction is a new feature for the flow and brings new difficulties. To handle these, we construct a new linearized system in terms of a combination of the solutions. Moreover, some optimal time-decay estimates of the solutions are derived when the initial perturbation is additionally bounded in \(L^1\). It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solution so that the long time behavior for the hyperbolic-parabolic system is exactly the same as that for the heat equation. As a byproduct, the above time-decay estimate at the highest order is also valid for compressible Navier-Stokes equations. The proof is accomplished by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity.Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6https://zbmath.org/1491.353242022-09-13T20:28:31.338867Z"Wu, Bian"https://zbmath.org/authors/?q=ai:wu.bianSummary: By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force \(f \in L_{\mathrm{loc}}^q \cap L^{3/2}\), \(q>3\) in general open subdomains of \(\mathbb{R}^6\). These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in \(\mathbb{R}^4 \times [0, \infty[\).Global solutions to 3D incompressible Navier-Stokes equations with some large initial datahttps://zbmath.org/1491.353252022-09-13T20:28:31.338867Z"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu.1"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. More precisely, we prove that if
\[
\begin{aligned}
\Bigg(&\| u_0^1 + u_0^2 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}} + \| u_0^3 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}}\Bigg) \Bigg(\| u_0^1 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}} + \| u_0^2 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}}\Bigg)\\
&\times \exp \Bigg(C \Big(\| u_0 \|_{\dot{B}_{\infty, 2}^{- 1}}^2 + \| u_0 \|_{\dot{B}_{\infty, 1}^{- 1}} \Big)\Bigg)
\end{aligned}
\]
is small enough, the Navier-Stokes equations have a unique global solution. As an application, we construct two examples of initial data satisfying the smallness condition, but whose \(\dot{B}_{\infty, \infty}^{- 1} (\mathbb{R}^3)\) norm can be arbitrarily large.Rayleigh-Taylor instability for viscous incompressible capillary fluidshttps://zbmath.org/1491.353262022-09-13T20:28:31.338867Z"Zhang, Zhipeng"https://zbmath.org/authors/?q=ai:zhang.zhipengSummary: We investigate the linear and nonlinear instability of a smooth Rayleigh-Taylor steady state solution to the three-dimensional incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution and find that for any capillary coefficient \(\kappa >0\), we can construct the solutions of the linearized problem that grow in time in Sobolev space \(H^m\), thus leading to the linear instability. However, with the help of the constructed unstable solutions of the linearized problem, we just establish the nonlinear instability for small enough capillary coefficient \(\kappa >0\).Conjugate points in \(\mathcal{D}_\mu^s(S^2)\)https://zbmath.org/1491.353272022-09-13T20:28:31.338867Z"Benn, J."https://zbmath.org/authors/?q=ai:benn.jamesSummary: Rossby-Haurwitz waves on the sphere \(S^2\) form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the \(L^2\) metric in the volume preserving diffeomorphism group of \(S^2\). Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves.Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problemshttps://zbmath.org/1491.353282022-09-13T20:28:31.338867Z"Wang, Guodong"https://zbmath.org/authors/?q=ai:wang.guodongSummary: This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in \(L^p\) norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem.On a higher integral invariant for closed magnetic lines, revisitedhttps://zbmath.org/1491.353292022-09-13T20:28:31.338867Z"Akhmet'ev, Peter M."https://zbmath.org/authors/?q=ai:akhmetev.petr-mSummary: We recall a definition of an asymptotic invariant of classical link, which is called \(M\)-invariant. \(M\)-invariant is a special Massey integral, this integral has an ergodic form and is generalized for magnetic fields with open magnetic lines in a bounded \(3D\)-domain. We present a proof that this integral is well defined. A combinatorial formula for \(M\)-invariant using the Conway polynomial is presented. The \(M\)-invariant is a higher invariant, it is not a function of pairwise linking numbers of closed magnetic lines. We discuss applications of \(M\)-invariant for MHD.Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaceshttps://zbmath.org/1491.353302022-09-13T20:28:31.338867Z"Azanzal, Achraf"https://zbmath.org/authors/?q=ai:azanzal.achraf"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Melliani, Said"https://zbmath.org/authors/?q=ai:melliani.saidSummary: This paper establishes the existence and uniqueness, and also presents a blow-up criterion, for solutions of the quasi-geostrophic (QG) equation in a framework of Fourier type, specifically Fourier-Besov-Morey spaces. If it is assumed that the initial data \(\theta_0\) is small and belonging to the critical Fourier-Besov-Morrey spaces \(\mathscr{F} {\mathscr{N}}_{p, \lambda, q}^{3-2 \alpha +\frac{\lambda -2}{p}} \), we get the global well-posedness results of the QG equation (1). Moreover, we prove that there exists a time \(T > 0\) such that the QG equation (1) admits a unique local solution for large initial data.Mixing solutions for the Muskat problemhttps://zbmath.org/1491.353312022-09-13T20:28:31.338867Z"Castro, A."https://zbmath.org/authors/?q=ai:castro.angel"Córdoba, D."https://zbmath.org/authors/?q=ai:cordoba.diego"Faraco, D."https://zbmath.org/authors/?q=ai:faraco.danielSummary: We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.Hamiltonian description of internal ocean waves with Coriolis forcehttps://zbmath.org/1491.353322022-09-13T20:28:31.338867Z"Cullen, Joseph D."https://zbmath.org/authors/?q=ai:cullen.joseph-d"Ivanov, Rossen I."https://zbmath.org/authors/?q=ai:ivanov.rossen-iSummary: The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a nearly Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.On uniqueness and helicity conservation of weak solutions to the electron-MHD systemhttps://zbmath.org/1491.353332022-09-13T20:28:31.338867Z"Dai, Mimi"https://zbmath.org/authors/?q=ai:dai.mimi"Krol, Jacob"https://zbmath.org/authors/?q=ai:krol.jacob"Liu, Han"https://zbmath.org/authors/?q=ai:liu.hanSummary: We study weak solutions to the electron-MHD system and obtain a conditional uniqueness result. In addition, we prove conservation of helicity for weak solutions to the electron-MHD system under a geometric condition.Travelling waves in the Boussinesq type systemshttps://zbmath.org/1491.353342022-09-13T20:28:31.338867Z"Dinvay, Evgueni"https://zbmath.org/authors/?q=ai:dinvay.evgueniSummary: Considered herein are a number of variants of the Boussinesq type systems modelling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified.Uniform regularity for a density-dependent incompressible Hall-MHD systemhttps://zbmath.org/1491.353352022-09-13T20:28:31.338867Z"Fan, Jishan"https://zbmath.org/authors/?q=ai:fan.jishan"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yong.1Summary: This paper proves uniform regularity for a density-dependent incompressible Hall-MHD system with positive density.Energy considerations for nonlinear equatorial water waveshttps://zbmath.org/1491.353362022-09-13T20:28:31.338867Z"Henry, David"https://zbmath.org/authors/?q=ai:henry.david.2|henry.david.1Summary: In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived.Continued gravitational collapse for gaseous star and pressureless Euler-Poisson systemhttps://zbmath.org/1491.353372022-09-13T20:28:31.338867Z"Huang, Feimin"https://zbmath.org/authors/?q=ai:huang.feimin"Yao, Yue"https://zbmath.org/authors/?q=ai:yao.yueGlobal well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial datahttps://zbmath.org/1491.353382022-09-13T20:28:31.338867Z"Huang, Xiangdi"https://zbmath.org/authors/?q=ai:huang.xiangdi"Li, Jing"https://zbmath.org/authors/?q=ai:li.jingOptimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balancehttps://zbmath.org/1491.353392022-09-13T20:28:31.338867Z"Ji, Ruihong"https://zbmath.org/authors/?q=ai:ji.ruihong"Yan, Li"https://zbmath.org/authors/?q=ai:yan.li"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahongSummary: This paper focuses on the three-dimensional (3D) incompressible anisotropic Boussinesq system with horizontal dissipation. The goal here is to assess the stability property and pinpoint the precise large-time behavior of perturbations near the hydrostatic balance. Important tools such as Schonbek's Fourier splitting method have been developed to understand the large-time behavior of PDE systems with full dissipation, but these tools may not apply directly when the systems are only partially dissipated. This paper solves the stability problem and designs an effective approach to obtain the optimal decay rates for the anisotropic Boussinesq system concerned here. The tool developed in this paper may be useful for many other partially dissipated systems.Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problemhttps://zbmath.org/1491.353402022-09-13T20:28:31.338867Z"Lepe, Felipe"https://zbmath.org/authors/?q=ai:lepe.felipe"Rivera, Gonzalo"https://zbmath.org/authors/?q=ai:rivera.gonzalo"Vellojin, Jesus"https://zbmath.org/authors/?q=ai:vellojin.jesusOrbital stability of the sum of smooth solitons in the Degasperis-Procesi equationhttps://zbmath.org/1491.353412022-09-13T20:28:31.338867Z"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.2"Liu, Yue"https://zbmath.org/authors/?q=ai:liu.yue"Wu, Qiliang"https://zbmath.org/authors/?q=ai:wu.qiliangSummary: The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the \(L^2 \cap L^\infty\) orbital stability of a wave train containing \(N\) smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the \(L^2\)-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce \textit{a priori} estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.On the effect of fast rotation and vertical viscosity on the lifespan of the \(3D\) Primitive equationshttps://zbmath.org/1491.353422022-09-13T20:28:31.338867Z"Lin, Quyuan"https://zbmath.org/authors/?q=ai:lin.quyuan"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.5|liu.xin.3|liu.xin.2|liu.xin|liu.xin.4|liu.xin.1"Titi, Edriss S."https://zbmath.org/authors/?q=ai:titi.edriss-salehSummary: We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation \(|\Omega|\), we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only \(L^2\) in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large \(|\Omega|\), we show that the existence time of the solution can be prolonged, with ``well-prepared'' initial data. Finally, in the case of two spatial dimensions with \(\Omega =0\), we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.Global well-posedness of 3d axisymmetric MHD-Boussinesq system with nonzero swirlhttps://zbmath.org/1491.353432022-09-13T20:28:31.338867Z"Liu, Qiao"https://zbmath.org/authors/?q=ai:liu.qiao"Yang, Yixin"https://zbmath.org/authors/?q=ai:yang.yixinSummary: In this paper, we consider the 3d axisymmetric MHD-Boussinesq system with nonzero swirl, and prove that the system, with initial data \((u_0, h_0, \rho_0) = (u^r_0 e_r + u^\theta_0 e_\theta + u^z_0 e_z, h^\theta_0 e_\theta, \rho_0)\) which satisfies some small nonlinear condition, admits a global unique solution \((u, h, \rho)\). Furthermore, some continuation criteria that imply regularity of axisymmetric solutions are also obtained.Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equationhttps://zbmath.org/1491.353442022-09-13T20:28:31.338867Z"Nazarov, Fedor"https://zbmath.org/authors/?q=ai:nazarov.fedor-l"Zumbrun, Kevin"https://zbmath.org/authors/?q=ai:zumbrun.kevin-rSummary: We establish an instantaneous smoothing property for decaying solutions on the half-line \((0, +\infty)\) of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \(H^1\) stable manifolds of such equations, showing that \(L^2_{loc}\) solutions that remain sufficiently small in \(L^\infty\) (i) decay exponentially, and (ii) are \(C^\infty\) for \(t>0 \), hence lie eventually in the \(H^1\) stable manifold constructed by Pogan and Zumbrun.Global existence in critical spaces for non Newtonian compressible viscoelastic flowshttps://zbmath.org/1491.353452022-09-13T20:28:31.338867Z"Pan, Xinghong"https://zbmath.org/authors/?q=ai:pan.xinghong"Xu, Jiang"https://zbmath.org/authors/?q=ai:xu.jiang"Zhu, Yi"https://zbmath.org/authors/?q=ai:zhu.yi|zhu.yi.1|zhu.yi.3|zhu.yi.2Summary: We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the ``div-curl'' structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from \textit{combination} of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations.A ternary Cahn-Hilliard-Navier-Stokes model for two-phase flow with precipitation and dissolutionhttps://zbmath.org/1491.353462022-09-13T20:28:31.338867Z"Rohde, Christian"https://zbmath.org/authors/?q=ai:rohde.christian"von Wolff, Lars"https://zbmath.org/authors/?q=ai:von-wolff.larsSharp convergence rates for Darcy's lawhttps://zbmath.org/1491.353472022-09-13T20:28:31.338867Z"Shen, Zhongwei"https://zbmath.org/authors/?q=ai:shen.zhongwei.1|shen.zhongweiSummary: This article is concerned with Darcy's law for an incompressible viscous fluid flowing in a porous medium. We establish the sharp \(O(\sqrt{\varepsilon})\) convergence rate in a periodically perforated and bounded domain in \(\mathbb{R}^d\) for \(d\geq 2\), where \(\varepsilon\) represents the size of solid obstacles. This is achieved by constructing two boundary layer correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values between the solution and the leading term in its asymptotic expansion. One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data.Compactness and large-scale regularity for Darcy's lawhttps://zbmath.org/1491.353482022-09-13T20:28:31.338867Z"Shen, Zhongwei"https://zbmath.org/authors/?q=ai:shen.zhongwei|shen.zhongwei.1Summary: This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior \(C^{1,\alpha}\) and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform \(W^{k,p}\) estimates for \(1<p<\infty\).The MHD equations in the Lorentz space with time dependent external forceshttps://zbmath.org/1491.353492022-09-13T20:28:31.338867Z"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong"Zhou, Jianfeng"https://zbmath.org/authors/?q=ai:zhou.jianfengSummary: We are concerned with the well-posedness of the incompressible Magneto-hydrodynamical (MHD) equations in \(\mathbb{R}^n\) (\(n\ge 3\)). First, by assuming the smallness of the external force in Lorentz spaces, we prove the existence, uniqueness and the time regularity of periodic mild solution of an integral form of MHD Eqs. (1.1). Next, we prove the local existence and uniqueness of mild solution of the Cauchy problem of MHD Eqs. (1.2). Finally, appealing to the existence and uniqueness of the mild solution of (1.2), we show that the obtained solution \((u, b)\) of (1.1) becomes the time periodic strong solution, derived from the strong solvability of the inhomogeneous Stokes equation and heat equation by an additional assumption of the external force.Inexact GMRES iterations and relaxation strategies with fast-multipole boundary element methodhttps://zbmath.org/1491.353502022-09-13T20:28:31.338867Z"Wang, Tingyu"https://zbmath.org/authors/?q=ai:wang.tingyu"Layton, Simon K."https://zbmath.org/authors/?q=ai:layton.simon-k"Barba, Lorena A."https://zbmath.org/authors/?q=ai:barba.lorena-aSummary: Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, \(p\). We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing \(p\). In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between \(1.5 \times\) and \(2.3 \times\) for Laplace problems and between \(2.7 \times\) and \(3.3 \times\) for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, \url{https://github.com/barbagroup/fmm-bem-relaxed}, and we share reproducibility packages for all results in \url{https://github.com/barbagroup/inexact-gmres/}.A stochastic approach to enhanced diffusionhttps://zbmath.org/1491.353512022-09-13T20:28:31.338867Z"Zelati, Michele Coti"https://zbmath.org/authors/?q=ai:coti-zelati.michele"Drivas, Theodore D."https://zbmath.org/authors/?q=ai:drivas.theodore-dSummary: We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are Hölder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. Our proof makes use of a probabilistic interpretation of the dissipation of solutions to the advection diffusion equation.Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow modelhttps://zbmath.org/1491.353522022-09-13T20:28:31.338867Z"Zhang, Qinglong"https://zbmath.org/authors/?q=ai:zhang.qinglongSummary: The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of \(\delta\)-waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a \(\delta\)-shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.Stabilization and exponential decay for 2D Boussinesq equations with partial dissipationhttps://zbmath.org/1491.353532022-09-13T20:28:31.338867Z"Zhong, Yueyuan"https://zbmath.org/authors/?q=ai:zhong.yueyuanSummary: This paper focuses on a special 2D Boussinesq equation with partial dissipation, for which the velocity equation involves no dissipation and there is only damping in the horizontal component equation. Without buoyancy force, the corresponding vorticity equation is a 2D Euler-like equation with an extra Calderon-Zygmund-type term. Its stability is an open problem. Our results reveal that the buoyancy force exactly stabilizes the fluids by the coupling and interaction between the velocity and temperature. In addition, we prove the solution decays exponentially to zero in Sobolev norm.Convergence towards the Vlasov-Poisson equation from the \(N\)-fermionic Schrödinger equationhttps://zbmath.org/1491.353542022-09-13T20:28:31.338867Z"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.1"Lee, Jinyeop"https://zbmath.org/authors/?q=ai:lee.jinyeop"Liew, Matthew"https://zbmath.org/authors/?q=ai:liew.matthewSummary: We consider the quantum dynamics of \(N\) interacting fermions in the large \(N\) limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to \(N\). From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.Anisotropic liquid drop modelshttps://zbmath.org/1491.353552022-09-13T20:28:31.338867Z"Choksi, Rustum"https://zbmath.org/authors/?q=ai:choksi.rustum"Neumayer, Robin"https://zbmath.org/authors/?q=ai:neumayer.robin"Topaloglu, Ihsan"https://zbmath.org/authors/?q=ai:topaloglu.ihsanSummary: We introduce and study certain variants of Gamow's liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.Nonrelativistic limit of ground state solutions for nonlinear Dirac-Klein-Gordon systemshttps://zbmath.org/1491.353562022-09-13T20:28:31.338867Z"Dong, Xiaojing"https://zbmath.org/authors/?q=ai:dong.xiaojing"Tang, Zhongwei"https://zbmath.org/authors/?q=ai:tang.zhongweiSummary: We study the nonrelativistic limit and some properties of the solutions
\[
(\psi,\phi):=(u,v,\phi) \in \mathbb{C}^2 \times \mathbb{C}^2 \times \mathbb{R}
\]
for the following nonlinear Dirac-Klein-Gordon systems:
\[
\begin{cases}
ic \displaystyle \sum_{k=1}^{3}a_k \partial_k \psi - mc^2 \beta \psi - \omega\psi -\lambda\phi\beta\psi = |\psi|^{p-2}\psi, \\
-\Delta\phi +c^2M^2\phi = 4\pi\lambda(\beta\psi) \cdot \psi,
\end{cases}
\]
where \(p \in [\frac{12}{5},\frac{8}{3}]\), \(c\) denotes the speed of light, \(m > 0\) is the mass of the electron. We show that the first component \(u\) and the last one \(\phi\) of ground state solutions for nonlinear Dirac-Klein-Gordon systems converge to zero and the second one \(v\) converges to corresponding solutions of a coupled system of nonlinear Schrödinger equations as the speed of light tends to infinity for electrons with small mass. Moreover, we also prove the uniform boundedness and the exponential decay properties
of the solutions for the nonlinear Dirac-Klein-Gordon systems with respect to the speed of light \(c\).Strichartz estimates for the Schrödinger equation with a measure-valued potentialhttps://zbmath.org/1491.353572022-09-13T20:28:31.338867Z"Erdoğan, M. Burak"https://zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Goldberg, Michael"https://zbmath.org/authors/?q=ai:goldberg.michael-joseph"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rIn the present paper, the authors obtain Strichartz estimates for the
Schrödinger equation in \(\mathbb{R}^{n},\) with the Hamiltonian
\(H=-\triangle+\mu,\) where the perturbation \(\mu\) is a compactly supported
measure in \(\mathbb{R}^{n}\) with dimension \(\alpha>n-\left( 1+\frac{1}
{n-1}\right) .\) The potentials considered in this paper are less singular
than a delta-function in \(\mathbb{R}^{n}\), but still not being absolutely
continuous with respect to Lebesgue measure. A canonical example of an
admissible potential considered in this paper is the surface measure of a
compact hypersurface \(\Sigma\) \(\subset\mathbb{R}^{n}\). More generally, the
authors work with compactly supported fractal measures (on \(\mathbb{R}^{n}\))
of a sufficiently high dimension.
Reviewer: Ivan Naumkin (Nice)Stabilization of the weakly coupled Schrödinger systemhttps://zbmath.org/1491.353582022-09-13T20:28:31.338867Z"Fu, Xiaoyu"https://zbmath.org/authors/?q=ai:fu.xiaoyu"Zhang, Hualei"https://zbmath.org/authors/?q=ai:zhang.hualei"Zhu, Xianzheng"https://zbmath.org/authors/?q=ai:zhu.xianzhengSummary: In this paper, we investigate the energy decay for solutions of the weakly coupled dissipative Schrödinger system. Among the \(m\)-coupled equations, only one equation is directly damped. Under some assumptions about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically with mixed boundary conditions, including the coupling of the Schrödinger system subject to Dirichlet and Robin type boundary conditions, respectively. The proof is based on some frequency estimates with an exponential loss on the resolvent operators, which will be solved by establishing an interpolation inequality for a suitable weakly coupled elliptic system.Perturbations of the Landau Hamiltonian: asymptotics of eigenvalue clustershttps://zbmath.org/1491.353592022-09-13T20:28:31.338867Z"Hernandez-Duenas, G."https://zbmath.org/authors/?q=ai:hernandez-duenas.gerardo"Pérez-Esteva, S."https://zbmath.org/authors/?q=ai:perez-esteva.salvador"Uribe, A."https://zbmath.org/authors/?q=ai:uribe.alejandro"Villegas-Blas, C."https://zbmath.org/authors/?q=ai:villegas-blas.carlosSummary: We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a short-range continuous potential. The spectrum of the operator forms eigenvalue clusters. We obtain a Szegő limit theorem for the eigenvalues in the clusters as the cluster index and the field strength \(B\) tend to infinity with a fixed ratio \(\mathcal{E}\). The answer involves the averages of the potential over circles of radius \(\sqrt{\mathcal{E}/2}\) (classical orbits). After rescaling, this becomes a semiclassical problem where the role of Planck's constant is played by \(2/B\). We also discuss a related inverse spectral result.Tightness of the solutions to approximating equations of the stochastic quantization equation associated with the weighted exponential quantum field model on the two-dimensional torushttps://zbmath.org/1491.353602022-09-13T20:28:31.338867Z"Hoshino, Masato"https://zbmath.org/authors/?q=ai:hoshino.masato"Kawabi, Hiroshi"https://zbmath.org/authors/?q=ai:kawabi.hiroshi"Kusuoka, Seiichiro"https://zbmath.org/authors/?q=ai:kusuoka.seiichiro-kusuokaSummary: We consider stochastic quantization associated with the weighted exponential quantum field model on the two-dimensional torus via a method of singular stochastic partial differential equations and show the tightness of the solutions to approximating equations of the stochastic quantization equation. If the model is not weighted, then the drift term of the stochastic quantization equation, which includes a renormalization term, is nonpositive or nonnegative. However, in the weighted case, generally the drift term is neither nonpositive nor nonnegative. We modify the argument in the case without weights and discuss the weighted model.
For the entire collection see [Zbl 1482.60003].How Lagrangian states evolve into random waveshttps://zbmath.org/1491.353612022-09-13T20:28:31.338867Z"Ingremeau, Maxime"https://zbmath.org/authors/?q=ai:ingremeau.maxime"Rivera, Alejandro"https://zbmath.org/authors/?q=ai:rivera.alejandroSummary: In this paper, we consider a compact connected manifold \((X,g)\) of negative curvature, and a family of semi-classical Lagrangian states \(f_h(x)=a(x)e^{i\phi (x)/h}\) on \(X\). For a wide family of phases \(\phi \), we show that \(f_h\), when evolved by the semi-classical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaronhttps://zbmath.org/1491.353622022-09-13T20:28:31.338867Z"Leopold, Nikolai"https://zbmath.org/authors/?q=ai:leopold.nikolai"Mitrouskas, David"https://zbmath.org/authors/?q=ai:mitrouskas.david"Rademacher, Simone"https://zbmath.org/authors/?q=ai:rademacher.simone"Schlein, Benjamin"https://zbmath.org/authors/?q=ai:schlein.benjamin"Seiringer, Robert"https://zbmath.org/authors/?q=ai:seiringer.robertSummary: We consider the Fröhlich Hamiltonian with large coupling constant \(\alpha \). For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm-approximation of the evolution, valid up to times of order \(\alpha^2\). The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order \(\alpha^2\).Concentration behavior of ground states for \( L^2\)-critical Schrödinger equation with a spatially decaying nonlinearityhttps://zbmath.org/1491.353632022-09-13T20:28:31.338867Z"Luo, Yong"https://zbmath.org/authors/?q=ai:luo.yong"Zhang, Shu"https://zbmath.org/authors/?q=ai:zhang.shuSummary: We consider ground states of the following time-independent nonlinear \(L^2\)-critical Schrödinger equation
\[
-\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{\frac{4-2b}{N}}u(x) = \mu u(x) \quad \text{in } \mathbb{R}^N,
\]
where \(\mu\in \mathbb{R}\), \(a>0\), \(N\geq 1\), \(0<b<\min\{2,N\}\), and \(V(x)\geq 0\) is an external potential. We get ground states of the above equation by solving the associated constrained minimization problem. In this paper, we prove that there is a threshold \(a^*>0\) such that minimizer exists for \(0<a<a^*\), and minimizer does not exist for any \(a>a^*\). However if \(a = a^*\), it is showed that whether minimizer exists depends sensitively on the value of \(V(0)\). Moreover if \(V(0) = 0\), we prove that minimizers must concentrate at the origin as \(a\nearrow a^*\) and give a detailed concentration behavior of minimizers as \(a\nearrow a^*\), based on which we finally prove that there is a unique minimizer when \(a\) is close enough to \(a^*\).Vortex-type solutions for magnetic pseudo-relativistic Hartree equationhttps://zbmath.org/1491.353642022-09-13T20:28:31.338867Z"Zhang, Guoqing"https://zbmath.org/authors/?q=ai:zhang.guoqing"Gao, Qian"https://zbmath.org/authors/?q=ai:gao.qianSummary: We deal with a class of magnetic pseudo-relativistic Hartree type equation
\[
\sqrt{(-i \nabla - A(x))^2 + m^2} u + W(x)u = (I_\alpha * |u|^p)|u|^{p-2} u, \quad x \in \mathbb{R}^N,
\]
where \(N \geq 3\), \(m>0\), \(A : \mathbb{R}^N \to \mathbb{R}^N\) is a continuous vector potential, \(W: \mathbb{R}^N \to \mathbb{R}\) is an external continuous scalar potential and \(I_\alpha (x) = (c_{N,\alpha} / |x|^{N-\alpha} (x \neq 0)\) is a convolution kernel, \(c_{N,\alpha}>0\) is a positive constant, \(2 \leq p < 2N / (N-1)\), \((N-1)p-N < \alpha < N\). Under the action of some subgroup of linear isometries on potential \(A\) and \(W\), and some assumptions on the decay of \(A\) and \(W\) at infinity, we prove the existence of vortex-type solutions to this problem by using variational methods and asymptotic estimates as \(p = 2\).The Cauchy problem for 3-evolution equations with data in Gelfand-Shilov spaceshttps://zbmath.org/1491.353652022-09-13T20:28:31.338867Z"Arias Junior, Alexandre"https://zbmath.org/authors/?q=ai:arias.alexandre-jun"Ascanelli, Alessia"https://zbmath.org/authors/?q=ai:ascanelli.alessia"Cappiello, Marco"https://zbmath.org/authors/?q=ai:cappiello.marcoThe authors consider the Cauchy problem for the 3-evolution operator $P = id/dt + A$ where $A$ is a third-order differential operator in the space variable $x$ with coefficients depending on $t$ and $x$. The principal symbol of $A$ is assumed real valued, to satisfy the assumptions of Lax-Mizohata, cf. [\textit{S. Mizohata}, J. Math. Kyoto Univ. 1, 109--127 (1961; Zbl 0104.31903)]. The authors first review results of well-posedness in different functional spaces, under corresponding assumptions on the imaginary part of the lower order symbols of $A$, cf. [\textit{M. Cicognani} and \textit{M. Reissig}, Evol. Equ. Control Theory 3, No. 1, 15--33 (2014; Zbl 1286.35086)]. The authors then present a new result in the frame of the Gelfand-Shilov spaces. Such functional frame allows very general assumptions, expressed in terms of the so-called SG-classes. Several examples are given to clarify the presentation.
Reviewer: Luigi Rodino (Torino)Growth of Sobolev norms for unbounded perturbations of the Schrödinger equation on flat torihttps://zbmath.org/1491.353662022-09-13T20:28:31.338867Z"Bambusi, Dario"https://zbmath.org/authors/?q=ai:bambusi.dario"Langella, Beatrice"https://zbmath.org/authors/?q=ai:langella.beatrice"Montalto, Riccardo"https://zbmath.org/authors/?q=ai:montalto.riccardoSummary: We prove a \(\langle t \rangle^\varepsilon\) upper bound on the growth of Sobolev norms for all solutions of Schrödinger equations on flat tori with a Hamiltonian which is an unbounded time dependent perturbation of the Laplacian.Mass propagation for electromagnetic Schrödinger evolutionshttps://zbmath.org/1491.353672022-09-13T20:28:31.338867Z"Barceló, Juan Antonio"https://zbmath.org/authors/?q=ai:barcelo.juan-antonio"Cassano, Biagio"https://zbmath.org/authors/?q=ai:cassano.biagio"Fanelli, Luca"https://zbmath.org/authors/?q=ai:fanelli.lucaSummary: We investigate the validity of Gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space-time cylinders is taken.A canonical model of the one-dimensional dynamical Dirac system with boundary controlhttps://zbmath.org/1491.353682022-09-13T20:28:31.338867Z"Belishev, Mikhail I."https://zbmath.org/authors/?q=ai:belishev.mikhail-igorevitch"Simonov, Sergey A."https://zbmath.org/authors/?q=ai:simonov.sergey-aSummary: The one-dimensional Dirac dynamical system \(\Sigma\) is
\[
iu_t+i\sigma_3\, u_x+Vu = 0, \quad x, t>0; \quad u|_{t = 0} = 0, \quad x>0; \quad u_1|_{x = 0} = f, \quad t>0,
\] where \(\sigma_3 = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix}\) is the Pauli matrix; \(V = \begin{pmatrix}0&p\\ \bar{p}&0\end{pmatrix}\) with \(p = p(x)\) is a potential; \(u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix}\) is the trajectory in \(\mathscr{H} = L_2(\mathbb{R}_+;\mathbb{C}^2); \, f\in\mathscr{F} = L_2([0, \infty);\mathbb{C})\) is a boundary control. System \(\Sigma\) is not controllable: the total reachable set \(\mathscr{U} = \mathrm{span}_{t>0}\{u^f(\cdot, t)\, |\, \, f \in \mathscr{F} \}\) is not dense in \(\mathscr{H}\), but contains a controllable part \(\Sigma_u\). We construct a dynamical system \(\Sigma_a\), which is controllable in \(L_2(\mathbb{R}_+;\mathbb{C})\) and connected with \(\Sigma_u\) via a unitary transform. The construction is based on geometrical optics relations: trajectories of \(\Sigma_a\) are composed of jump amplitudes that arise as a result of projecting in \(\overline{\mathscr{U}}\) onto the reachable sets \(\mathscr{U}^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr{F}\}\). System \(\Sigma_a\), which we call the \textit{amplitude model} of the original \(\Sigma\), has the same input/output correspondence as system \(\Sigma\). As such, \(\Sigma_a\) provides a canonical completely reachable realization of the Dirac system.Local in time Strichartz estimates for the Dirac equation on spherically symmetric spaceshttps://zbmath.org/1491.353692022-09-13T20:28:31.338867Z"Cacciafesta, Federico"https://zbmath.org/authors/?q=ai:cacciafesta.federico"de Suzzoni, Anne-Sophie"https://zbmath.org/authors/?q=ai:de-suzzoni.anne-sophieSummary: We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.Sharp exponential decay for solutions of the stationary perturbed Dirac equationhttps://zbmath.org/1491.353702022-09-13T20:28:31.338867Z"Cassano, Biagio"https://zbmath.org/authors/?q=ai:cassano.biagioWell-posedness and stability for Schrödinger equations with infinite memoryhttps://zbmath.org/1491.353712022-09-13T20:28:31.338867Z"Cavalcanti, M. M."https://zbmath.org/authors/?q=ai:cavalcanti.marcelo-moreira"Domingos Cavalcanti, V. N."https://zbmath.org/authors/?q=ai:domingos-cavalcanti.valeria-neves"Guesmia, A."https://zbmath.org/authors/?q=ai:guesmia.aissa"Sepúlveda, M."https://zbmath.org/authors/?q=ai:sepulveda.mauricio-aSummary: We study in this paper the well-posedness and stability for two linear Schrödinger equations in \(d\)-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sense of semigroup theory. Then, a decay estimate depending on the smoothness of initial data and the arbitrarily growth at infinity of the relaxation function is established for each equation with the help of multipliers method and some arguments devised in [\textit{A. Guesmia}, J. Math. Anal. Appl. 382, No. 2, 748--760 (2011; Zbl 1225.45005); Appl. Anal. 94, No. 1, 184--217 (2015; Zbl 1311.35026)].Pointwise convergence of the fractional Schrödinger equation in \(\mathbb{R}^2\)https://zbmath.org/1491.353722022-09-13T20:28:31.338867Z"Cho, Chu-Hee"https://zbmath.org/authors/?q=ai:cho.chu-hee"Ko, Hyerim"https://zbmath.org/authors/?q=ai:ko.hyerimSummary: We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in \(\mathbb{R}^2\). By establishing \(H^s(\mathbb{R}^2) - L^3(\mathbb{R}^2)\) estimates for the associated maximal operator provided that \(s > 1/3\), we improve the previous result obtained by \textit{C. Miao} et al. [Stud. Math. 230, No. 2, 121--165 (2015; Zbl 1343.42027)]. Our estimates extend the refined Strichartz estimates obtained by \textit{X. Du} et al. [Ann. Math. (2) 186, No. 2, 607--640 (2017; Zbl 1378.42011)] to a general class of elliptic functions.Global solution to the cubic Dirac equation in two space dimensionshttps://zbmath.org/1491.353732022-09-13T20:28:31.338867Z"Dong, Shijie"https://zbmath.org/authors/?q=ai:dong.shijie"Li, Kuijie"https://zbmath.org/authors/?q=ai:li.kuijieSummary: We are interested in the cubic Dirac equation with mass \(m \in [0, 1]\) in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the cubic Dirac equation, which is uniform-in-mass in the sense that the smallness condition on the initial data is independent of the mass parameter \(m\). In addition, we derive a unified pointwise decay result valid for all \(m \in [0, 1]\). Last but not least, we prove solution to the cubic Dirac equation scatters linearly. When the mass \(m = 0\), we can show an improved pointwise decay result.The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimateshttps://zbmath.org/1491.353742022-09-13T20:28:31.338867Z"Erdoğan, M. Burak"https://zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Goldberg, Michael"https://zbmath.org/authors/?q=ai:goldberg.michael-joseph"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rSummary: We investigate \(L^1\to L^\infty\) dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural \(t^{-\frac{1}{2}}\) decay rate, which may be improved to \(t^{-\frac{1}{2} - \gamma}\) for any \(0\leq \gamma < \frac{3}{2}\) at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.Hardy uncertainty principle for the linear Schrödinger equation on regular quantum treeshttps://zbmath.org/1491.353752022-09-13T20:28:31.338867Z"Fernández Bertolin, Aingeru"https://zbmath.org/authors/?q=ai:fernandez-bertolin.aingeru"Grecu, Andreea"https://zbmath.org/authors/?q=ai:grecu.andreea"Ignat, Liviu I."https://zbmath.org/authors/?q=ai:ignat.liviu-iSummary: In this paper we consider the linear Schrödinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the real line with a piece-wise constant coefficient and uses this result in the context of regular trees. The second part treats the case of a LSE with a real potential in the framework of a star-shaped graph.Spectral properties and time decay of the wave functions of Pauli and Dirac operators in dimension twohttps://zbmath.org/1491.353762022-09-13T20:28:31.338867Z"Kovařík, Hynek"https://zbmath.org/authors/?q=ai:kovarik.hynekSummary: We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows us to compute explicitly all the singular terms in the associated resolvent expansions.
As an application we show how the magnetic field influences the time decay of the associated wave-functions quantifying thereby the paramagnetic and diamagnetic effects of the spin.Growth of Sobolev norms for linear Schrödinger operatorshttps://zbmath.org/1491.353772022-09-13T20:28:31.338867Z"Thomann, Laurent"https://zbmath.org/authors/?q=ai:thomann.laurentSummary: We give an example of a linear, time-dependent, Schrödinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.Global in time self-interacting Dirac fields in the de Sitter spacehttps://zbmath.org/1491.353782022-09-13T20:28:31.338867Z"Yagdjian, Karen"https://zbmath.org/authors/?q=ai:yagdjian.karenSummary: In this paper the semilinear equation of the spin-\(\frac{1}{2}\) fields in the de Sitter space is investigated. We prove the existence of the global in time small data solution in the expanding de Sitter universe. Then, under the Lochak-Majorana condition, we prove the existence of the global in time solution with large data. The sufficient conditions for the solutions to blow up in finite time are given for large data in the expanding and contracting de Sitter spacetimes. The influence of the Hubble constant on the lifespan is estimated.Ground states for Dirac equation with singular potential and asymptotically periodic conditionhttps://zbmath.org/1491.353792022-09-13T20:28:31.338867Z"Yang, Gang"https://zbmath.org/authors/?q=ai:yang.gang"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian|zhang.jian.4|zhang.jian.2|zhang.jian.5|zhang.jian.1|zhang.jian.7|zhang.jian.6|zhang.jian.3Summary: In this paper we study the existence and asymptotic analysis of ground states for nonlinear Dirac equation with singular potential. Under the asymptotically periodic condition, using variational tools from non-Nehari manifold method, we establish a global compactness result and we prove that the existence of ground state solution and the continuous dependence of ground state energy about parameter as well as the asymptotic convergence of solutions.Convergence toward the steady state of a collisionless gas with Cercignani-Lampis boundary conditionhttps://zbmath.org/1491.353802022-09-13T20:28:31.338867Z"Bernou, Armand"https://zbmath.org/authors/?q=ai:bernou.armandSummary: We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence toward it, in the \(L^1\) norm. The strategy is an application of a deterministic version of Harris' subgeometric theorem, in the spirit of the recent results of Cañizo-Mischler and of the previous study of Bernou. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem.Smoothing does not give a selection principle for transport equations with bounded autonomous fieldshttps://zbmath.org/1491.353812022-09-13T20:28:31.338867Z"De Lellis, Camillo"https://zbmath.org/authors/?q=ai:de-lellis.camillo"Giri, Vikram"https://zbmath.org/authors/?q=ai:giri.vikramSummary: We give an example of a bounded divergence free autonomous vector field in \({\mathbb{R}}^3\) (and of a nonautonomous bounded divergence free vector field in \({\mathbb{R}}^2)\) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.Classical approximation of a linearized three waves kinetic equationhttps://zbmath.org/1491.353822022-09-13T20:28:31.338867Z"Escobedo, M."https://zbmath.org/authors/?q=ai:escobedo.miguel|escobedo.miguel-angelSummary: The purpose of this work is to solve the Cauchy problem for the classical approximation of an isotropic linearized three waves kinetic equation that appears in the kinetic theory of a condensed gas of bosons near the critical temperature. The fundamental solution is obtained, it is proved to be unique in a suitable space of distributions, and some of its regularity and integrability properties are described. The initial value problem for integrable and locally bounded initial data is then solved. Classical solutions are obtained as functions, whose regularity depends on time and that satisfy the expected conservation of energy.On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobilityhttps://zbmath.org/1491.353832022-09-13T20:28:31.338867Z"Fagioli, Simone"https://zbmath.org/authors/?q=ai:fagioli.simone"Tse, Oliver"https://zbmath.org/authors/?q=ai:tse.oliverSummary: We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic system of interacting particles that exhibits a gradient flow structure. At the same time, we expose a rigorous gradient flow structure for this class of equations in terms of an Energy-Dissipation balance, which we obtain via the asymptotic convergence of functionals.A new commutator method for averaging lemmashttps://zbmath.org/1491.353842022-09-13T20:28:31.338867Z"Jabin, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:jabin.pierre-emmanuel"Lin, Hsin-Yi"https://zbmath.org/authors/?q=ai:lin.hsinyi"Tadmor, Eitan"https://zbmath.org/authors/?q=ai:tadmor.eitanSummary: This document corresponds to the talk that the first author gave at the Laurent Schwartz seminar on March 10th 2020. It introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in \(L^2([0,T], H^s_x)\). The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.Existence results for nonlinear mono-energetic singular transport equations: \(L^p\)-spaceshttps://zbmath.org/1491.353852022-09-13T20:28:31.338867Z"Latrach, Khalid"https://zbmath.org/authors/?q=ai:latrach.khalid"Oummi, Hssaine"https://zbmath.org/authors/?q=ai:oummi.hssaine"Zeghal, Ahmed"https://zbmath.org/authors/?q=ai:zeghal.ahmedSummary: We establish some results regarding the existence of solutions to a nonlinear mono-energetic singular transport equation in slab geometry on \(L^p\)-spaces with \(p\in (1,+\infty)\). Both the cases where the boundary conditions are specular reflections and periodic are discussed.Transport equations with inflow boundary conditionshttps://zbmath.org/1491.353862022-09-13T20:28:31.338867Z"Scott, L. Ridgway"https://zbmath.org/authors/?q=ai:scott.larkin-ridgway"Pollock, Sara"https://zbmath.org/authors/?q=ai:pollock.saraSummary: We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late \textit{C. J. Amick} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 473--513 (1977; Zbl 0367.76027)], we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions.Monotone decomposition of the Cauchy problem for a hyperbolic equation based on transport equationshttps://zbmath.org/1491.353872022-09-13T20:28:31.338867Z"Shishkin, G. I."https://zbmath.org/authors/?q=ai:shishkin.grigorii-i"Shishkina, L. P."https://zbmath.org/authors/?q=ai:shishkina.lidia-pSummary: For the Cauchy problem for a hyperbolic equation, a multiplicative approach is developed: a monotone decomposition of the problem is constructed since the hyperbolic operator can be represented by a product of transport operators. The problem for the hyperbolic equation is reduced to a system of problems for transport equations -- transport in the direction of the axis \(x\) and transport in the opposite direction of the axis \(x\). Conditions for the monotonicity of each problem for the transport equations and for the entire multiplicative problem are found. Such a decomposition of the Cauchy problem based on transport problems solved one after the other significantly simplifies the solution of the hyperbolic equation, and the problems for the transport equations are monotone thus ensuring the monotonicity of the decomposition of the Cauchy problem for the hyperbolic equation.Internal rapid stabilization of a 1-D linear transport equation with a scalar feedbackhttps://zbmath.org/1491.353882022-09-13T20:28:31.338867Z"Zhang, Christophe"https://zbmath.org/authors/?q=ai:zhang.christopheSummary: We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \((0,L)\), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.Unstable kink and anti-kink profile for the sine-Gordon equation on a \({\mathcal{Y}} \)-junction graphhttps://zbmath.org/1491.353892022-09-13T20:28:31.338867Z"Angulo Pava, Jaime"https://zbmath.org/authors/?q=ai:angulo-pava.jaime"Plaza, Ramón G."https://zbmath.org/authors/?q=ai:plaza.ramon-gSummary: The sine-Gordon equation on a metric graph with a structure represented by a \(\mathcal{Y} \)-junction, is considered. The model is endowed with boundary conditions at the graph-vertex of \(\delta^{\prime}\)-interaction type, expressing continuity of the derivatives of the wave functions plus a Kirchhoff-type rule for the self-induced magnetic flux. It is shown that particular stationary, kink and kink/anti-kink soliton profile solutions to the model are linearly (and nonlinearly) unstable. To that end, a recently developed linear instability criterion for evolution models on metric graphs by \textit{J. Angulo Pava} and \textit{M. Cavalcante} [Nonlinearity 34, No. 5, 3373--3410 (2021; Zbl 1468.35164)], which provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is applied. This leads to the spectral study to the linearized operator and of its Morse index. The analysis is based on analytic perturbation theory, Sturm-Liouville oscillation results and the extension theory of symmetric operators. The methods presented in this manuscript have prospect for the study of the dynamics of solutions for the sine-Gordon model on metric graphs with finite bounds or on metric tree graphs and/or loop graphs.Spin generalizations of the Benjamin-Ono equationhttps://zbmath.org/1491.353902022-09-13T20:28:31.338867Z"Berntson, Bjorn K."https://zbmath.org/authors/?q=ai:berntson.bjorn-k"Langmann, Edwin"https://zbmath.org/authors/?q=ai:langmann.edwin"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatanSummary: We present new soliton equations related to the \(A\)-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.Almost \(\eta\)-Ricci and almost \(\eta\)-Yamabe solitons with torse-forming potential vector fieldhttps://zbmath.org/1491.353912022-09-13T20:28:31.338867Z"Blaga, Adara M."https://zbmath.org/authors/?q=ai:blaga.adara-monica"Özgür, Cihan"https://zbmath.org/authors/?q=ai:ozgur.cihanSummary: We provide properties of almost \(\eta\)-Ricci and almost \(\eta\)-Yamabe solitons on submanifolds isometrically immersed into a Riemannian manifold \(\widetilde{M}, \tilde{g}\) whose potential vector field is the tangential component of a torse-forming vector field on \(\widetilde{M}\), treating also the case of a minimal or pseudo quasi-umbilical hypersurface. Moreover, we give necessary and sufficient conditions for an orientable hypersurface of the unit sphere to be an almost \(\eta\)-Ricci or an almost \(\eta\)-Yamabe soliton in terms of the second fundamental tensor field.Kinetic equation for soliton gas: integrable reductionshttps://zbmath.org/1491.353922022-09-13T20:28:31.338867Z"Ferapontov, E. V."https://zbmath.org/authors/?q=ai:ferapontov.evgeny-vladimirovich"Pavlov, M. V."https://zbmath.org/authors/?q=ai:pavlov.maxim-vSummary: Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several \(2\times 2\) Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev's theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.Darboux transformation and exact solutions of the variable coefficient nonlocal Newell-Whitehead equationhttps://zbmath.org/1491.353932022-09-13T20:28:31.338867Z"Hu, Yuru"https://zbmath.org/authors/?q=ai:hu.yuru"Zhang, Feng"https://zbmath.org/authors/?q=ai:zhang.feng"Xin, Xiangpeng"https://zbmath.org/authors/?q=ai:xin.xiangpeng"Liu, Hanze"https://zbmath.org/authors/?q=ai:liu.hanzeSummary: In this article, the integrable nonlocal nonlinear variable coefficient Newell-Whitehead (NW) equation is investigated for the first time. First, the variable coefficient nonlocal NW equation is constructed with the aid of symmetry reduction and Lax pair. On this basis, the Darboux transformation of the variable coefficient nonlocal NW equation is studied. Then, some exact solutions are obtained by applying the Darboux transformation. The results show that the variable coefficient equation has more general solutions than its constant coefficient form. Finally, the solutions of the variable coefficient nonlocal NW equation are given when the coefficient function takes on special values, and the structural features of the solutions are visualized in images.Integrability, modulational instability and mixed localized wave solutions for the generalized nonlinear Schrödinger equationhttps://zbmath.org/1491.353942022-09-13T20:28:31.338867Z"Li, Xinyue"https://zbmath.org/authors/?q=ai:li.xinyue"Han, Guangfu"https://zbmath.org/authors/?q=ai:han.guangfu"Zhao, Qiulan"https://zbmath.org/authors/?q=ai:zhao.qiulanSummary: Under investigation in this paper is the generalized nonlinear Schrödinger (g-NLS) equation which has extensive applications in various physical fields. Firstly, we prove Liouville integrability of this equation by deriving its bi-Hamiltonian structures applying the variational identity. Nextly, we calculate the modulational instability for the possible reason of the formation of the rogue wave. Moreover, based on the generalized \((2, N-2)\)-fold Darboux transformation (DT), we can derive several mixed localized wave solutions such as breathers, rogue waves and semi-rational solitons for this equation, and accurately analyze a lot of important physical quantities. Finally, we present these solutions graphically by choosing appropriate parameters and discuss their dynamic behavior. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity, and useful to explain some physical phenomena in nonlinear optics.Abundant explicit non-traveling wave solutions for the (2+1)-dimensional breaking soliton equationhttps://zbmath.org/1491.353952022-09-13T20:28:31.338867Z"Shang, Yadong"https://zbmath.org/authors/?q=ai:shang.yadongSummary: By combining the generalized variable separation method with the extended homoclinic test approach (EHTA) explicit exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are constructed. With the aid of symbolic computation, a series of new non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are expressed explicitly. These non-traveling wave solutions are new solutions with three arbitrary functions, which have a more general form than that in the previous literatures. The result obtained here reveals the complex structure of the solutions of the (2+1)-dimensional breaking soliton equation. The previous results obtained in literatures can be regarded as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, cross soliton-like wave, periodic cross-kink wave, periodic two-solitary wave are presented.On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regionshttps://zbmath.org/1491.353962022-09-13T20:28:31.338867Z"Yang, Yiling"https://zbmath.org/authors/?q=ai:yang.yiling"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.enguiSummary: We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation in the solitonic regions
\[
m_t + (m (u^2 - u_x^2))_x + \kappa u_x = 0, \quad m = u - u_{x x}, \quad u(x, 0) = u_0(x) \in H^{4 , 2}(\mathbb{R}),
\]
where \(\kappa\) is a positive constant characterizing the effect of the linear dispersion. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale \((y, t)\). Further using the \(\overline{\partial}\) generalization of the Deift-Zhou steepest descent method, we derive different long time asymptotic expansions of the solution \(u(y, t)\) in different space-time solitonic regions of \(\xi = y / t\). We divide the half-plane \(\{(y, t) : - \infty < y < \infty, \, t > 0 \}\) into four asymptotic regions: The phase function \(\theta(z)\) has no stationary phase point on the jump contour in the space-time solitonic regions \(\xi \in(- \infty, - 1 / 4) \cup(2, + \infty)\), corresponding asymptotic approximations can be characterized with an \(N(\Lambda)\)-solitons with diverse residual error order \(\mathcal{O}(t^{- 1 + 2 \rho})\); The phase function \(\theta(z)\) has four phase points and eight phase points on the jump contour in the space-time solitonic regions \(\xi \in(0, 2)\) and \(\xi \in(- 1 / 4, 0)\), respectively. The corresponding asymptotic approximations can be characterized with an \(N(\Lambda)\)-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order \(\mathcal{O}(t^{- 3 / 4})\). Our results also confirm the soliton resolution conjecture and asymptotically stability of the N-soliton solutions for the mCH equation.Optical solitons in fiber Bragg gratings with quadratic-cubic law of nonlinear refractive index and cubic-quartic dispersive reflectivityhttps://zbmath.org/1491.353972022-09-13T20:28:31.338867Z"Zayed, Elsayed M. E."https://zbmath.org/authors/?q=ai:zayed.elsayed-m-e"Alngar, Mohamed E. M."https://zbmath.org/authors/?q=ai:alngar.mohamed-e-m"Biswas, Anjan"https://zbmath.org/authors/?q=ai:biswas.anjan"Ekici, Mehmet"https://zbmath.org/authors/?q=ai:ekici.mehmet"Khan, Salam"https://zbmath.org/authors/?q=ai:khan.salam"Alzahrani, Abdullah K."https://zbmath.org/authors/?q=ai:alzahrani.abdullah-khamis-hassan"Belic, Milivoj R."https://zbmath.org/authors/?q=ai:belic.milivoj-rSummary: This paper recovers cubic-quartic perturbed solitons in fiber Bragg gratings with quadratic-cubic law nonlinear refractive index. The unified Riccati equation expansion method and the modified Kudryashov's approach make this retrieval of soliton solutions possible. The parameter constraints, for the existence of such solitons, are also presented.\(2N\) parameter solutions to the Burgers' equationhttps://zbmath.org/1491.353982022-09-13T20:28:31.338867Z"Gaillard, Pierre"https://zbmath.org/authors/?q=ai:gaillard.pierre.1|gaillard.pierre|gaillard.pierre-yvesSummary: We construct \(2N\) real parameter solutions to the Burgers' equation in terms of determinant of order \(N\) and we call these solutions, \(N\) order solutions. We deduce general expressions of these solutions in terms of exponentials and study the patterns of these solutions in functions of the parameters for \(N=1\) until \(N=4\).Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider modelhttps://zbmath.org/1491.353992022-09-13T20:28:31.338867Z"Krichever, I."https://zbmath.org/authors/?q=ai:krichever.igor-moiseevich"Zabrodin, A."https://zbmath.org/authors/?q=ai:zabrodin.anton-vSummary: We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint \(\bar{{\mathcal{L}}}={{\mathcal{L}}}^{\dag}\) on the two Lax operators (in the symmetric gauge). We prove the existence of the tau function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau function. In this and some other respects, the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchanges the marked points in which the Baker-Akhiezer function has essential singularities.Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed termshttps://zbmath.org/1491.354002022-09-13T20:28:31.338867Z"Chen, Guanwei"https://zbmath.org/authors/?q=ai:chen.guanwei"Schechter, Martin"https://zbmath.org/authors/?q=ai:schechter.martinSummary: In infinite \(m\)-dimensional lattices, we obtain the existence of two nontrivial solutions for a class of non-periodic Schrödinger lattice systems with perturbed terms, where the potentials are coercive and the nonlinearities are asymptotically linear at infinity. In addition, examples are given to illustrate our results.On a stochastic Hardy-Littlewood-Sobolev inequality with application to Strichartz estimates for a noisy dispersionhttps://zbmath.org/1491.354012022-09-13T20:28:31.338867Z"Duboscq, Romain"https://zbmath.org/authors/?q=ai:duboscq.romain"Réveillac, Anthony"https://zbmath.org/authors/?q=ai:reveillac.anthonySummary: In this paper, we investigate a stochastic Hardy-Littlewood-Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.An orthogonalization-free parallelizable framework for all-electron calculations in density functional theoryhttps://zbmath.org/1491.354022022-09-13T20:28:31.338867Z"Gao, Bin"https://zbmath.org/authors/?q=ai:gao.bin"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Kuang, Yang"https://zbmath.org/authors/?q=ai:kuang.yang"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.1Existence of global solutions to a quasilinear Schrödinger equation with general nonlinear optimal control conditionshttps://zbmath.org/1491.354032022-09-13T20:28:31.338867Z"Hu, Yisheng"https://zbmath.org/authors/?q=ai:hu.yisheng"Qin, Songhai"https://zbmath.org/authors/?q=ai:qin.songhai"Liu, Zhibin"https://zbmath.org/authors/?q=ai:liu.zhibin"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.8|wang.yi.10|wang.yi.3|wang.yi.5|wang.yi.4|wang.yi.1|wang.yi.2|wang.yi.9|wang.yi.6|wang.yi.7The authors consider the global solutions of a fourth-order equation, involving the second power of the Laplacian and odd power nonlinearities. Solutions can be regarded as standing waves to a corresponding fourth-order Schrodinger equation, of relevance in Mathematical Physics. In the discussion, the authors use an interesting modified version of the classical maximum principle approach.
The editors-in-chief of the journal have retracted this article, for the motivations, see the retraction note [ibid. 2021, Paper No. 24, 1 p. (2021; Zbl 07509868)].
Reviewer: Luigi Rodino (Torino)Stochastic nonlinear Schrödinger equation on half-line with boundary noisehttps://zbmath.org/1491.354042022-09-13T20:28:31.338867Z"Kaikina, Elena I."https://zbmath.org/authors/?q=ai:kaikina.elena-igorevna"Sotelo-Garcia, Norma"https://zbmath.org/authors/?q=ai:sotelo-garcia.norma"Vázquez-Esquivel, Alexis V."https://zbmath.org/authors/?q=ai:vazquez-esquivel.alexis-vSummary: We consider the stochastic nonlinear Schrödinger equations on the half-line with Neumann brown-noise boundary conditions. We establish the global existence and uniqueness of solutions to initial-boundary value problem with values in \(\mathbf{H}^1\). We are also interested in the regularity behavior of the first spatial derivative of the solutions near the origin, where the boundary data are highly irregular. To obtain optimal estimate of the stochastic boundary response we propose new method based on Laplace transform and Cauchy theory of complex analysis. Also we adopt Sthriharts estimates and the Gagliardo-Nirenberg interpolation inequalities for the case of stochastic equations on a half-line.Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditionshttps://zbmath.org/1491.354052022-09-13T20:28:31.338867Z"Lin, Genghong"https://zbmath.org/authors/?q=ai:lin.genghong"Yu, Jianshe"https://zbmath.org/authors/?q=ai:yu.jian-sheNormalized multi-bump solutions of nonlinear Schrödinger equations via variational approachhttps://zbmath.org/1491.354062022-09-13T20:28:31.338867Z"Zhang, Chengxiang"https://zbmath.org/authors/?q=ai:zhang.chengxiang"Zhang, Xu"https://zbmath.org/authors/?q=ai:zhang.xu.2|zhang.xu|zhang.xu.3|zhang.xu.4|zhang.xu.1Summary: We investigate the existence and concentration behavior of the multi-bump solutions for the nonlinear Schrödinger equation \(-\hbar^2\Delta v-K(x)|v|^{2\sigma}v=-\lambda v\) with an \(L^2\)-constraint in the \(L^2\)-subcritical case \(\sigma\in (0, \frac{2}{N})\) and the \(L^2\)-supercritical case \(\sigma \in (\frac{2}{N}, \frac{2^*}{N})\), where \(N\ge 1\) is the dimension, \(\hbar >0\) is a small parameter and \(K>0\) possesses several local maximum points. By variational approach, we construct normalized multi-bump solutions concentrating at a finite set of local maximum points of \(K\). The construction combines the variational gluing arguments of \textit{E. Séré} [Math. Z. 209, No. 1, 27--42 (1992; Zbl 0725.58017)] and \textit{V. Coti Zelati} and \textit{P. H. Rabinowitz} [J. Am. Math. Soc. 4, No. 4, 693--727 (1991; Zbl 0744.34045); Commun. Pure Appl. Math. 45, No. 10, 1217--1269 (1992; Zbl 0785.35029)] and a penalization technique which is developed in order not to solve local minimization problems on the \(L^2\) sphere. Our approach is robust without imposing any nondegeneracy assumptions on \(K\).Existence and non-existence of global solutions for the semilinear complex Ginzburg-Landau type equation in homogeneous and isotropic spacetimehttps://zbmath.org/1491.354072022-09-13T20:28:31.338867Z"Nakamura, Makoto"https://zbmath.org/authors/?q=ai:nakamura.makoto"Sato, Yuya"https://zbmath.org/authors/?q=ai:sato.yuyaSummary: The Cauchy problem for the semilinear complex Ginzburg-Landau type equation is considered in homogeneous and isotropic spacetime. Global solutions and their asymptotic behaviours for small initial data are obtained. The non-existence of non-trivial global solutions is also shown. The effects of spatial expansion and contraction are studied through the problem.Time-dependent electromagnetic scattering from thin layershttps://zbmath.org/1491.354082022-09-13T20:28:31.338867Z"Nick, Jörg"https://zbmath.org/authors/?q=ai:nick.jorg"Kovács, Balázs"https://zbmath.org/authors/?q=ai:kovacs.balazs"Lubich, Christian"https://zbmath.org/authors/?q=ai:lubich.christianSummary: The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell's equations. The time-dependent boundary integral equation is discretized with Runge-Kutta based convolution quadrature in time and Raviart-Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge-Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.Coupled domain-boundary variational formulations for Hodge-Helmholtz operatorshttps://zbmath.org/1491.354092022-09-13T20:28:31.338867Z"Schulz, Erick"https://zbmath.org/authors/?q=ai:schulz.erick"Hiptmair, Ralf"https://zbmath.org/authors/?q=ai:hiptmair.ralfSummary: We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a ``rough'' surface. The low-frequency robustness of the potential formulation of Maxwell's equations makes this model a promising starting point for Galerkin discretization.Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3Dhttps://zbmath.org/1491.354102022-09-13T20:28:31.338867Z"Jonov, Boyan"https://zbmath.org/authors/?q=ai:jonov.boyan"Kessenich, Paul"https://zbmath.org/authors/?q=ai:kessenich.paul"Sideris, Thomas C."https://zbmath.org/authors/?q=ai:sideris.thomas-cSummary: The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulushttps://zbmath.org/1491.354112022-09-13T20:28:31.338867Z"Jang, Jin Woo"https://zbmath.org/authors/?q=ai:jang.jin-woo"Strain, Robert M."https://zbmath.org/authors/?q=ai:strain.robert-m"Wong, Tak Kwong"https://zbmath.org/authors/?q=ai:wong.tak-kwongSummary: Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the \textit{Vlasov-Maxwell} system in a two-dimensional annulus when a huge (\textit{but finite-in-time}) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite \textit{within a finite time interval} and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the \textit{Vlasov-Maxwell} system.Kinetic description of stable white dwarfshttps://zbmath.org/1491.354122022-09-13T20:28:31.338867Z"Jang, Juhi"https://zbmath.org/authors/?q=ai:jang.juhi"Seok, Jinmyoung"https://zbmath.org/authors/?q=ai:seok.jinmyoungSummary: In this paper, we study fermion ground states of the relativistic Vlasov-Poisson system arising in the semiclassical limit from relativistic quantum theory of white dwarfs. We show that fermion ground states of the three dimensional relativistic Vlasov-Poisson system exist for subcritical mass, the mass density of such fermion ground states satisfies the Chandrasekhar equation for white dwarfs, and that they are orbitally stable as long as solutions exist.A second look at the Kurth solution in galactic dynamicshttps://zbmath.org/1491.354132022-09-13T20:28:31.338867Z"Kunze, Markus"https://zbmath.org/authors/?q=ai:kunze.markus|kunze.markus-christianSummary: The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state \(Q(x, v) = \tilde{Q}(e_Q, \beta)\), depending upon the particle energy \(e_Q\) and \(\beta = \ell^2 = |x\wedge v|^2 \), the question arises if solutions \(f\) could be generated that are of the form
\[
f(t) = \tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big)
\] for suitable functions \(R, P\) and \(B \), all depending on \((t, r, p_r, \beta)\) for \(r = |x|\) and \(p_r = \frac{x\cdot v}{|x|}\). We are going to show that, under some mild assumptions, basically if \(R\) and \(P\) are independent of \(\beta \), and if \(B = \beta\) is constant, then \(Q\) already has to be the Kurth solution. This paper is dedicated to the memory of Professor Robert Glassey.Linear instability of Vlasov-Maxwell systems revisited -- a Hamiltonian approachhttps://zbmath.org/1491.354142022-09-13T20:28:31.338867Z"Lin, Zhiwu"https://zbmath.org/authors/?q=ai:lin.zhiwuSummary: We consider linear stability of steady states of \(1\frac{1}{2}\) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.The Fokker-Planck equation with subcritical confinement forcehttps://zbmath.org/1491.354152022-09-13T20:28:31.338867Z"Kavian, Otared"https://zbmath.org/authors/?q=ai:kavian.otared"Mischler, Stéphane"https://zbmath.org/authors/?q=ai:mischler.stephane"Ndao, Mamadou"https://zbmath.org/authors/?q=ai:ndao.mamadouSummary: We consider the Fokker-Planck equation with subcritical confinement force field which may not derive from a potential function. We prove the existence of a unique positive equilibrium of mass one and we establish some subgeometric, or geometric, rate of convergence to a multiple of this equilibrium (depending on the space to which belongs the initial datum) in many spaces. Our results generalize similar results introduced by \textit{G. Toscani} and \textit{C. Villani} [J. Stat. Phys. 98, No. 5--6, 1279--1309 (2000; Zbl 1034.82032)] and \textit{M. Röckner} and \textit{F.-Y. Wang} [J. Funct. Anal. 185, No. 2, 564--603 (2001; Zbl 1009.47028)] for some forces associated to a potential and extended by \textit{R. Douc} et al. [Stochastic Processes Appl. 119, No. 3, 897--923 (2009; Zbl 1163.60034)] and \textit{D. Bakry} et al. [J. Funct. Anal. 254, No. 3, 727--759 (2008; Zbl 1146.60058)] for some general forces: however in our approach the spaces are more general, and the rates of convergence to equilibrium are sharper.The Einstein-Vlasov system in maximal areal coordinates-local existence and continuationhttps://zbmath.org/1491.354162022-09-13T20:28:31.338867Z"Günther, Sebastian"https://zbmath.org/authors/?q=ai:gunther.sebastian"Rein, Gerhard"https://zbmath.org/authors/?q=ai:rein.gerhardSummary: We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system, but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.SIS reaction-diffusion model with risk-induced dispersal under free boundaryhttps://zbmath.org/1491.354172022-09-13T20:28:31.338867Z"Choi, Wonhyung"https://zbmath.org/authors/?q=ai:choi.wonhyung"Lin, Zhigui"https://zbmath.org/authors/?q=ai:lin.zhigui"Ahn, Inkyung"https://zbmath.org/authors/?q=ai:ahn.inkyungSummary: A spatial susceptible-infected-susceptible epidemic model with a free boundary, where infected individuals disperse non-uniformly, is investigated in this study. Spatial heterogeneity and movement of individuals are essential factors that affect pandemics and the eradication of infectious diseases. Our goal is to investigate the effect of a dispersal strategy for infected individuals, known as risk-induced dispersal (RID), which represents the motility of infected individuals induced by risk depending on whether they are in a high- or a low-risk region. We first construct the basic reproduction number and then understand the manner in which a nonuniform movement of infected individuals affects the spreading-vanishing dichotomy of a disease in a one-dimensional domain. We conclude that even though the infected individuals reside in a high-risk initial domain, the disease can be eradicated from the region if the infected individuals move with a high sensitivity of RID as they disperse. Finally, we demonstrate our results via simulations for a one-dimensional case.Sobolev spaces and \(\nabla\)-differential operators on manifolds. I: Basic properties and weighted spaceshttps://zbmath.org/1491.354182022-09-13T20:28:31.338867Z"Kohr, Mirela"https://zbmath.org/authors/?q=ai:kohr.mirela"Nistor, Victor"https://zbmath.org/authors/?q=ai:nistor.victorSummary: We study \textit{covariant Sobolev spaces} and \(\nabla\)-\textit{differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted \(\nabla)\). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the \(\nabla\)-differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection \(\nabla\), as long as the new connection is obtained using a totally bounded perturbation. We also introduce the \textit{Fréchet finiteness condition} (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our \(\nabla\)-differential operators. We also introduce and study the notion of a \(\nabla\)-\textit{bidifferential} operator (a bilinear version of differential operators), obtaining results similar to those obtained for \(\nabla\)-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.The wave equation with symmetric velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional spherehttps://zbmath.org/1491.354192022-09-13T20:28:31.338867Z"Shafarevich, A. I."https://zbmath.org/authors/?q=ai:shafarevich.andrei-i"Tsvetkova, A. V."https://zbmath.org/authors/?q=ai:tsvetkova.anna-vSummary: In the paper, the Cauchy problem for the wave equation with variable (symmetric) velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere is considered. It is assumed that the initial conditions are localized on the ray and the velocity on the sphere depends only on the geodesic distance to the gluing point. The asymptotic series of the solution of the problem as parameter characterizing the initial perturbation tends to zero is given. Since the sphere is compact, then the wave propagating over the sphere is reflected at the pole opposite to the gluing point and returns to the ray. Thus, the question of the distribution of wave energy at every moment of time is also interested and discussed in this work.Existence of radial solutions for a weighted \(p\)-biharmonic problem with Navier boundary condition on the Heisenberg grouphttps://zbmath.org/1491.354202022-09-13T20:28:31.338867Z"Safari, Farzaneh"https://zbmath.org/authors/?q=ai:safari.farzaneh"Razani, Abdolrahman"https://zbmath.org/authors/?q=ai:razani.abdolrahmanSummary: The existence of at least one positive radial solution of the \(p\)-biharmonic problem
\[
\Delta_{\mathbb{H}^n}\big(w(\xi)|\Delta_{\mathbb{H}^n}u|^{p-2}\Delta_{\mathbb{H}^n}u\big)+ R (\xi)w(\xi) |u|^{p-2}u=\sum\limits_{i=1}^ma_i (|\xi|_{\mathbb{H}^n})|u|^{q_i-2} u -\sum\limits_{j=1}^kb_j (|\xi|_{\mathbb{H}^n})|u|^{r_j-2}u,
\]
with Navier boundary condition on a Korányi ball is proved, where \(w \in A_s\) is a Muckenhoupt weight function and \(\Delta^2_{\mathbb{H}^n,p}\) is the Heisenberg \(p\)-biharmonic operator.Nonlinear parabolic equations for measureshttps://zbmath.org/1491.354212022-09-13T20:28:31.338867Z"Manita, O. A."https://zbmath.org/authors/?q=ai:manita.oxana-a"Shaposhnikov, S. V."https://zbmath.org/authors/?q=ai:shaposhnikov.stanislav-v(no abstract)Existence, uniqueness and mass conservation for Safronov-Dubovski coagulation equationhttps://zbmath.org/1491.354222022-09-13T20:28:31.338867Z"Kaushik, Sonali"https://zbmath.org/authors/?q=ai:kaushik.sonali"Kumar, Rajesh"https://zbmath.org/authors/?q=ai:kumar.rajesh-sSummary: In this work, the existence and uniqueness of solutions for Safronov-Dubovski coagulation equation are studied for unbounded coagulation kernel, namely \(\min\{i,j\}V_{i,j} \leq (i+j)\) \(\forall i,j \geq 1\). For the existence of solution, classical approach of \textit{J. M. Ball} and \textit{J. Carr} [J. Stat. Phys. 61, No. 1--2, 203--234 (1990; Zbl 1217.82050)] for solving discrete coagulation-fragmentation equations is adopted. Further, a contraction mapping theorem is used to establish the uniqueness. Moreover, it is also shown that the obtained unique solution is mass conserving.\(W^{\sigma, p}\) a priori estimates for fully nonlinear integro-differential equationshttps://zbmath.org/1491.354232022-09-13T20:28:31.338867Z"Kitano, Shuhei"https://zbmath.org/authors/?q=ai:kitano.shuheiSummary: \(W^{\sigma, p}\) estimates are studied for a class of fully nonlinear integro-differential equations of order \(\sigma\), which are analogues of \(W^{2, p}\) estimates by Caffarelli. We also present Aleksandrov-Bakelman-Pucci maximum principles, which are improvements of estimates proved by Guillen-Schwab, depending only on \(L^p\) norms of inhomogeneous terms.A nonlinear fractional problem with a second kind integral condition for time-fractional partial differential equationhttps://zbmath.org/1491.354242022-09-13T20:28:31.338867Z"Abdelouahab, Benbrahim"https://zbmath.org/authors/?q=ai:abdelouahab.benbrahim"Oussaeif, Taki-Eddine"https://zbmath.org/authors/?q=ai:taki-eddine.oussaeif"Ouannas, Adel"https://zbmath.org/authors/?q=ai:ouannas.adel"Saad, Khaled M."https://zbmath.org/authors/?q=ai:saad.khaled-mohammed"Jahanshahi, Hadi"https://zbmath.org/authors/?q=ai:jahanshahi.hadi"Diar, Ahmed"https://zbmath.org/authors/?q=ai:diar.ahmed"Aljuaid, Awad M."https://zbmath.org/authors/?q=ai:aljuaid.awad-m"Aly, Ayman A."https://zbmath.org/authors/?q=ai:aly.ayman-a(no abstract)Solution of fractional partial differential equations using fractional power series methodhttps://zbmath.org/1491.354252022-09-13T20:28:31.338867Z"Ali, Asif Iqbal"https://zbmath.org/authors/?q=ai:ali.asif-iqbal"Kalim, Muhammad"https://zbmath.org/authors/?q=ai:kalim.muhammad"Khan, Adnan"https://zbmath.org/authors/?q=ai:khan.adnan-qadir|khan.adnan-a(no abstract)Novel analysis of fuzzy fractional Klein-Gordon model via semianalytical methodhttps://zbmath.org/1491.354262022-09-13T20:28:31.338867Z"Alshammari, Mohammad"https://zbmath.org/authors/?q=ai:alshammari.mohammad"Mohammed, Wael W."https://zbmath.org/authors/?q=ai:mohammed.wael-w"Yar, Mohammed"https://zbmath.org/authors/?q=ai:yar.mohammed(no abstract)Exact solutions of the 3D fractional Helmholtz equation by fractional differential transform methodhttps://zbmath.org/1491.354272022-09-13T20:28:31.338867Z"Alshammari, Saleh"https://zbmath.org/authors/?q=ai:alshammari.saleh"Abuasad, Salah"https://zbmath.org/authors/?q=ai:abuasad.salah(no abstract)Multiple solutions for a binonlocal fractional \(p(x,\cdot)\)-Kirchhoff type problemhttps://zbmath.org/1491.354282022-09-13T20:28:31.338867Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Benkirane, Abdelmoujib"https://zbmath.org/authors/?q=ai:benkirane.abdelmoujib"Shimi, Mohammed"https://zbmath.org/authors/?q=ai:shimi.mohammed"Srati, Mohammed"https://zbmath.org/authors/?q=ai:srati.mohammedSummary: We are interested in the multiplicity of weak solutions for a binonlocal fractional \(p(x,\cdot)\)-Kirchhoff type problems. Our technical approach is based on the general three critical points theorem obtained by B. Ricceri.Existence of infinitely many solutions for fractional \(p\)-Laplacian Schrödinger-Kirchhof-type equations with general potentialshttps://zbmath.org/1491.354292022-09-13T20:28:31.338867Z"Benhamida, Ghania"https://zbmath.org/authors/?q=ai:benhamida.ghania"Moussaoui, Toufik"https://zbmath.org/authors/?q=ai:moussaoui.toufikAnalytical solutions for the equal width equations containing generalized fractional derivative using the efficient combined methodhttps://zbmath.org/1491.354302022-09-13T20:28:31.338867Z"Derakhshan, Mohammadhossein"https://zbmath.org/authors/?q=ai:derakhshan.mohammadhossein(no abstract)Boundary-value problem for the Aller-Lykov nonlocal moisture transfer equationhttps://zbmath.org/1491.354312022-09-13T20:28:31.338867Z"Gekkieva, S. Kh."https://zbmath.org/authors/?q=ai:gekkieva.sakinat-khadanovna|gekkieva.sakinat-khasanovna"Kerefov, M. A."https://zbmath.org/authors/?q=ai:kerefov.marat-aslanbievichSummary: In this paper, a boundary-value problem for the inhomogeneous Aller-Lykov moisture transfer equation with a fractional Riemann-Liouville time derivative is examined. The equation considered is a generalization of the Aller-Lykov equation obtained by introducing the fractal rate of change of humidity, which explains the appearance of flows directed against the potential of humidity. The existence of a solution to the first boundary-value problem is proved by the Fourier method. Using the method of energy inequalities for solutions of the problem, we obtain an a priori estimate in terms of the fractional Riemann-Liouville derivative, which implies the uniqueness of the solution.Reference tracking and observer design for space fractional partial differential equation modeling gas pressures in fractured mediahttps://zbmath.org/1491.354322022-09-13T20:28:31.338867Z"Ghaffour, Lilia"https://zbmath.org/authors/?q=ai:ghaffour.lilia"Laleg-Kirati, Taous-Meriem"https://zbmath.org/authors/?q=ai:laleg-kirati.taous-meriemTime-fractional Moore-Gibson-Thompson equationshttps://zbmath.org/1491.354332022-09-13T20:28:31.338867Z"Kaltenbacher, Barbara"https://zbmath.org/authors/?q=ai:kaltenbacher.barbara"Nikolić, Vanja"https://zbmath.org/authors/?q=ai:nikolic.vanjaFractional McKean-Vlasov and Hamilton-Jacobi-Bellman-Isaacs equationshttps://zbmath.org/1491.354342022-09-13T20:28:31.338867Z"Kolokoltsov, V. N."https://zbmath.org/authors/?q=ai:kolokoltsov.vassili-n"Troeva, M. S."https://zbmath.org/authors/?q=ai:troeva.marianna-sSummary: We study a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the McKean-Vlasov type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman-Isaacs (HJB-Isaacs) equations of stochastic control and games. This approach allows us to develop a unified analysis of these equations. We establish their well-posedness in the sense of classical solutions and prove the continuous dependence of the solutions on the initial data. The obtained results are extended to the case of generalized fractional equations.The obstacle problem and the Perron method for nonlinear fractional equations in the Heisenberg grouphttps://zbmath.org/1491.354352022-09-13T20:28:31.338867Z"Piccinini, Mirco"https://zbmath.org/authors/?q=ai:piccinini.mircoSummary: We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron-Wiener-Brelot generalized solution by proving its existence for very general boundary data.Green function of the first boundary-value problem for the fractional diffusion-wave equation in a multidimensional rectangular domainhttps://zbmath.org/1491.354362022-09-13T20:28:31.338867Z"Pskhu, A. V."https://zbmath.org/authors/?q=ai:pskhu.arsen-vladimirovichSummary: In this paper, the Green functions of the first boundary-value problem for the fractional diffusion-wave equation in multidimensional (bounded and unbounded) hyper-rectangular domains are constructed.Novel evaluation of fuzzy fractional Cauchy reaction-diffusion equationhttps://zbmath.org/1491.354372022-09-13T20:28:31.338867Z"Shah, Nehad Ali"https://zbmath.org/authors/?q=ai:shah.nehad-ali"El-Zahar, Essam R."https://zbmath.org/authors/?q=ai:el-zahar.essam-roshdy"Dutt, Hina M."https://zbmath.org/authors/?q=ai:dutt.hina-m"Arefin, Mohammad Asif"https://zbmath.org/authors/?q=ai:arefin.mohammad-asif(no abstract)Local existence and nonexistence for fractional in time reaction-diffusion equations and systems with rapidly growing nonlinear termshttps://zbmath.org/1491.354382022-09-13T20:28:31.338867Z"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: We study the fractional in time reaction-diffusion equation
\[
\begin{cases}
\partial_t^\alpha u = \Delta u + f (u) & \text{in } \mathbb{R}^N \times (0, T), \\
u (x, 0) = u_0 (x) & \text{in } \mathbb{R}^N,
\end{cases}
\] where \(0 < \alpha < 1\), \(N \geq 1\), \(T > 0\) and \(u_0 \geq 0\). The fractional derivative \(\partial_t^\alpha\) is meant in a generalized Caputo sense. We mainly consider the case where \(f\) has an exponential or a superexponential growth, and \(u_0\) has a singularity. We obtain integrability conditions on \(u_0\) which explicitly determine local in time existence/nonexistence of a nonnegative mild solution. Moreover, our analysis can be applied to time fractional systems.A unified three-dimensional extended fractional analytical solution for air pollutants dispersionhttps://zbmath.org/1491.354392022-09-13T20:28:31.338867Z"Sylvain, Tankou Tagne Alain"https://zbmath.org/authors/?q=ai:sylvain.tankou-tagne-alain"Patrice, Ele Abiama"https://zbmath.org/authors/?q=ai:patrice.ele-abiama"Marie, Ema'a Ema'a Jean"https://zbmath.org/authors/?q=ai:marie.emaa-emaa-jean"Pierre, Owono Ateba"https://zbmath.org/authors/?q=ai:pierre.owono-ateba"Hubert, Ben-Bolie Germain"https://zbmath.org/authors/?q=ai:hubert.ben-bolie-germainShort-time behavior for a class of semilinear nonlocal evolution equations in Hilbert spaceshttps://zbmath.org/1491.354402022-09-13T20:28:31.338867Z"Tuan, Tran Van"https://zbmath.org/authors/?q=ai:tuan.tran-vanSummary: This paper studies the global solvability and finite-time attractivity of solutions to a class of semilinear evolution equations in Hilbert spaces. Our analysis is based on the theory of integral equations with completely positive kernel, the fixed point theory and local estimates of solutions. An application to semilinear integro-differential equations of parabolic type will be shown.Infinitely many vector solutions of a fractional nonlinear Schrödinger system with strong competitionhttps://zbmath.org/1491.354412022-09-13T20:28:31.338867Z"Xu, Ruijin"https://zbmath.org/authors/?q=ai:xu.ruijin"Tian, Rushun"https://zbmath.org/authors/?q=ai:tian.rushunSummary: In this paper, we prove the existence of infinitely many nonnegative vector solutions to a symmetric fractional nonlinear Schrödinger system, where the coupling parameter is sufficiently large (strong competition). Our arguments can be used to refine the multiplicity results of the corresponding local system, in the sensed that more general exponents and space dimensions are allowed.Novel evaluation of fuzzy fractional Helmholtz equationshttps://zbmath.org/1491.354422022-09-13T20:28:31.338867Z"Alesemi, Meshari"https://zbmath.org/authors/?q=ai:alesemi.meshari"Iqbal, Naveed"https://zbmath.org/authors/?q=ai:iqbal.naveed-h"Wyal, Noorolhuda"https://zbmath.org/authors/?q=ai:wyal.noorolhuda(no abstract)Regularization of the backward stochastic heat conduction problemhttps://zbmath.org/1491.354432022-09-13T20:28:31.338867Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Lesnic, Daniel"https://zbmath.org/authors/?q=ai:lesnic.daniel"Thach, Tran Ngoc"https://zbmath.org/authors/?q=ai:thach.tran-ngoc"Ngoc, Tran Bao"https://zbmath.org/authors/?q=ai:ngoc.tran-baoSummary: In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.A new coupled complex boundary method (CCBM) for an inverse obstacle problemhttps://zbmath.org/1491.354442022-09-13T20:28:31.338867Z"Afraites, Lekbir"https://zbmath.org/authors/?q=ai:afraites.lekbirSummary: In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion \(\omega\) contained in a larger bounded domain \(\Omega\) via boundary measurements on \(\partial\Omega\). In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [\textit{X. Cheng} et al., Inverse Probl. 30, No. 5, Article ID 055002, 20 p. (2014; Zbl 1290.35324)]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion \(\omega\). Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain \(\omega\). We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditionshttps://zbmath.org/1491.354452022-09-13T20:28:31.338867Z"Ait Ben Hassi, El Mustapha"https://zbmath.org/authors/?q=ai:benhassi.el-mustapha-ait"Chorfi, Salah-Eddine"https://zbmath.org/authors/?q=ai:chorfi.salah-eddine"Maniar, Lahcen"https://zbmath.org/authors/?q=ai:maniar.lahcenSummary: We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.Lorentzian Calderón problem under curvature boundshttps://zbmath.org/1491.354462022-09-13T20:28:31.338867Z"Alexakis, Spyros"https://zbmath.org/authors/?q=ai:alexakis.spyros"Feizmohammadi, Ali"https://zbmath.org/authors/?q=ai:feizmohammadi.ali"Oksanen, Lauri"https://zbmath.org/authors/?q=ai:oksanen.lauriSummary: We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior region of double null cones. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.Solution of inverse Euler-Bernoulli problem with integral overdetermination and periodic boundary conditionshttps://zbmath.org/1491.354472022-09-13T20:28:31.338867Z"Baglan, Irem"https://zbmath.org/authors/?q=ai:baglan.irem-sakinc"Kanca, Fatma"https://zbmath.org/authors/?q=ai:kanca.fatma"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this work, we tried to find the inverse coefficient in the Euler problem with over determination conditions. It showed the existence, stability of the solution by iteration method and linearization method was used for this problem in numerical part. Also two examples are presented with figures.Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operatorshttps://zbmath.org/1491.354482022-09-13T20:28:31.338867Z"Bhattacharyya, Sombuddha"https://zbmath.org/authors/?q=ai:bhattacharyya.sombuddha"de Hoop, Maarten V."https://zbmath.org/authors/?q=ai:de-hoop.maarten-v"Katsnelson, Vitaly"https://zbmath.org/authors/?q=ai:katsnelson.vitaly"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-aSummary: We revisit the problem of recovering wave speeds and density across a curved interface from reflected wave amplitudes. Such amplitudes have been exploited for decades in (exploration) seismology in this context. However, the analysis in seismology has been based on linearization and mostly flat interfaces. Here, we present an analysis without linearization and allow curved interfaces, establish uniqueness and provide a reconstruction, while making the notion of amplitude precise through a procedure rooted in microlocal analysis.Stability for quantitative photoacoustic tomography revisitedhttps://zbmath.org/1491.354492022-09-13T20:28:31.338867Z"Bonnetier, Eric"https://zbmath.org/authors/?q=ai:bonnetier.eric"Choulli, Mourad"https://zbmath.org/authors/?q=ai:choulli.mourad"Triki, Faouzi"https://zbmath.org/authors/?q=ai:triki.faouziSummary: This paper deals with the issue of stability in determining the absorption and the diffusion coefficients in quantitative photoacoustic imaging. We establish a global conditional Hölder stability inequality from the knowledge of two internal data obtained from optical waves, generated by two point sources in a region where the optical coefficients are known.On an inverse boundary value problem for a nonlinear time-harmonic Maxwell systemhttps://zbmath.org/1491.354502022-09-13T20:28:31.338867Z"Cârstea, Cătălin I."https://zbmath.org/authors/?q=ai:carstea.catalin-ionSummary: This paper considers a class of nonlinear time-harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form \(\mathscr{X} (x,|\vec{E}(x)|^2)\vec{E}(x), \mathscr{Y}(x,|\vec{H}(x)|^2)\vec{H}(x)\) such that \(\mathscr{X}(x,s)\), \(\mathscr{Y}(x,s)\) are both real analytic in \(s\). Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions for \(\mathcal{X}\) and \(\mathcal{Y} \), it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonlinear terms.The linear sampling method for penetrable cylinder with inclusions for obliquely incident polarized electromagnetic waveshttps://zbmath.org/1491.354512022-09-13T20:28:31.338867Z"Deng, Xia"https://zbmath.org/authors/?q=ai:deng.xia"Guo, Jun"https://zbmath.org/authors/?q=ai:guo.jun"Li, Jin"https://zbmath.org/authors/?q=ai:li.jin|li.jin.4|li.jin.1|li.jin.3|li.jin.5|li.jin.2"Yan, Guozheng"https://zbmath.org/authors/?q=ai:yan.guozhengSummary: Consider the scattering of electromagnetic waves by a penetrable homogeneous cylinder at oblique incident. The Maxwell equations are then reduced to a system of a pair of the two-dimensional Helmholtz equations for \(z\)-components of the electric and magnetic field through coupled oblique boundary conditions. This paper studies an inverse problem of recovering the penetrable obstacle from the far-field pattern of the electric field. The well-known linear sampling method is used to solve this problem. Compared with the usual inverse scattering problem, the coupled system and the oblique derivative boundary condition bring difficulties in theoretical analysis. Some numerical examples are presented to illustrate the validity and feasibility of the proposed method.Simultaneous recovery of Robin boundary and coefficient for the Laplace equation by shape derivativehttps://zbmath.org/1491.354522022-09-13T20:28:31.338867Z"Fang, Weifu"https://zbmath.org/authors/?q=ai:fang.weifuSummary: We study the simultaneous recovery of the boundary and coefficient of the Robin boundary condition for the Laplace equation from a pair of solution measurements on another part of the boundary. We derive the variational derivatives of the data-fitting objective functional with respect to the Robin boundary and coefficient, which are then used to device a nonlinear conjugate gradient iterative scheme for the numerical recovery of both the Robin boundary and coefficient together. Numerical examples are presented to illustrate the effectiveness of the recovery algorithms.Identifications for general degenerate problems of hyperbolic type in Hilbert spaceshttps://zbmath.org/1491.354532022-09-13T20:28:31.338867Z"Favini, A."https://zbmath.org/authors/?q=ai:favini.angelo"Marinoschi, G."https://zbmath.org/authors/?q=ai:marinoschi.gabriela"Tanabe, H."https://zbmath.org/authors/?q=ai:tanabe.hiroki"Yakubov, Ya."https://zbmath.org/authors/?q=ai:yakubov.yakovSummary: In a Hilbert space X, we consider the abstract problem
\[{M}^{\ast}\frac{d}{dt}\left( My(t)\right)= Ly(t)+f(t)z,\quad 0\le t\le \tau,\]
\[My(0)={My}_0,\]
where \(L\) is a closed linear operator in \(X\) and \(M \in \mathcal{L} (X)\) is not necessarily invertible, \(z \in X\). Given the additional information \(\Phi [ My (t)] = g(t)\) with \(\Phi \in X^*\), \(g \in C^1([0, \tau ];\mathbb{C})\), we are concerned with the determination of the conditions under which we can identify \(f \in C([0, \tau ];\mathbb{C} )\) such that \(y\) be a strict solution to the abstract problem, i.e., \( My \in C^1([0, \tau ];X)\), \(Ly \in C([0, \tau ];X)\). A similar problem is considered for general second-order equations in time. Various examples of these general problems are given.Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurementshttps://zbmath.org/1491.354542022-09-13T20:28:31.338867Z"Favini, Angelo"https://zbmath.org/authors/?q=ai:favini.angelo"Mola, Gianluca"https://zbmath.org/authors/?q=ai:mola.gianluca"Romanelli, Silvia"https://zbmath.org/authors/?q=ai:romanelli.silviaSummary: Let \((H,\langle\cdot,\cdot\rangle)\) be a separable Hilbert space and \(A_i:D(A_i)\to H(i=1,\cdots,n)\) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function \(u:[0,T]\to H\) and \(n\) time-dependent \textit{diffusion coefficients} \(\alpha_1,\cdots,\alpha_n:[s,T]\to\mathbb{R}_+\) that fulfill the initial-value problem
\[
u'(t)+\alpha_1(t)A_1u(t)+\cdots+\alpha_n(t)A_nu(t)=0,\quad s\leq t\leq T,\quad u(s)=x,
\]
and the additional conditions
\[
\langle A_1 u(t), u(t)\rangle=\varphi_1(t),\quad\cdots\quad,\langle A_n u(t), u(t)\rangle=\varphi_n(t),\quad s\leq t\leq T.
\]
Under suitable assumptions on the operators \(A_i\), \(i=1,\cdots, n\), on the initial data \(x\in H\) and on the given functions \(\varphi_i\), \(i=1,\cdots,n\), we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lamé parameters in an elasticity problem on a plate.Inverse linear problems for a certain class of degenerate fractional evolution equationshttps://zbmath.org/1491.354552022-09-13T20:28:31.338867Z"Fedorov, V. E."https://zbmath.org/authors/?q=ai:fedorov.v-e"Nagumanova, A. V."https://zbmath.org/authors/?q=ai:nagumanova.anna-viktorovnaSummary: In this paper, we study the unique solvability of linear inverse coefficient problems with a time-independent unknown coefficient for evolution equations in Banach spaces with degenerate operators acting on the Gerasimov-Caputo fractional derivative. We apply abstract results obtained in the paper to the study of inverse problems with undetermined coefficients depending only on spatial variables for equations with polynomials on a self-adjoint, elliptic differential operator with respect to spatial variables. Also, we apply general results to the study of the unique solvability of inverse problems for time-fractional Sobolev systems.Series reversion in Calderón's problemhttps://zbmath.org/1491.354562022-09-13T20:28:31.338867Z"Garde, Henrik"https://zbmath.org/authors/?q=ai:garde.henrik"Hyvönen, Nuutti"https://zbmath.org/authors/?q=ai:hyvonen.nuuttiSummary: This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is \(\nabla \cdot (A\nabla u)=0\) in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends \(A\) to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Fréchet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model.Inverse scattering for three-dimensional quasi-linear biharmonic operatorhttps://zbmath.org/1491.354572022-09-13T20:28:31.338867Z"Harju, Markus"https://zbmath.org/authors/?q=ai:harju.markus"Kultima, Jaakko"https://zbmath.org/authors/?q=ai:kultima.jaakko"Serov, Valery"https://zbmath.org/authors/?q=ai:serov.valery-sSummary: We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito's formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.Parameter identification for elliptic boundary value problems: an abstract framework and applicationshttps://zbmath.org/1491.354582022-09-13T20:28:31.338867Z"Hoffmann, Heiko"https://zbmath.org/authors/?q=ai:hoffmann.heiko.1|hoffmann.heiko"Wald, Anne"https://zbmath.org/authors/?q=ai:wald.anne"Nguyen, Tram Thi Ngoc"https://zbmath.org/authors/?q=ai:nguyen.tram-thi-ngocAnalysis of the inverse Born series: an approach through geometric function theoryhttps://zbmath.org/1491.354592022-09-13T20:28:31.338867Z"Hoskins, Jeremy G."https://zbmath.org/authors/?q=ai:hoskins.jeremy-g"Schotland, John C."https://zbmath.org/authors/?q=ai:schotland.john-cDetermination of the initial density in nonlocal diffusion from final time measurementshttps://zbmath.org/1491.354602022-09-13T20:28:31.338867Z"Hrizi, Mourad"https://zbmath.org/authors/?q=ai:hrizi.mourad"Bensalah, Mohamed"https://zbmath.org/authors/?q=ai:bensalah.mohamed-oudi"Hassine, Maatoug"https://zbmath.org/authors/?q=ai:hassine.maatougSummary: This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.Revisiting the probe and enclosure methodshttps://zbmath.org/1491.354612022-09-13T20:28:31.338867Z"Ikehata, Masaru"https://zbmath.org/authors/?q=ai:ikehata.masaruDetermining damping terms in fractional wave equationshttps://zbmath.org/1491.354622022-09-13T20:28:31.338867Z"Kaltenbacher, Barbara"https://zbmath.org/authors/?q=ai:kaltenbacher.barbara"Rundell, William"https://zbmath.org/authors/?q=ai:rundell.williamSimultaneous determination of different class of parameters for a diffusion equation from a single measurementhttps://zbmath.org/1491.354632022-09-13T20:28:31.338867Z"Kian, Yavar"https://zbmath.org/authors/?q=ai:kian.yavarHyperbolic equation with rapidly oscillating data: reconstruction of the small lowest order coefficient and the right-hand side from partial asymptotics of the solutionhttps://zbmath.org/1491.354642022-09-13T20:28:31.338867Z"Levenshtam, V. B."https://zbmath.org/authors/?q=ai:levenshtam.valerii-borisovichSummary: We consider the Cauchy problem for a one-dimensional hyperbolic equation. The lowest order coefficient and the right-hand side oscillate in time at a high frequency, and the amplitude of the lowest order coefficient is small. The way of reconstructing these oscillating functions from partial solution asymptotics given at a certain point of the domain is studied.Determination of a time-dependent potential in a Rayleigh-Love equation with non-classical boundary conditionhttps://zbmath.org/1491.354652022-09-13T20:28:31.338867Z"Tekin, İbrahim"https://zbmath.org/authors/?q=ai:tekin.ibrahimSummary: Mathematical model of the longitudinal vibration of bars includes higher-order derivatives in the equation of motion under considering the effect of the lateral motion of a relatively thick bar. This paper considers such an inverse coefficient problem of determining time-dependent potential of a linear source together with the unknown longitudinal displacement from a Rayleigh-Love equation (containing the fourth-order space derivative) by using an additional measurement. Existence and uniqueness theorem of the considered inverse coefficient problem is proved for small times by using contraction principle.Optimal regularity for the no-sign obstacle problemhttps://zbmath.org/1491.354662022-09-13T20:28:31.338867Z"Andersson, John"https://zbmath.org/authors/?q=ai:andersson.john-erik"Lindgren, Erik"https://zbmath.org/authors/?q=ai:lindgren.erik"Shahgholian, Henrik"https://zbmath.org/authors/?q=ai:shahgholian.henrikSummary: In this paper we prove the optimal \(C^{1,1}(B_{1/2})\)-regularity for a general obstacle-type problem
\[
\Delta u=f_{\chi\{u\neq 0\}}\text{ in }B_1,
\]
under the assumption that \(f*N\) is \(C^{1,1}(B_1)\), where \(N\) is the Newtonian potential. This is the weakest assumption for which one can hope to get \(C^{1,1}\)-regularity. As a by-product of the \(C^{1,1}\)-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point \(x^0\), the free boundary is locally a \(C^1\)-graph close to \(x^0\) provided \(f\) is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.Nonexistence for nonlinear hyperbolic inequalities in an annulushttps://zbmath.org/1491.354672022-09-13T20:28:31.338867Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: This paper is concerned with nonexistence results for a class of nonlinear hyperbolic inequalities with a potential function \(V=V(x)\), posed in \(A\), where \(A\) is the annulus domain given by \(A = \{x\in\mathbb{R}^N: 1<|x|\le 2\}\). Two types of non-homogeneous boundary conditions depending on time and space are investigated: Neumann type boundary condition and Dirichlet type boundary condition. In particular, we investigate the combined effects of the considered boundary conditions, the behavior of the potential function near the boundary and the power nonlinearity, on the nonexistence of solutions. Moreover, in certain special cases, we show that our results are sharp.Nonexistence criteria for systems of parabolic inequalities in an annulushttps://zbmath.org/1491.354682022-09-13T20:28:31.338867Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We are concerned with nonexistence results for a class of systems of parabolic inequalities in \((0, \infty) \times A\), where \(A = \{x \in \mathbb{R}^N : 1 < | x | \leq 2 \} \). The considered systems involve a singular potential function \(V(x) = ( |x| - 1)^{- \rho}\), \(\rho > 0\), in front of the power nonlinearities. Two types of inhomogeneous boundary conditions on \(\partial B_2 = \{x \in \mathbb{R}^N : | x | = 2 \}\) are discussed: Neumann type conditions and Dirichlet type conditions. Using a unified approach, an optimal criterium of nonexistence is obtained for both cases. Our study yields naturally optimal nonexistence results for the corresponding stationary systems.On a stochastic nonclassical diffusion equation with standard and fractional Brownian motionhttps://zbmath.org/1491.354692022-09-13T20:28:31.338867Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Ngoc, Tran Bao"https://zbmath.org/authors/?q=ai:ngoc.tran-bao"Thach, Tran Ngoc"https://zbmath.org/authors/?q=ai:thach.tran-ngoc"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.The non-Lipschitz stochastic Cahn-Hilliard-Navier-Stokes equations in two space dimensionshttps://zbmath.org/1491.354702022-09-13T20:28:31.338867Z"Sun, Chengfeng"https://zbmath.org/authors/?q=ai:sun.chengfeng"Huang, Qianqian"https://zbmath.org/authors/?q=ai:huang.qianqian"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.3|liu.hui.2|liu.hui.1|liu.hui.4Large deviation principles for a 2D stochastic Allen-Cahn-Navier-Stokes driven by jump noisehttps://zbmath.org/1491.354712022-09-13T20:28:31.338867Z"Tachim Medjo, Theodore"https://zbmath.org/authors/?q=ai:tachim-medjo.theodoreEllipticity and Fredholmness of pseudo-differential operators on \(\ell^2(\mathbb{Z}^n)\)https://zbmath.org/1491.354722022-09-13T20:28:31.338867Z"Dasgupta, Aparajita"https://zbmath.org/authors/?q=ai:dasgupta.aparajita"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishveshSummary: The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on \(\mathbb{Z}^n\times\mathbb{T}^n\) are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on \(\mathbb{Z}^n\). The index of an elliptic pseudo-differential operator on \(\mathbb{Z}^n\) is also computed.On discrete boundary value problemshttps://zbmath.org/1491.354732022-09-13T20:28:31.338867Z"Vasilyev, Vladimir"https://zbmath.org/authors/?q=ai:vasilyev.vladimir-b|vasilev.vladimir-andreevich|vasilev.vladimir-aleksandrovichSummary: We consider discrete pseudo-differential equations and related discrete boundary value problems in appropriate discrete spaces. First we study simplest types of operators acting in canonical domains like a half-space and a cone. We try to describe solvability conditions for such equations and boundary value problems and further to compare the discrete and continuous cases. We use a concept of periodic factorization for elliptic symbols to obtain a form of solution for such equations in canonical domains.
For the entire collection see [Zbl 1436.46003].On certain operator familieshttps://zbmath.org/1491.354742022-09-13T20:28:31.338867Z"Vasilyev, V. B."https://zbmath.org/authors/?q=ai:vasilyev.vladimir-bSummary: In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev-Slobodecky spaces.Ergodic theory for energetically open compressible fluid flowshttps://zbmath.org/1491.370042022-09-13T20:28:31.338867Z"Fanelli, Francesco"https://zbmath.org/authors/?q=ai:fanelli.francesco"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Hofmanová, Martina"https://zbmath.org/authors/?q=ai:hofmanova.martinaSummary: The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier-Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff-Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any \(t \in R\)). In particular, the ergodic averages converge for \textit{any} trajectory provided its \(\omega\)-limit set in the trajectory space supports a unique (in law) stationary solution.Long-time asymptotics for an integrable evolution equation with a \(3 \times 3\) Lax pairhttps://zbmath.org/1491.370612022-09-13T20:28:31.338867Z"Charlier, C."https://zbmath.org/authors/?q=ai:charlier.christophe"Lenells, J."https://zbmath.org/authors/?q=ai:lenells.jonatanSummary: We derive a Riemann-Hilbert representation for the solution of an integrable nonlinear evolution equation with a \(3 \times 3\) Lax pair. We use the derived representation to obtain formulas for the long-time asymptotics.Universal rogue wave patterns associated with the Yablonskii-Vorob'ev polynomial hierarchyhttps://zbmath.org/1491.370622022-09-13T20:28:31.338867Z"Yang, Bo"https://zbmath.org/authors/?q=ai:yang.bo.3|yang.bo.1|yang.bo.4|yang.bo.2|yang.bo.5|yang.bo"Yang, Jianke"https://zbmath.org/authors/?q=ai:yang.jiankeSummary: We show that universal rogue wave patterns exist in integrable systems. These rogue patterns comprise fundamental rogue waves arranged in shapes such as a triangle, pentagon and heptagon, with a possible lower-order rogue wave at the center. These patterns appear when one of the internal parameters in bilinear expressions of rogue waves gets large. Analytically, these patterns are determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy through a linear transformation. Thus, the induced rogue patterns in the space-time plane are simply the root structures of Yablonskii-Vorob'ev hierarchy polynomials under actions such as dilation, rotation, stretch, shear and translation. Which level of the Yablonskii-Vorob'ev hierarchy is determined by which internal parameter is chosen to be large, and which polynomial at that level of the hierarchy is determined by the order of the underlying rogue wave. As examples, these universal rogue patterns are explicitly determined and graphically illustrated for the generalized derivative nonlinear Schrödinger equations, the Boussinesq equation, and the Manakov system. Similarities and differences between these rogue patterns and those reported earlier in the nonlinear Schrödinger equation are discussed.Continuation of spatially localized periodic solutions in discrete NLS lattices via normal formshttps://zbmath.org/1491.370632022-09-13T20:28:31.338867Z"Danesi, Veronica"https://zbmath.org/authors/?q=ai:danesi.veronica"Sansottera, Marco"https://zbmath.org/authors/?q=ai:sansottera.marco"Paleari, Simone"https://zbmath.org/authors/?q=ai:paleari.simone"Penati, Tiziano"https://zbmath.org/authors/?q=ai:penati.tizianoSummary: We consider the problem of the continuation with respect to a small parameter \(\varepsilon\) of spatially localized and time periodic solutions in 1-dimensional dNLS lattices, where \(\varepsilon\) represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localized periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.Subharmonic dynamics of wave trains in reaction-diffusion systemshttps://zbmath.org/1491.370642022-09-13T20:28:31.338867Z"Johnson, Mathew A."https://zbmath.org/authors/?q=ai:johnson.mathew-a"Perkins, Wesley R."https://zbmath.org/authors/?q=ai:perkins.wesley-rSummary: We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction-diffusion systems. For each \(N \in \mathbb{N}\), such \(T\)-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to \(NT\)-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on \(N\), and, in particular, they tend to zero as \(N \to \infty\), leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in \(N\). Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one pointhttps://zbmath.org/1491.370652022-09-13T20:28:31.338867Z"Kechiche, Wided"https://zbmath.org/authors/?q=ai:kechiche.widedAuthor's abstract: We consider the nonlinear Schrödinger equation in dimension one with a nonlinearity concentrated in one point. We prove that this equation provides an infinite dimensional dynamical system. We also study the asymptotic behavior of the dynamics. We prove the existence of a global attractor for the system.
Reviewer: Bixiang Wang (Socorro)Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in \(\mathbb{R}^3\) nonlinear KGS systemhttps://zbmath.org/1491.370662022-09-13T20:28:31.338867Z"Missaoui, Salah"https://zbmath.org/authors/?q=ai:missaoui.salahSummary: The main goal of this paper is to study the asymptotic behavior of a coupled Klein-Gordon-Schrödinger system in three dimensional unbounded domain. We prove the existence of a global attractor of the systems of the nonlinear Klein-Gordon-Schrödinger (KGS) equations in \(H^1 (\mathbb{R}^3)\times H^1 (\mathbb{R}^3)\times L^2 (\mathbb{R}^3)\) and more particularly that this attractor is in fact a compact set of \(H^2 (\mathbb{R}^3)\times H^2 (\mathbb{R}^3)\times H^1 (\mathbb{R}^3)\).Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domainshttps://zbmath.org/1491.370672022-09-13T20:28:31.338867Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Wang, Renhai"https://zbmath.org/authors/?q=ai:wang.renhaiAuthors' abstract: This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space \(\mathbb{R}^n\). The existence, uniqueness, time-semi-uniform compactness and \textit{asymptotically autonomous robustness} of pullback random attractors are proved in \(H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)\) when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the \textit{time-semi-uniform} pullback asymptotic compactness of the solutions in \(H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)\) is caused by the lack of compact Sobolev embeddings on \(\mathbb{R}^n\), as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by \textit{S. Wang} and \textit{Y. Li} [Physica D 382--383, 46--57 (2018; Zbl 1415.37073)].
Reviewer: Bixiang Wang (Socorro)Emergent behaviors of relativistic flocks on Riemannian manifoldshttps://zbmath.org/1491.370772022-09-13T20:28:31.338867Z"Ahn, Hyunjin"https://zbmath.org/authors/?q=ai:ahn.hyunjin"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Kang, Myeongju"https://zbmath.org/authors/?q=ai:kang.myeongju"Shim, Woojoo"https://zbmath.org/authors/?q=ai:shim.woojooSummary: We present a relativistic counterpart of the Cucker-Smale (CS) model on Riemannian manifolds (manifold RCS model in short) and study its collective behavior. For Euclidean space, the \textit{relativistic Cucker-Smale} (RCS) model was introduced in [\textit{S.-Y. Ha} et al., Arch. Ration. Mech. Anal. 235, No. 3, 1661--1706 (2020; Zbl 1439.35397)] via the method of a rational reduction from the relativistic gas mixture equations by assuming space-homogeneity, suitable ansatz for entropy and principle of subsystem. In this work, we extend the RCS model on Euclidean space to connected, complete and smooth Riemannian manifolds by replacing usual time derivative of velocity and relative velocity by suitable geometric quantities such as covariant derivative and parallel transport along length-minimizing geodesics. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using LaSalle's invariance principle. As concrete examples, we further analyze the RCS models on the unit sphere \(\mathbb{S}^d\) and the hyperbolic space \(\mathbb{H}^d\). More precisely, we show that the RCS model on \(\mathbb{S}^d\) exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in \(\mathbb{H}^d\). For the hyperbolic space \(\mathbb{H}^d\), we also rigorously justify smooth transition from the RCS model to the CS model in any finite time interval, as speed of light tends to infinity.Harmonic analysishttps://zbmath.org/1491.420012022-09-13T20:28:31.338867Z"Varadhan, S. R. S."https://zbmath.org/authors/?q=ai:varadhan.s-r-srinivasaThis book presents some topics related to Harmonic Analysis in various chapters not always connected to each other.
Below we make some comments on the content of the book and its presentation.
For the Fourier series of a given function, the convergence in the $L^p$ norm of the Cesaro means of the partial sums of the series is proved by using elementary arguments. The proof of the convergence of the Fourier series is based on the boundedness of the Hilbert operator, that is the fact that convolution by the kernel $1/x$ is a bounded operator in $L^p$. This turns out to be a special case of operators whose kernel has a bounded Fourier transform as well as bounded integral increments for the truncated operator.
Previously the Fourier transform is considered and proved to be a bijection between functions in the Schwartz space and a unitary map of $L^2(R^d)$. The functions that are Fourier transforms of nonnegative functions are also characterized.
As a generalization of the Hilbert transform, the author deals with convolution in $R^d$ with the kernel $x_j/|x|^(d+1)$, which gives rise to the Riesz transform. This operator is bounded in $L^p(R^d)$ and this fact is applied to prove the existence of solutions of the partial differential equation $u-Lu=f$, for $f$ in $L^p$ and $L$ being a differential operator of the second order with variable coefficients that satisfy some boundedness conditions. The result is that the equation above has a solution in the Sobolev space $W_{2,p}$ and that the $L^p$ norm of the solution is bounded by the $L^p$ norm of $f$.
The Sobolev spaces are introduced, and a particular case of the Sobolev embedding theorem is proved as well as the classical boundedness of the $L^{p^\prime}$ norm of a function by the $L^p$ norm of its gradient, where $p^\prime=pd/(d-p)$.
Two more topics that the author deals with are Hardy spaces and BMO spaces. For the first one, the factorization of a function in $H_p$ as a product of an inner function and an outer function is established. This fact is applied to prediction theory, which in mathematical terms consists in approximating a function in $L^2$ with respect to some measure by linear combinations of a sequence of stationary functions, calculating the minimum of the $L^2$ norm of the error, and finding the minimizer.
Concerning BMO spaces, a deep theorem due to Charles Fefferman is proved: the fact that BMO is the dual space of $H_1$. However, this result is not applied in the rest of the book.
After proving that the inverse of a function with an absolutely convergent Fourier series has also an absolutely convergent Fourier series, based on Gelfand's theory on Banach algebras, the author finishes the book with some considerations about representations of a compact group equipped with the Haar measure. The main result presented is the Peter-Weyl theorem about finite-dimensional irreducible representations, and two examples concerning the permutation group and SO(3) are given.
In the end, some references are provided but they are not mentioned in the text.
Sometimes the author uses some notation that is not previously introduced. This and some minor writing errors should be corrected in a forthcoming edition.
Reviewer: Julià Cufí (Bellaterra)Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functionshttps://zbmath.org/1491.420032022-09-13T20:28:31.338867Z"Isaev, Mikhail"https://zbmath.org/authors/?q=ai:isaev.mikhail-ismailovitch"Novikov, Roman G."https://zbmath.org/authors/?q=ai:novikov.roman-gSummary: We give new formulas for finding a compactly supported function \(v\) on \(\mathbb{R}^d\), \(d\geq 1\), from its Fourier transform \(\mathcal{F}v\) given within the ball \(B_r\). For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.On \(L^p\)-boundedness of Fourier integral operatorshttps://zbmath.org/1491.420262022-09-13T20:28:31.338867Z"Yang, Jie"https://zbmath.org/authors/?q=ai:yang.jie.3|yang.jie.1|yang.jie.2|yang.jie.4"Wang, Guangqing"https://zbmath.org/authors/?q=ai:wang.guangqing"Chen, Wenyi"https://zbmath.org/authors/?q=ai:chen.wenyiSummary: In this paper, we get an \(L^p\) boundedness of Fourier integral operators with rough amplitude \(a\in L^{\infty} S^m_{\varrho}\), and phase \(\varphi \in L^{\infty}{\Phi}^2\) for \(1\leq p\leq +\infty\). This is an improvement of the corresponding results in [\textit{D. Dos Santos Ferreira} and \textit{W. Staubach}, Global and local regularity of Fourier integral operators on weighted and unweighted spaces. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1323.35237)].An Ambrosetti-Prodi type result for integral equations involving dispersal operatorshttps://zbmath.org/1491.450132022-09-13T20:28:31.338867Z"de Assis Lima, Natan"https://zbmath.org/authors/?q=ai:de-assis-lima.natan"Souto, Marco A. S."https://zbmath.org/authors/?q=ai:souto.marco-aurelio-sThe authors derive necessary conditions on the function \(g(x)\) for the non-existence of solutions, the existence of at least one solution, and the existence of at least two distinct solutions for the following nonlocal problem: \[ L_0 u=f(x,u)+g(x),\qquad x\in\Omega, \] where \(L_0\) is a nonlocal dispersal operator and \(\Omega\) is a bounded domain in \(\mathbb{R}^n\).
Reviewer: Leonid Berezanski (Be'er Sheva)Multiperiodic solutions of quasilinear systems of integro-differential equations with \(D_c\)-operator and \(\epsilon \)-period of hereditarityhttps://zbmath.org/1491.450152022-09-13T20:28:31.338867Z"Sartabanov, Zhaishylyk Almaganbetovich"https://zbmath.org/authors/?q=ai:sartabanov.zh-a"Aitenova, Gulsezim Muratovna"https://zbmath.org/authors/?q=ai:aitenova.gulsezim-muratovna"Abdikalikova, Galiya Amirgalievna"https://zbmath.org/authors/?q=ai:abdikalikova.galiya-amirgalievnaSummary: We investigate a quasilinear system of partial integro-differential equations with the operator of differentiation in the direction of a vector field, which describes the process of hereditary propagation with an \(\epsilon \)-period of heredity. Under some conditions on the input data, conditions for the solvability of the initial problem for a quasilinear system of integro-differential equations are obtained. On this basis, sufficient conditions for the existence of multiperiodic solutions of integro-differential systems are found under the exponential dichotomy additional assumption on the corresponding homogeneous integro-differential system. The unique solvability of an operator equation in the space of smooth multiperiodic functions is proved, to which the main question under consideration reduces. Thus, sufficient conditions are established for the existence of a unique multiperiodical in all time variables solution of a quasilinear system of integro-differential equations with the differentiation operator in the directions of a vector field and a finite period of hereditarity.Method of orthogonal polynomials for an approximate solution of singular integro-differential equations as applied to two-dimensional diffraction problemshttps://zbmath.org/1491.450162022-09-13T20:28:31.338867Z"Rasol'ko, G. A."https://zbmath.org/authors/?q=ai:rasolko.galina-alekseevna"Volkov, V. M."https://zbmath.org/authors/?q=ai:volkov.vasilii-mikhailovichSummary: We consider a mathematical model of scattering of \(H \)-polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based on the representation of the unknown function in the form of a linear combination of orthogonal Chebyshev polynomials and spectral relations, which permit one to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the basis of Chebyshev polynomials are calculated as a solution of the corresponding system of linear algebraic equations. The results of numerical experiments show that the error in the approximate solution on a grid of 20--30 nodes does not exceed the roundoff error.Threshold phenomenon and traveling waves for heterogeneous integral equations and epidemic modelshttps://zbmath.org/1491.450172022-09-13T20:28:31.338867Z"Ducasse, Romain"https://zbmath.org/authors/?q=ai:ducasse.romainThe article is devoted to the study of mathematical models of epidemiology described by the nonlinear equations of Volterra, Volterra-Fredholm and systems of ordinary differential equations. It consists of an introduction and three sections.
The introduction provides an overview of mathematical models of epidemiology, including Kermack-McKendrick models, such as
\[
u(t)=\int_0^t \Gamma (\tau)g(u(t-\tau))d\tau+f(t),\qquad t>0,
\]
SIR models and Diekmann-Thieme models, such as
\[
u(t,x)=\int_0^t \int_{y \in \mathbb{R}^N} \Gamma(\tau,x,y)g(u(t-\tau,y))dyd\tau+f(t,x),\qquad t>0, \quad x \in \mathbb{R}^N,
\]
and the Diekmann model:
\[
u(t,x)=\int_0^\infty \int_{y \in \mathbb{R}^N} \Gamma(\tau,x,y)g(u(t-\tau,y))dyd\tau, \qquad t \in \mathbb{R},\quad x \in \mathbb{R}^N.
\]
Here \(u>0\) is the cumulative strength of the infection, \(\Gamma\) characterizes the nature of the infection, \(g\) is the nonlinear growth of the infection.
A qualitative analysis of the spread of infection is given.
In the first section, under various assumptions, threshold phenomena are studied in the models represented by the above equations.
In the second section, the occurrence of traveling wave solutions is investigated. Finally, in the third section, the results of the previous sections are applied to the study of spatially inhomogeneous SIR models.
Reviewer: Ilia V. Boikov (Penza)Elliptic and parabolic problems for a class of operators with discontinuous coefficientshttps://zbmath.org/1491.470372022-09-13T20:28:31.338867Z"Metafune, Giorgio"https://zbmath.org/authors/?q=ai:metafune.giorgio"Sobajima, Motohiro"https://zbmath.org/authors/?q=ai:sobajima.motohiro"Spina, Chiara"https://zbmath.org/authors/?q=ai:spina.chiaraSummary: We study elliptic and parabolic problems associated to the second order elliptic operator \[L=\Delta + (a-1) \sum^N_{i,j=1}\frac{x_ix_j}{|x|^2}D_{ij}+c \frac{x}{|x|^2}\cdot \nabla -b |x|^{-2}\] with \({a}>0\) and\({b}, {c}\) real coefficients. We prove generation of analytic semigroup and domain characterization.Optimal distributed control for a coupled phase-field systemhttps://zbmath.org/1491.490042022-09-13T20:28:31.338867Z"Chen, Bosheng"https://zbmath.org/authors/?q=ai:chen.bosheng"Li, Huilai"https://zbmath.org/authors/?q=ai:li.huilai"Liu, Changchun"https://zbmath.org/authors/?q=ai:liu.chein-shanThis paper considered a distributed optimal control for a coupled phase-field system. The control acts in the whole domain and the objective functional is of tracking type. The authors proved the existence of a weak solution to the governing state equantion and the existence of a solution to the nonlinear optimal control problem. The differentiability of the control-to-state mapping was studied which allows to derive a first order necessary optimality condition with the aid of an adjoint state system.
Reviewer: Wei Gong (Beijing)Pontryagin's maximum principle for distributed optimal control of two dimensional tidal dynamics system with state constraints of integral typehttps://zbmath.org/1491.490052022-09-13T20:28:31.338867Z"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: In this paper, we consider two dimensional tidal dynamics equations in a bounded domain and address a distributed optimal control problem of minimizing a suitable cost functional with state constraints of integral type (on the velocity field). It is well known that the Pontryagin maximum principle provides the first-order necessary conditions of optimality. We show the existence of an optimal control and establish Pontryagin's maximum principle for the state constrained optimization problem for the tidal dynamics system using Ekeland's variational principle and characterize optimal control through the adjoint variable.Strong solution and optimal control problems for a class of fractional linear equationshttps://zbmath.org/1491.490062022-09-13T20:28:31.338867Z"Plekhanova, M. V."https://zbmath.org/authors/?q=ai:plekhanova.marina-vasilevnas|plekhanova.marina-vasilevnaSummary: In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactnesshttps://zbmath.org/1491.490112022-09-13T20:28:31.338867Z"Novack, Michael"https://zbmath.org/authors/?q=ai:novack.michael-r"Yan, Xiaodong"https://zbmath.org/authors/?q=ai:yan.xiaodongSummary: We consider the 3D smectic energy
\[
\mathcal{E}_\varepsilon(u) = \frac{1}{2}\int_\Omega \frac{1}{\varepsilon} \left( \partial_z u-\frac{(\partial_x u)^2+(\partial_y u)^2}{2}\right)^2 +\varepsilon \left(\partial_x^2u + \partial_y^2u\right)^2dx\,dy\,dz.
\]
The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on \(\mathcal{E}_\varepsilon\) as \(\varepsilon \rightarrow 0\) by introducing 3D analogues of the Jin-Kohn entropies [\textit{W. Jin} and \textit{R. V. Kohn}, J. Nonlinear Sci. 10, No. 3, 355--390 (2000; Zbl 0973.49009)]. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for \(\varepsilon_n\rightarrow 0\) and an energy-bounded sequence \(\{u_n\}\) with \(\Vert\nabla u_n\Vert_{L^p(\Omega)}\), \(\Vert \nabla u_n\Vert_{L^2(\partial\Omega)}\le C\) for some \(p>6\), we obtain compactness of \(\nabla u_n\) in \(L^2\) assuming that \(\Delta_{xy}u_n\) has constant sign for each \(n\).A numerical construction of the universal feedback control in problems of nonlinear controls under disturbancehttps://zbmath.org/1491.490222022-09-13T20:28:31.338867Z"Ri, Kuk Hwan"https://zbmath.org/authors/?q=ai:ri.kuk-hwan"Sonu, Kuk Hyon"https://zbmath.org/authors/?q=ai:sonu.kuk-hyonSummary: This paper deals with problem of control under disturbance where dynamics of control system is fully nonlinear. Formulation of the problem follows first player's problem in Krasovskii-Subbotin framework for the differential games. In a single cubic grid of time-state space the value function and the control values are simultaneously calculated, based on minmax structure with the use of multilinear interpolation. The control function at each instant of temporal partition is constructed by constant interpolation of the control values calculated at spatial nodes in the same instant. Convergence of the approximation scheme to the value function and universal suboptimality of the proposed feedback control are shown. Through several examples, the correctness of the schemes is illustrated.The average distance problem with perimeter-to-area ratio penalizationhttps://zbmath.org/1491.490332022-09-13T20:28:31.338867Z"Du, Qiang"https://zbmath.org/authors/?q=ai:du.qiang"Lu, Xin Yang"https://zbmath.org/authors/?q=ai:lu.xinyang.1|lu.xinyang"Wang, Chong"https://zbmath.org/authors/?q=ai:wang.chong|wang.chong.7|wang.chong.4|wang.chong.6Quantitative stability in the geometry of semi-discrete optimal transporthttps://zbmath.org/1491.490352022-09-13T20:28:31.338867Z"Bansil, Mohit"https://zbmath.org/authors/?q=ai:bansil.mohit"Kitagawa, Jun"https://zbmath.org/authors/?q=ai:kitagawa.junSummary: We show quantitative stability results for the geometric ``cells'' arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma-Trudinger-Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.Optimal control of thin elastic inclusion in an elastic bodyhttps://zbmath.org/1491.490382022-09-13T20:28:31.338867Z"Karnaev, Vyacheslav Mikhailovich"https://zbmath.org/authors/?q=ai:karnaev.vyacheslav-mikhailovichSummary: The article deals with the inverse problem of the location of a thin elastic inclusion in an elastic body. A thin inclusion is considered to be soldered. The body is fixed on one part of the outer border, while external surface forces act on the other part. The inverse problem of identification of the inclusion is considered as the problem of minimizing the target functional. The existence of a solution to the inverse problem is proved. The first variations of the solution of the direct problem with respect to the shape of the domain and the derivative of the functional with respect to the shape are calculated. A numerical algorithm for solving this problem is proposed and numerical results are presented.Boundary behavior of rotationally symmetric prescribed mean curvature hypersurfaces in \(\mathbb{R}^4\)https://zbmath.org/1491.530102022-09-13T20:28:31.338867Z"Khanfer, Ammar"https://zbmath.org/authors/?q=ai:khanfer.ammar"Lancaster, Kirk E."https://zbmath.org/authors/?q=ai:lancaster.kirk-eugeneIn this paper under review, the authors consider prescribed mean curvature boundary value problems in $\mathbb{R}^4$ which are rotationally symmetric, and investigate the behavior of variational solutions near a conical point. Let $\Omega \subset\mathbb{R}^3$ be a locally Lipschitz domain whose boundary has a `crease' (or `edge') ${\mathcal C}\subset \Omega$ such that $\partial \Omega \setminus {\mathcal C}$ is $C^2$. Consider the boundary value problem
\[
\left\{\begin{array}{ll}
\mathrm{div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right) = n H(\cdot, f) \quad \mbox{in }\Omega\\
f = \phi\quad \mbox{ on }\partial \Omega.
\end{array}\right.
\]
In the previous work [J. Geom. Anal. 30, No. 2, 2241--2252 (2020; Zbl 1441.53007)], the authors studied the behavior of a nonparametric minimal hypersurface in $\Omega \times\mathbb{R} \subset\mathbb{R}^4$ near the nonconvex conical point ${\mathcal O} = (0,0,0)\in \partial \Omega$ when the graph of the variational solution of the problem mentioned above is rotationally symmetric with respect to the axis of the cone and $\mathcal C$ has codimension $2$ in $\partial \Omega$.
In the present paper, the authors consider the prescribed mean curvature boundary value problem when $\mathcal C$ has codimension one. From a domain $U \subset\mathbb{R}^2$ with locally Lipschitz boundary $\partial U$ satisfying some conditions, the authors obtain the domain $$ \Omega = \{(x \cos t, x\sin t, y)\in\mathbb{R}^3 : (x, y) \in U, t \in [0, 2\pi)\} $$ by rotating $U$ about the vertical axis $\{(0.0)\}\times \mathbb{R}$, and crease(edge) is $\mathcal C = \{(x_0 \cos t, x_0 \sin t, 0) : t \in [0, 2\pi)\}$. The authors prove that if $f \in C^2(\Omega) \cap L^\infty(\Omega)$ is a variational solution of the prescribed mean curvature boundary value problem, then the radial limit $Rf(\omega, P)$ of $f$ at each point $P \in \mathcal C$ exists, where the radial limt is defined as $$ Rf(\omega, P) = \lim_{r\downarrow 0} f(P+r \omega)\quad \mbox{for }\omega \in T_P $$ with $T_P = \{\omega \in\mathbb{R}^3 : |\omega| = 1$ and $\{P+r\omega : 0 < r < \epsilon \} \subset \Omega$ for some $\epsilon >0\}$. The authors also obtain properties on the behavior of radial limits in various cases.
Reviewer: Gabjin Yun (Yongin)A sharp bound for the growth of minimal graphshttps://zbmath.org/1491.530122022-09-13T20:28:31.338867Z"Weitsman, Allen"https://zbmath.org/authors/?q=ai:weitsman.allenSummary: We consider minimal graphs \(u = u(x,y) > 0\) over unbounded domains \(D \subset R^2\) bounded by a Jordan arc \(\gamma\) on which \(u = 0\). We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if \(D\) contains a sector
\[
S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi,
\]
then the rate of growth is at most \(r^{\pi /\lambda }\).Slant curves in the Lorentzian warped product manifold \(-I \times_f{\mathbb{E}}^2\)https://zbmath.org/1491.530142022-09-13T20:28:31.338867Z"Dursun, Uğur"https://zbmath.org/authors/?q=ai:dursun.ugurSummary: In this work, we study slant curves in the 3-dimensional Lorentzian warped product \(-I \times_f{\mathbb{E}}^2\), where \({\mathbb{E}}^2\) is a 2-dimensional Euclidean plane, \(I \subseteq{\mathbb{R}}\) is an open interval equipped with the metric \(dt^2\), and \(f\) is a positive smooth function on \(I\). First we give a characterization of slant curves, and then we obtain a classification of all slant curves in \(-I \times_f{\mathbb{E}}^2\). We also compute their curvature and torsion, and we obtaine some results on slant curves and helices in the de Sitter space \(\mathbb S^3_1(1)\) and in the Minkowski space \({\mathbb{E}}^3_1\). Moreover we determined some biharmonic slant curves in \({\mathbb{S}}^3_1(1)\).De Lellis-Topping inequalities on weighted manifolds with boundaryhttps://zbmath.org/1491.530482022-09-13T20:28:31.338867Z"Cruz, F. Jr."https://zbmath.org/authors/?q=ai:cruz.f-jun"Freitas, A."https://zbmath.org/authors/?q=ai:freitas.ana-t|freitas.allan-g|freitas.adelaide-valente|freitas.antonio-a|freitas.augusto-s|freitas.ana-cristina-moreira|freitas.a-g-c|freitas.amauri-a|freitas.a-r-r|freitas.alex-alves|freitas.ayres|freitas.andre|freitas.a-d|freitas.a-b|freitas.ana-p-c"Santos, M."https://zbmath.org/authors/?q=ai:santos.michael-r|santos.marcelino-b|santos.marcelo-f|santos.marta|santos.micael|santos.m-l-o|santos.melina-erica|santos.mauro-lima|santos.marco-antonio-cetale|santos.marcio-c|santos.m-a-f|santos.maria-celia|santos.maria-emma|santos.m-madalena|santos.manoel-j-dos|santos.makson-s|santos.m-franca|santos.manuel-s|santos.m-b-l|santos.marcus-vinicius|santos.marcos-a-c|santos.moises-s|santos.m-t|santos.marta-d|santos.maria-augusta|santos.mauricio-cardoso|santos.mila|santos.marcio-s|santos.maria-luiza-f|santos.maristela-oliveira|santos.mariel|santos.marcelo-m|santos.marcelo-p|santos.maria-r-b|santos.marcelo-r|santos.matheus-c|santos.miguel-aThe De Lellis-Topping inequality implies that if a closed Riemannian manifold with nonnegative Ricci curvature is close to being an Einstein manifold, then its scalar curvature is close to being constant [\textit{C. De Lellis} and \textit{P. M. Topping}, Calc. Var. Partial Differ. Equ. 43, No. 3--4, 347--354 (2012; Zbl 1236.53036)]. In this paper, the authors prove a type of De Lellis-Topping inequality for a symmetric \((0,2)\)-tensor field \(T\) on a compact weighted Riemannian manifold with a convex boundary whose Bakry-Émery Ricci tensor is bounded from below by a negative constant. It is also assumed that the divergence of \(T\) is a constant multiple of \(\nabla(\operatorname{tr}T)\) and that \(T(\nu, -)\) is nonnegative on the boundary, where \(\nu\) is the outward unit normal.
Reviewer: James Hebda (St. Louis)Heavenly metrics, BPS indices and twistorshttps://zbmath.org/1491.530632022-09-13T20:28:31.338867Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:alexandrov.sergei-yu"Pioline, Boris"https://zbmath.org/authors/?q=ai:pioline.borisSummary: Recently, \textit{T. Bridgeland} [``Geometry from Donaldson-Thomas invariants'', Preprint, \url{arXiv:1912.06504}] defined a complex hyperkähler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function \(W(z,\theta)\) satisfying a heavenly equation, or a potential \(F(z,\theta)\) satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both \(W\) and \(F\) in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of \(F\) as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called \(\tau\) function for arbitrary values of the fiber coordinates \(\theta\), in terms of a suitable two-variable generalization of Barnes' \(G\) function.A maximum principle for free boundary minimal varieties of arbitrary codimensionhttps://zbmath.org/1491.530722022-09-13T20:28:31.338867Z"Li, Martin Man-chun"https://zbmath.org/authors/?q=ai:li.martin-man-chun"Zhou, Xin"https://zbmath.org/authors/?q=ai:zhou.xin.1Summary: We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.Liouville hypersurfaces and connect sum cobordismshttps://zbmath.org/1491.530812022-09-13T20:28:31.338867Z"Avdek, Russell"https://zbmath.org/authors/?q=ai:avdek.russellIn symplectic/contact geometry, the Moser/Gray trick to prove existence of the unique local model (up to symplectomorphism), constructions of isotopies, the flow of a natural vector field provide powerful tools to find nicer local shapes (or the model structure), to decompose a symplectic manifold into building blocks of the model structure, to glue pieces and to assemble topological/homological/categorical data.
In [\textit{M. Abouzaid}, Ann. Math. (2) 175, No. 1, 71--185 (2012; Zbl 1244.53089)], the author defined a gluing operation for contact manifolds along nice neighborhoods of Liouville hypersurfaces, called the Liouville connected sum, which generalizes the Weinstein handle attachment for Weinstein (symplectic) manifolds. To do it, he introduced the concept of Liouville hypersurfaces (more generally, Liouville submanifolds) in contact manifolds which generalize ribbons of Legendrian graphs and pages of open book decomposition of contact manifolds. A fundamental construction is that any Liouville hypersurface \(\Sigma\) of a contact manifold can have a standard tubular neighborhood \(\mathcal{N}(\Sigma)\) with smooth, convex boundary, which can be considered as a flattening of \(\Sigma\) by the Reeb vector fields. A local model of Liouville connected sum is convex gluing of unit-cotangent bundles of two smooth compact manifolds with non-empty boundaries with orientation-reversing diffeomorphism on the boundary of the base manifolds (Example 2.14).
A Liouville domain is a smooth compact symplectic manifold with boundary with exact symplectic form with Liouville vector field trasversely outward along the boundary. A Liouville hypersurface of a contact manifold \((M^{2n+1}, \xi\subset \text{Ker}(\alpha))\) is the image of an embedding \(i:\Sigma \to M\) of a Liouville domain \((\Sigma^{2n}, d\beta)\) of codimension \(1\) with \(i^{*}\alpha=\beta\). Since a Liouville hypersurface is transverse to the Reeb vector field \(R\) for \(\alpha\) (\(\alpha(R)=1\) and \(d\alpha(R,\cdot)=0\)), a map \(\Phi:[-\epsilon,\epsilon]\times \Sigma\to M\), defined by a time \((z \in [-\epsilon,\epsilon])\) flow of \(R\) starting from \(x\in \Sigma\), gives a small tubular neighborhood \(N(\Sigma)\) of \(\Sigma\) inside \(M\) satisfying \(\alpha|_{N(\Sigma)}=dz+\beta\). After edge-rounding on the neighborhood \(N(\Sigma)\), one get a standard neighborhood \(\mathcal{N}(\Sigma)\) of the Liouville hypersurface (Figure 3, Definition 3.6). The author also proves a neighborhood theorem for Liouville submanifolds of high codimension.
The second half of the paper consists of applications of the gluing construction, for example, the construction of a Weinstein handle attachment from a standard neighborhood of a Liouville hypersurface, the symplectic handle attachment to the positive boundary of a weak symplectic cobordism, fillabilities preserved under contact connected sum, the monoid structure on non-vanishing contact homology, extension of contact \((1/k)\)-surgeries to arbitrary dimension, and certain generalized Dehn twists that create exotic structures.
The main idea of [\textit{R. Avdek}, J. Symplectic Geom. 19, No. 4, 865--957 (2021; Zbl 07455583)], that was released in the earlier arXiv preprint version, was adapted to glue of Weinstein pairs in [\textit{Y. Eliashberg}, Proc. Symp. Pure Math. 99, 59--82 (2018; Zbl 1448.53083), Section 3.1], where the skeleta of Weinstein pairs are glued along the skeleta of glued hypersurfaces. It would also be interesting to consider the concept of sutured Liouville manifolds and Liouville sectors in [\textit{S. Ganatra} et al., Publ. Math., Inst. Hautes Étud. Sci. 131, 73--200 (2020; Zbl 07209675); ``Sectorial descent for wrapped Fukaya categories'', Preprint, \url{arXiv:1809.03427}; ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}].
Reviewer: Dahye Cho (Stony Brook)Sasakian 3-metric as a generalized Ricci-Yamabe solitonhttps://zbmath.org/1491.530842022-09-13T20:28:31.338867Z"Dey, Dibakar"https://zbmath.org/authors/?q=ai:dey.dibakar"Majhi, Pradip"https://zbmath.org/authors/?q=ai:majhi.pradipSummary: In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if \((g, V, \lambda, \alpha, \beta, \gamma)\) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold \(M\) with potential function \(f\), then \(M\) is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field \(V\) is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field \(\xi\). Further, if \(h\) is the Hodge-de Rham potential for \(V\), then, upto a constant, \(f = h\).Best proximity point theory on vector metric spaceshttps://zbmath.org/1491.541472022-09-13T20:28:31.338867Z"Şahin, Hakan"https://zbmath.org/authors/?q=ai:sahin.hakanSummary: In this paper, we first give a new definition of \(\Omega\)-Dedekind complete Riesz space \((E,\leq)\) in the frame of vector metric space \((\Omega,\rho,E)\) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called \(\alpha\)-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings in vector metric spaces \((\Omega,\rho,E)\) where \((E,\leq)\) is \(\Omega\)-Dedekind complete Riesz space. Thus, for the first time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some fixed point results proved in both vector metric spaces and partially ordered vector metric spaces. Further, we provide nontrivial and comparative examples to show the effectiveness of our main results.The mean-field limit of the Cucker-Smale model on complete Riemannian manifoldshttps://zbmath.org/1491.580112022-09-13T20:28:31.338867Z"Ahn, Hyunjin"https://zbmath.org/authors/?q=ai:ahn.hyunjin"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Kim, Doheon"https://zbmath.org/authors/?q=ai:kim.doheon"Schlöder, Franz Wilhelm"https://zbmath.org/authors/?q=ai:schloder.franz-wilhelm"Shim, Woojoo"https://zbmath.org/authors/?q=ai:shim.woojooIn this paper the authors apply the particle-in-cell-method to establish the existence of a mean-field limit of the abstract Cucker-Smale model which describes the flocking dynamics (i.e., featuring some collective behaviour) of \(N\) interacting particles on a complete Riemannian manifold.
More precisely the following questions are addressed: (i) the existence and properties of a model in the infinite particle limit \(N\rightarrow +\infty\); (ii) the dependence on the model's parameters (the communication weight, initial data and further \textit{a priori} assumptions) of the emergence of a coherent behaviour of the participating individual particles.
Reviewer: Gabor Etesi (Budapest)Partial smoothing of delay transition semigroups acting on special functionshttps://zbmath.org/1491.600872022-09-13T20:28:31.338867Z"Masiero, Federica"https://zbmath.org/authors/?q=ai:masiero.federica"Tessitore, Gianmario"https://zbmath.org/authors/?q=ai:tessitore.gianmarioSummary: It is well known that the transition semigroup of an Ornstein Uhlenbeck process with delay is not strong Feller for small times, so it has no regularizing effects when acting on bounded and continuous functions. In this paper we study regularizing properties of this transition semigroup when acting on special functions of the past trajectory. With this regularizing property, we are able to prove existence and uniqueness of a mild solution for a special class of semilinear Kolmogorov equations; we apply these results to a stochastic optimal control problem.Nonlinear parabolic stochastic evolution equations in critical spaces. II: Blow-up criteria and instantaneous regularizationhttps://zbmath.org/1491.600932022-09-13T20:28:31.338867Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cSummary: This paper is a continuation of Part I [the authors, ``Nonlinear parabolic stochastic evolution equations in critical spaces I: Stochastic maximal regularity and local existence'', Preprint, \url{arXiv:2001.00512}] of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Simonett-Wilke [\textit{J. Prüss} et al., J. Differ. Equations 264, No. 3, 2028--2074 (2018; Zbl 1377.35176)] to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical \(L^2\)-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier-Stokes equations, reaction -- diffusion equations and the Allen-Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.Analysing differential equations with uncertainties via the Liouville-Gibbs theorem: theory and applicationshttps://zbmath.org/1491.600942022-09-13T20:28:31.338867Z"Bevia, V."https://zbmath.org/authors/?q=ai:bevia.v"Burgos, C."https://zbmath.org/authors/?q=ai:burgos.clara"Cortés, J.-C."https://zbmath.org/authors/?q=ai:cortes.juan-carlos"Navarro-Quiles, A."https://zbmath.org/authors/?q=ai:navarro-quiles.ana"Villanueva, R.-J."https://zbmath.org/authors/?q=ai:villanueva.rafael-jacintoSummary: In this contribution, we revisit the Liouville-Gibbs theorem for dynamical systems. This theorem states that a partial differential equation that is satisfied by the probability density function of the solution stochastic process of an initial value problem with uncertainties in its initial condition, forcing term and coefficients. We show its key role in the setting of dynamical systems with uncertainties by means of a variety of illustrative models appearing in several scientific realms that include physics and biology. Specifically, we deal with the undamped and damped linear oscillator, and the logistic model. These models are formulated via random differential equations with a finite degree of randomness. Numerical simulations and computations are carried out to illustrate the capability of the Liouville-Gibbs theorem.
For the entire collection see [Zbl 1464.65006].Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity. I: measureshttps://zbmath.org/1491.600952022-09-13T20:28:31.338867Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjornThe author of this study shows the Gibbs measure invariance for a three-dimensional wave equation with a Hartree nonlinearity. The singularity of the Gibbs measure with respect to the Gaussian free field is the primary innovation. In both measure-theoretic and dynamical aspects, the singularity has various repercussions. In this study, the author primarily creates and investigates the Gibbs measure. The technique used in this research is based on previous work by Barashkov and Gubinelli for the \(\Phi^4_3\)-model. In addition, the author creates new techniques to cope with the Hartree interaction's nonlocality. Depending on the regularity of the interaction potential, the exact threshold between singularity and absolute continuity of the Gibbs measure are also calculated.
Reviewer: Udhayakumar Ramalingam (Vellore)On ill-posedness of nonlinear stochastic wave equations driven by rough noisehttps://zbmath.org/1491.600962022-09-13T20:28:31.338867Z"Deya, Aurélien"https://zbmath.org/authors/?q=ai:deya.aurelienSummary: We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide a systematic interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index \(H \leq \frac{1}{4} \).
Our results clarify and extend previous similar properties exhibited in [\textit{A. Deya}, Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 1, 477--501 (2020; Zbl 1434.60152)] or in [\textit{T. Oh} and \textit{M. Okamoto}, Electron. J. Probab. 26, Paper No. 9, 44 p. (2021; Zbl 1469.35270)].Moderate deviations for stochastic tidal dynamics equations with multiplicative Gaussian noisehttps://zbmath.org/1491.600982022-09-13T20:28:31.338867Z"Haseena, A."https://zbmath.org/authors/?q=ai:haseena.a"Suvinthra, M."https://zbmath.org/authors/?q=ai:suvinthra.murugan"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-t"Balachandran, K."https://zbmath.org/authors/?q=ai:balachandran.krishnanThe paper deals with the stochastic version of the tidal dynamics model proposed by Marchuk and Kagan. The authors establish a central limit theorem and moderate deviation principle for the tidal dynamics equations perturbed by multiplicative Gaussian noise. The proof relies on a variational method obtained by \textit{A. Budhiraja} and \textit{P. Dupuis} [Probab. Math. Stat. 20, No. 1, 39--61 (2000; Zbl 0994.60028)].
Reviewer: Ivan Podvigin (Novosibirsk)Correction to: ``The stochastic Gierer-Meinhardt system''https://zbmath.org/1491.600992022-09-13T20:28:31.338867Z"Hausenblas, Erika"https://zbmath.org/authors/?q=ai:hausenblas.erika"Panda, Akash Ashirbad"https://zbmath.org/authors/?q=ai:panda.akash-ashirbadFrom the text: In the original article [the authors, ibid. 85, No. 2, Paper No. 11, 49 p. (2022; Zbl 1486.60077)] the ESM file has some errors.
Now, it has been resolved and the updated ESM file has been included.Kolmogorov's theory of turbulence and its rigorous 1d modelhttps://zbmath.org/1491.601032022-09-13T20:28:31.338867Z"Kuksin, Sergei"https://zbmath.org/authors/?q=ai:kuksin.sergei-bThe author summarizes the main results (with some sketched proofs) of the book [One-dimensional turbulence and the stochastic Burgers equation. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1486.60002)] coauthored with \textit{A. Boritchev} and \textit{S. Kuksin}. The author considers the 1D viscous Burgers equation with periodic boundary condition and additive noise which is spatially smooth. When the viscosity is small enough (equivalently, the Reynolds number is sufficiently big), he is able to rigorously estimate some quantities like dissipation scale, structure function and energy spectrum; the purpose is to compare these quantities with the predictions of Kolmogorov's turbulence theory, abbreviated as the K41 theory. The author concludes that the statistical properties of stochastic 1D Burgers equation with small viscosity are close analogues of the main laws of the K41 theory, which supports the belief that K41 theory is ``close to the truth''.
Reviewer: Dejun Luo (Beijing)An efficient jet marcher for computing the quasipotential for 2D SDEs. Enhancing accuracy and efficiency of quasipotential solvershttps://zbmath.org/1491.601062022-09-13T20:28:31.338867Z"Paskal, Nicholas"https://zbmath.org/authors/?q=ai:paskal.nicholas"Cameron, Maria"https://zbmath.org/authors/?q=ai:cameron.maria-kourkinaSummary: We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths, EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet-Reygner sharp estimate for the expected escape time for the Maier-Stein SDE. Our codes are available on GitHub.A stochastic fractional Laplace equation driven by colored noise on bounded domain, and its covariance functionalhttps://zbmath.org/1491.601072022-09-13T20:28:31.338867Z"Piña, Nicolás"https://zbmath.org/authors/?q=ai:pina.nicolas"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Porcu, Emilio"https://zbmath.org/authors/?q=ai:porcu.emilioSummary: The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived.Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemeshttps://zbmath.org/1491.601092022-09-13T20:28:31.338867Z"van Neerven, Jan"https://zbmath.org/authors/?q=ai:van-neerven.jan-m-a-m"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cSummary: We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if \((S(t,s))_{0\leq s\leq t\leq T}\) is a \(C_0\)-evolution family of contractions on a 2-smooth Banach space \(X\) and \((W_t)_{t\in [0,T]}\) is a cylindrical Brownian motion on a probability space \((\Omega, \mathbb{P})\) adapted to some given filtration, then for every \(0<p<\infty\) there exists a constant \(C_{p,X}\) such that for all progressively measurable processes \(g: [0,T]\times \Omega \rightarrow X\) the process \((\int_0^t S(t,s) g_s \,\text{d}W_s)_{t\in [0,T]}\) has a continuous modification and
\[
\mathbb{E}\sup_{t\in [0,T]}\Big\Vert \int_0^t S(t,s)g_s\,\text{d}W_s \Big\Vert^p\leq C_{p,X}^p \mathbb{E} \Big(\int_0^T \Vert g_t\Vert^2_{\gamma (H,X)}\,\text{d}t\Big)^{p/2}.
\]
Moreover, for \(2\leq p<\infty\) one may take \(C_{p,X} = 10 D \sqrt{p}\), where \(D\) is the constant in the definition of 2-smoothness for \(X\). The order \(O(\sqrt{p})\) coincides with that of Burkholder's inequality and is therefore optimal as \(p\rightarrow \infty\). Our result improves and unifies several existing maximal estimates and is even new in case \(X\) is a Hilbert space. Similar results are obtained if the driving martingale \(g_t\,\text{d} W_t\) is replaced by more general \(X\)-valued martingales \(\text{d} M_t\). Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs
\[
\text{d} u_t = A(t)u_t\,\text{d}t + g_t\,\text{d} W_t, \quad u_0 = 0,
\]
where the family \((A(t))_{t\in [0,T]}\) is assumed to generate a \(C_0\)-evolution family \((S(t,s))_{0\leq s\leq t\leq T}\) of contractions on a 2-smooth Banach spaces \(X\). Under spatial smoothness assumptions on the inhomogeneity \(g\), contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.The exponential behavior and stabilizability of quasilinear parabolic stochastic partial differential equationhttps://zbmath.org/1491.601102022-09-13T20:28:31.338867Z"Yin, Xiuwei"https://zbmath.org/authors/?q=ai:yin.xiuwei"Shen, Guangjun"https://zbmath.org/authors/?q=ai:shen.guangjun"Wu, Jiang-Lun"https://zbmath.org/authors/?q=ai:wu.jianglunA parabolic Harnack principle for balanced difference equations in random environmentshttps://zbmath.org/1491.601122022-09-13T20:28:31.338867Z"Berger, Noam"https://zbmath.org/authors/?q=ai:berger.noam"Criens, David"https://zbmath.org/authors/?q=ai:criens.davidSummary: We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting diffusionshttps://zbmath.org/1491.601132022-09-13T20:28:31.338867Z"Jourdain, Benjamin"https://zbmath.org/authors/?q=ai:jourdain.benjamin"Tse, Alvin"https://zbmath.org/authors/?q=ai:tse.alvinSummary: In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the \textit{master equation}, a parabolic PDE involving \(L\)-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.Coupled FBSDEs with measurable coefficients and its application to parabolic PDEshttps://zbmath.org/1491.601142022-09-13T20:28:31.338867Z"Nam, Kihun"https://zbmath.org/authors/?q=ai:nam.kihun"Xu, Yunxi"https://zbmath.org/authors/?q=ai:xu.yunxiSummary: Using purely probabilistic methods, we prove the existence and the uniqueness of solutions for a system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous, coefficients. As a corollary, we obtain the well-posedness of semilinear parabolic partial differential equations (PDEs)
\[
\mathcal{L} u(t, x) + F(t, x, u, \partial_x u) = 0; \quad u(T, x) = h(x)
\] in the natural domain of the second order linear parabolic operator \(\mathcal{L}\) when \(F\) and \(h\) are not necessarily continuous with respect to \(x\). We also provide a sufficient condition for this solution to be in a Sobolev space. Finally, we apply the result to optimal policymaking for pandemics and pricing of carbon emission allowances.Long-time influence of small perturbations and motion on the simplex of invariant probability measureshttps://zbmath.org/1491.601402022-09-13T20:28:31.338867Z"Freidlin, Mark I."https://zbmath.org/authors/?q=ai:freidlin.mark-iSummary: We present a general approach to a broad class of asymptotic problems related to the long-time influence of small perturbations, of both the deterministic and stochastic type. The main characteristic of this influence is a limiting motion on the simplex of invariant probability measures of the non-perturbed system in an appropriate time scale. We consider perturbations of dynamical systems in \(\mathbb{R}^n\), linear and nonlinear perturbations of PDEs, wave fronts in the reaction-diffusion equations, homogenization problems and perturbations caused by small time delay. The main tools we use in these problems are limit theorems for large deviations, modified averaging principle and diffusion approximation.Sticky Bessel diffusionshttps://zbmath.org/1491.601432022-09-13T20:28:31.338867Z"Peskir, Goran"https://zbmath.org/authors/?q=ai:peskir.goranSummary: We consider a Bessel process \(X\) of dimension \(\delta \in (0, 2)\) having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter \(1 / \mu \in (0, \infty)\). We show that (i) the process \(X\) can be characterised through its square \(Y = X^2\) as a unique weak solution to the SDE system
\[ \begin{aligned}
&d Y_t = \delta I (Y_t > 0) d t + 2 \sqrt{Y_t} d B_t \\
&I (Y_t = 0) d t = \frac{ 1}{ 2 \mu} d \ell_t^0 (Y)
\end{aligned} \] where \(B\) is a standard Brownian motion and \(\ell^0 (Y)\) is a diffusion local time process of \(Y\) at 0, and (ii) the transition density function of \(X\) can be expressed in the closed form as a convolution integral involving a Mittag-Leffler function and a modified Bessel function of the second kind. Appearance of the Mittag-Leffler function is novel in this context. We determine a (sticky) boundary condition at zero that characterises the transition density function of \(X\) as a unique solution to the Kolmogorov backward/forward equation of \(X\). We also show that the convolution integral can be characterised as a unique solution to the generalised Abel equation of the second kind. Letting \(\mu \downarrow 0\) (absorption) and \(\mu \uparrow \infty \) (instantaneous reflection) the closed-form expression for the transition density function of \(X\) reduces to the ones found by
\textit{W. Feller} [Ann. Math. (2) 54, 173--182 (1951; Zbl 0045.04901)] and \textit{S. A. Molchanov} [Theory Probab. Appl. 12, 307--310 (1967; Zbl 0308.60044); translation from Teor. Veroyatn. Primen. 12, 358--362 (1967)] respectively.Age evolution in the mean field forest fire model via multitype branching processeshttps://zbmath.org/1491.601732022-09-13T20:28:31.338867Z"Crane, Edward"https://zbmath.org/authors/?q=ai:crane.edward"Ráth, Balázs"https://zbmath.org/authors/?q=ai:rath.balazs"Yeo, Dominic"https://zbmath.org/authors/?q=ai:yeo.dominicSummary: We study the distribution of ages in the mean field forest fire model introduced by \textit{B. Rath} and \textit{B. Toth} [Electron. J. Probab. 14, 1290--1327 (2009; Zbl 1197.60093)]. This model is an evolving random graph whose dynamics combine Erdős-Rényi edge-addition with a Poisson rain of \textit{lightning strikes}. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The \textit{age} of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of \textit{B. Bollobàs} et al. [Random Struct. Algorithms 31, No. 1, 3--122 (2007; Zbl 1123.05083)]. We then study the evolution of the ages via the multitype Galton-Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.Handbook of fractional calculus for engineering and sciencehttps://zbmath.org/1491.650012022-09-13T20:28:31.338867ZPublisher's description: Fractional calculus is used to model many real-life situations from science and engineering. The book includes different topics associated with such equations and their relevance and significance in various scientific areas of study and research. In this book readers will find several important and useful methods and techniques for solving various types of fractional-order models in science and engineering. The book should be useful for graduate students, PhD students, researchers and educators interested in mathematical modelling, physical sciences, engineering sciences, applied mathematical sciences, applied sciences, and so on.
This Handbook:
\begin{itemize}
\item Provides reliable methods for solving fractional-order models in science and engineering.
\item Contains efficient numerical methods and algorithms for engineering-related equations.
\item Contains comparison of various methods for accuracy and validity.
\item Demonstrates the applicability of fractional calculus in science and engineering.
\item Examines qualitative as well as quantitative properties of solutions of various types of science- and engineering-related equations.
\end{itemize}
Readers will find this book to be useful and valuable in increasing and updating their knowledge in this field and will be it will be helpful for engineers, mathematicians, scientist and researchers working on various real-life problems.
The articles of this volume will not be indexed individually.Analysis of a multilevel Markov chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusionshttps://zbmath.org/1491.650102022-09-13T20:28:31.338867Z"Hoang, Viet Ha"https://zbmath.org/authors/?q=ai:hoang.viet-ha"Quek, Jia Hao"https://zbmath.org/authors/?q=ai:quek.jia-hao"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophAbout one method of numerical solution Schrödinger equationhttps://zbmath.org/1491.650122022-09-13T20:28:31.338867Z"Plokhotnikov, K. E."https://zbmath.org/authors/?q=ai:plokhotnikov.k-ehSummary: The paper considers the method of numerical solution of the Schrödinger equation, which, in part, can be attributed to the class of Monte Carlo methods. The method is presented and simultaneously illustrated by the examples of solving the one-dimensional and multidimensional Schrödinger equation in the problems of linear one-dimensional oscillator, hydrogen atom and benzene.Approximating the solution of the differential equations with fractional operatorshttps://zbmath.org/1491.650612022-09-13T20:28:31.338867Z"Ahmed, Rana Talha"https://zbmath.org/authors/?q=ai:ahmed.rana-talha"Sohail, Ayesha"https://zbmath.org/authors/?q=ai:sohail.ayeshaSummary: We described a brief history of fractional calculus from sixteenth century to twentieth century. Basic functions like gamma function and Mittag-Leffler function are defined which help to understand fractional calculus. Popular fractional integral and derivative operators are also defined. Fractional order ordinary and partial differential equations are also introduced. In this research, we study and proposed two basic methods to solve fractional order ordinary differential equations like semi-analytical method which is followed by Laplace transform and a numerical method which is followed by backward (implicit) Euler's method. To support these methods we also solved a few examples for better understanding.Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systemshttps://zbmath.org/1491.650772022-09-13T20:28:31.338867Z"Allwright, Amy"https://zbmath.org/authors/?q=ai:allwright.amy"Atangana, Abdon"https://zbmath.org/authors/?q=ai:atangana.abdonSummary: The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.Computer turbulence as a tunnelling effecthttps://zbmath.org/1491.650792022-09-13T20:28:31.338867Z"Sharkovsky, A. N."https://zbmath.org/authors/?q=ai:sharkovskyi.oleksandr-m"Romanenko, E. Yu."https://zbmath.org/authors/?q=ai:romanenko.elena-yu"Akbergenov, A. A."https://zbmath.org/authors/?q=ai:akbergenov.a-aIn this paper several specific features of computer turbulence is discussed. This phenomena occurs in problems of mathematical physics where the exact solutions have smooth dynamics, but their discrete analogs (numerical approximations) lead to the emergence, at a certain time, of the ``Brownian'' dynamics. This is usually considered as the result of incorrect calculations caused by the discretization of the original (continuous) problems. On the other hand these models may give a more adequate description of the material world then the continuous models, and in this case this phenomena is called computer turbulence. In particular, it can be understand as the phenomenon that occurs in discrete models of mathematical physics characterized by the emergence of Brownian-like dynamics at a certain time. In this paper several examples are given and discussed. The computer tunnelling effect is addressed.
Reviewer: Petr Sváček (Praha)Error estimation of the Besse relaxation scheme for a semilinear heat equationhttps://zbmath.org/1491.650802022-09-13T20:28:31.338867Z"Zouraris, Georgios E."https://zbmath.org/authors/?q=ai:zouraris.georgios-eSummary: The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse Relaxation Scheme in time [\textit{C. Besse}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 12, 1427--1432 (1998; Zbl 0911.65072)] with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete \(L_t^\infty (H_x^2)\)-norm at the time-nodes and in the discrete \(L_t^\infty (H_x^1)\)-norm at the intermediate time-nodes. It is the first time in the literature where the Besse Relaxation Scheme is applied and analysed in the context of parabolic equations.Non-overlapping Schwarz algorithms for the incompressible Navier-Stokes equations with DDFV discretizationshttps://zbmath.org/1491.650812022-09-13T20:28:31.338867Z"Goudon, Thierry"https://zbmath.org/authors/?q=ai:goudon.thierry"Krell, Stella"https://zbmath.org/authors/?q=ai:krell.stella"Lissoni, Giulia"https://zbmath.org/authors/?q=ai:lissoni.giuliaThe authors consider the numerical resolution of the unsteady incompressible Navier-Stokes problem. They first establish the well-posedness of DDFV (Discrete Duality Finite Volume) schemes on the whole spatial domain with general convection fluxes defined by \(B\)-schemes. Subsequently, they propose two non-overlapping DDFV Schwarz algorithms. DDFV discretizations are constructed with suitable transmission conditions. When using standard convection fluxes in the domain decomposition method, the iterative process converges to a system with modified fluxes at the interface. However, it is possible to modify the fluxes of the domain decomposition algorithm so that it converges to the reference scheme on the entire domain. Some numerical tests are presented to illustrate the behavior and the performances of the algorithms
Reviewer: Abdallah Bradji (Annaba)Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth modelhttps://zbmath.org/1491.650822022-09-13T20:28:31.338867Z"Remesan, Gopikrishnan Chirappurathu"https://zbmath.org/authors/?q=ai:remesan.gopikrishnan-chirappurathuThe author considers the finite volume approximation, on nonuniform Cartesian grids, of the nonlinear scalar conservation law \(\partial_t \alpha +\operatorname{div}(u f(\alpha )) = 0\) in two and three spatial dimensions with an initial data of bounded variation. A uniform estimate on total variation of discrete solutions is proved. The standard assumption which states that the advecting velocity vector is divergence free is relaxed. Since the underlying meshes are nonuniform Cartesian, it is possible to adaptively refine the mesh on regions where the solution is expected to have sharp fronts. A uniform BV estimate is also obtained for finite volume approximations of conservation laws that has a fully nonlinear flux on nonuniform Cartesian grids. Some numerical tests are presented to support the theoretical results.
Reviewer: Abdallah Bradji (Annaba)Variable piecewise interpolation solution of the transport equationhttps://zbmath.org/1491.650832022-09-13T20:28:31.338867Z"Romm, Ya. E."https://zbmath.org/authors/?q=ai:romm.ya-e"Dzhanunts, G. A."https://zbmath.org/authors/?q=ai:dzhanunts.g-aSummary: In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed.A fully discrete low-regularity integrator for the nonlinear Schrödinger equationhttps://zbmath.org/1491.650852022-09-13T20:28:31.338867Z"Ostermann, Alexander"https://zbmath.org/authors/?q=ai:ostermann.alexander"Yao, Fangyan"https://zbmath.org/authors/?q=ai:yao.fangyanSummary: For the solution of the one dimensional cubic nonlinear Schrödinger equation on the torus, we propose and analyze a fully discrete low-regularity integrator. The considered scheme is explicit. Its implementation relies on the fast Fourier transform with a complexity of \(\mathcal{O}(N\log N)\) operations per time step, where \(N\) denotes the degrees of freedom in the spatial discretization. We prove that the new scheme provides an \(\mathcal{O}(\tau^{\frac{3}{2}\gamma -\frac{1}{2}-\varepsilon}+N^{-\gamma})\) error bound in \(L^2\) for any initial data in \(H^\gamma\), \(\frac{1}{2}<\gamma \leq 1\), where \(\tau\) denotes the temporal step size. Numerical examples illustrate this convergence behavior.Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equationshttps://zbmath.org/1491.650862022-09-13T20:28:31.338867Z"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie"Xu, Jie"https://zbmath.org/authors/?q=ai:xu.jieSummary: We develop a set of numerical schemes for the Poisson-Nernst-Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and the second-order scheme preserves the maximum principle for cases where the equation satisfies the maximum principle.A noniterative domain decomposition method for the forward-backward heat equationhttps://zbmath.org/1491.650872022-09-13T20:28:31.338867Z"Banei, S."https://zbmath.org/authors/?q=ai:banei.siamak"Shanazari, K."https://zbmath.org/authors/?q=ai:shanazari.kamalSummary: A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous attempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system of equations. In addition, the convergence and stability of the proposed method are investigated. Some numerical experiments are given to show the effectiveness of the proposed method.Parallel tridiagonal matrix inversion with a hybrid multigrid-Thomas algorithm methodhttps://zbmath.org/1491.650882022-09-13T20:28:31.338867Z"Parker, J. T."https://zbmath.org/authors/?q=ai:parker.jason-t"Hill, P. A."https://zbmath.org/authors/?q=ai:hill.p-a"Dickinson, D."https://zbmath.org/authors/?q=ai:dickinson.detta|dickinson.david-l"Dudson, B. D."https://zbmath.org/authors/?q=ai:dudson.b-dSummary: Tridiagonal matrix inversion is an important operation with many applications. It arises frequently in solving discretized one-dimensional elliptic partial differential equations, and forms the basis for many algorithms for block tridiagonal matrix inversion for discretized PDEs in higher-dimensions. In such systems, this operation is often the scaling bottleneck in parallel computation. In this paper, we derive a hybrid multigrid-Thomas algorithm designed to efficiently invert tridiagonal matrix equations in a highly-scalable fashion in the context of time evolving partial differential equation systems. We decompose the domain between processors, using multigrid to solve on a grid consisting of the boundary points of each processor's local domain. We then reconstruct the solution on each processor using a direct solve with the Thomas algorithm. This algorithm has the same theoretical optimal scaling as cyclic reduction and recursive doubling. We use our algorithm to solve Poisson's equation as part of the spatial discretization of a time-evolving PDE system. Our algorithm is faster than cyclic reduction per inversion and retains good scaling efficiency to twice as many cores.Stability and error estimates for non-linear Cahn-Hilliard-type equations on evolving surfaceshttps://zbmath.org/1491.650892022-09-13T20:28:31.338867Z"Beschle, Cedric Aaron"https://zbmath.org/authors/?q=ai:beschle.cedric-aaron"Kovács, Balázs"https://zbmath.org/authors/?q=ai:kovacs.balazsSummary: In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix-vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.Semi-discrete finite-element approximation of nonlocal hyperbolic problemhttps://zbmath.org/1491.650902022-09-13T20:28:31.338867Z"Chaudhary, Sudhakar"https://zbmath.org/authors/?q=ai:chaudhary.sudhakar"Srivastava, Vimal"https://zbmath.org/authors/?q=ai:srivastava.vimalSummary: In this paper, we investigate a semi-discrete finite-element approximation of nonlocal hyperbolic problem. A priori error estimate for the semi-discrete scheme is derived. A fully discrete scheme based on backward difference method is constructed. We discuss the existence-uniqueness of the solution for fully discrete problem. In order to linearize the nonlinear fully discrete problem, we use Newton's method. Numerical results based on the usual finite-element method are provided to confirm the theoretical estimate.Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn-Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosityhttps://zbmath.org/1491.650912022-09-13T20:28:31.338867Z"Chen, Chuanjun"https://zbmath.org/authors/?q=ai:chen.chuanjun"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: We construct a fully-discrete finite element numerical scheme for the Cahn-Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier-Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn-Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh-Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noisehttps://zbmath.org/1491.650922022-09-13T20:28:31.338867Z"Feng, Xiaobing"https://zbmath.org/authors/?q=ai:feng.xiaobing"Qiu, Hailong"https://zbmath.org/authors/?q=ai:qiu.hailongSummary: This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite element methods.A space-time Trefftz discontinuous Galerkin method for the linear Schrödinger equationhttps://zbmath.org/1491.650932022-09-13T20:28:31.338867Z"Gómez, Sergio"https://zbmath.org/authors/?q=ai:gomez.sergio-alejandro"Moiola, Andrea"https://zbmath.org/authors/?q=ai:moiola.andreaThis paper proposes and analyzes a space-time Trefftz discontinuous Galerkin method for the Schrödinger equation with piecewise-constant potential. The main feature of Trefftzmethods is that they seek approximations in spaces spanned by local solutions ofthe PDE considered. This typically requires nonpolynomial basis functions. Trefftz scheme allows for much faster convergence in terms of degrees of freedom than classical polynomial DG schemes.This approach to the Trefftz approximation theory is completelydifferent from that used for the Helmholtz equation. The well-posedness and quasi-optimality of theTrefftz-DG approximation for arbitrary dimensions and discrete Trefftz subspaces are proved.The error analysis for discrete subspaces spanned by complexexponentials satisfying the Schrödinger equation is presented. Some numerical experiments validatethe theoretical results presented.
Reviewer: Yan Xu (Hefei)Numerical analysis for Maxwell obstacle problems in electric shieldinghttps://zbmath.org/1491.650942022-09-13T20:28:31.338867Z"Hensel, Maurice"https://zbmath.org/authors/?q=ai:hensel.maurice"Yousept, Irwin"https://zbmath.org/authors/?q=ai:yousept.irwinThis paper discusses a finite element method for a Maxwell obstacle problem in electric shielding. The approach relies on the leapfrog time-stepping and the Nedelec edge elements in which no additional nonlinear solver is required for the computation of the discrete evolutionary variational inequality of Ampere-Maxwell type. \(L^1\) and \(L^2\) stability are discussed and several numerical experiments are included.
Reviewer: Marius Ghergu (Dublin)Local transparent boundary conditions for wave propagation in fractal trees. I: Method and numerical implementationhttps://zbmath.org/1491.650952022-09-13T20:28:31.338867Z"Joly, Patrick"https://zbmath.org/authors/?q=ai:joly.patrick"Kachanovska, Maryna"https://zbmath.org/authors/?q=ai:kachanovska.marynaA nonsymmetric approach and a quasi-optimal and robust discretization for the Biot's modelhttps://zbmath.org/1491.650962022-09-13T20:28:31.338867Z"Khan, Arbaz"https://zbmath.org/authors/?q=ai:khan.arbaz"Zanotti, Pietro"https://zbmath.org/authors/?q=ai:zanotti.pietroThe paper analyzes the numerical method for Biot's model describing the elastic wave propagation inside a porous medium saturated with a fluid. Variables in this model represent the displacement of the medium and the fluid pressure. In addition, there are several material parameters. However, spurious oscillations or volumetric locking may occur for specific values of these parameters. The authors focus on overcoming this problem and propose a method that is robust in the sense that it is uniformly stable with respect to all parameters.
First, the authors establish a novel nonsymmetric variational setting, where the norm measuring the data is not dual to the norm for measuring the solution. Then, they show the well-posedness of the setting and derive stability estimates. Furthermore, the authors propose a method that uses the backward Euler scheme for temporal discretization combined with the finite element method using first-order nonconforming Crouzeix-Raviart elements for the displacement and first-order discontinuous piecewise affine functions for the fluid pressure. The presented analysis of stability and error estimates leads to the conclusion that the method is robust and quasi-optimal. Finally, possible generalizations of the results are discussed.
Reviewer: Dana Černá (Liberec)Second-order convergence of the linearly extrapolated Crank-Nicolson method for the Navier-Stokes equations with \(H^1\) initial datahttps://zbmath.org/1491.650972022-09-13T20:28:31.338867Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyang"Ma, Shu"https://zbmath.org/authors/?q=ai:ma.shu"Wang, Na"https://zbmath.org/authors/?q=ai:wang.naSummary: This article concerns the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with \(H^1\) initial data. By utilizing special locally refined temporal stepsizes, we prove that the linearly extrapolated Crank-Nicolson scheme, with the usual stabilized Taylor-Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to support the theoretical analysis.Local and parallel efficient BDF2 and BDF3 rotational pressure-correction schemes for a coupled Stokes/Darcy systemhttps://zbmath.org/1491.650982022-09-13T20:28:31.338867Z"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.1"Wang, Xue"https://zbmath.org/authors/?q=ai:wang.xue"Al Mahbub, Md. Abdullah"https://zbmath.org/authors/?q=ai:al-mahbub.md-abdullah"Zheng, Haibiao"https://zbmath.org/authors/?q=ai:zheng.haibiao"Chen, Zhangxin"https://zbmath.org/authors/?q=ai:chen.zhangxinThis paper extends authors earlier work [\textit{J. Li} et al., Comput. Math. with Appl. 79, 337--353 (2020; Zbl 1443.65187); Numer. Methods Partial Differential Equations 35, 1873--1889 (2019; Zbl 1423.76253)] where first- and second-order (in time) BE (backward Euler) and BDF2 schemes with the rotational pressure-correction methods introduced in [\textit{J. Guermond} et al., SIAM J. Numer. Anal. 43, 239--258 (2005; Zbl 1083.76044)] are studied for a coupled Stokes/Darcy system. These temporal schemes are developed, and the BDF2/BDF3 rotational pressure-correction methods are studied for the Stokes/Darcy system. It was proven that the BDF2/BDF3 rotational pressure-correction methods are unconditionally stable, long-time accurate with a uniform-in-time error bound, and efficient in that only two decoupled equations are required to solve at each time step. At each time step, only one linear system of equations has to be solved, which thus significantly reduces the computational time and memory costs in practice. The presented projection methods are combined with the local and parallel methods based on full overlapping decoupled techniques for the coupled Stokes/Darcy system which increases the computational efficiency further. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed methods.
Reviewer: Bülent Karasözen (Ankara)A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergencehttps://zbmath.org/1491.650992022-09-13T20:28:31.338867Z"Masri, Rami"https://zbmath.org/authors/?q=ai:masri.rami"Liu, Chen"https://zbmath.org/authors/?q=ai:liu.chen"Riviere, Beatrice"https://zbmath.org/authors/?q=ai:riviere.beatrice-mSummary: A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the proposed scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problemshttps://zbmath.org/1491.651002022-09-13T20:28:31.338867Z"Singh, Gautam"https://zbmath.org/authors/?q=ai:singh.gautam-b"Natesan, Srinivasan"https://zbmath.org/authors/?q=ai:natesan.srinivasanSummary: A parabolic convection-diffusion-reaction problem is discretized by the non-symmetric interior penalty Galerkin (NIPG) method in space and discontinuous Galerkin (DG) method in time. To improve the order of convergence of the numerical scheme, we have used piecewise Lagrange interpolation at Gauss points and estimated the error bound in the discrete energy norm. We have shown superconvergence properties of the DG method, i.e., \((k + 1)\)-order convergence in space and \((l + 1)\)-order convergence in time, where \(k\) and \(l\) are the degrees of piecewise polynomials in the finite element space used in spatial and temporal variables, respectively. Numerical results are given to verify our theoretical findings.New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous mediahttps://zbmath.org/1491.651012022-09-13T20:28:31.338867Z"Sun, Weiwei"https://zbmath.org/authors/?q=ai:sun.weiweiSummary: Numerical methods and analysis for compressible miscible flow in porous media have been investigated extensively in the last several decades. Amongst those methods, the lowest-order mixed method is the most popular one in practical applications. The method is based on the linear Lagrange approximation for the concentration and the lowest order (zero-order) Raviart-Thomas mixed approximation for the Darcy velocity/pressure. However, the existing error analysis only provides the first-order accuracy in \(L^2\)-norm for all three physical components in spatial direction, which was proved under certain extra restrictions on both time step and spatial meshes. The analysis is not optimal for the concentration mainly due to the strong coupling of the system and the drawback of the traditional approach which leads to serious pollution to the numerical concentration in analysis. The main task of this paper is to present a new analysis and establish the optimal error estimate of the commonly-used linearized lowest-order mixed FEM. In particular, the second-order accuracy for the concentration in spatial direction is proved unconditionally. Moreover, we propose a simple recovery technique to obtain a new numerical Darcy velocity/pressure of second-order accuracy by re-solving an elliptic pressure equation. Also we extend our analysis to a second-order time discrete scheme to obtain optimal error estimates in both spatial and temporal directions. Numerical results are provided to confirm our theoretical analysis and show the efficiency of the method.A two-grid combined mixed finite element and discontinuous Galerkin method for an incompressible miscible displacement problem in porous mediahttps://zbmath.org/1491.651042022-09-13T20:28:31.338867Z"Yang, Jiming"https://zbmath.org/authors/?q=ai:yang.jiming"Su, Yifan"https://zbmath.org/authors/?q=ai:su.yifanSummary: An incompressible miscible displacement problem is investigated. A two-grid algorithm of a full-discretized combined mixed finite element and discontinuous Galerkin approximation to the miscible displacement in porous media is proposed. The error estimate for the concentration in \(H^1\)-norm and the error estimates for the pressure and the velocity in \(L^2\)-norm are obtained. The analysis shows that the asymptotically optimal approximation can be achieved as long as the mesh size satisfies \(h = O(H^2)\), where \(H\) and \(h\) are the sizes of the coarse mesh and the fine mesh, respectively. Meanwhile, the effectiveness of the presented algorithm is verified by numerical experiments, from which it can be seen that the algorithm is spent much less time.Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cellhttps://zbmath.org/1491.651052022-09-13T20:28:31.338867Z"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``zero-energy-contribution'' feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.The convergence analysis of semi- and fully-discrete projection-decoupling schemes for the generalized Newtonian modelshttps://zbmath.org/1491.651072022-09-13T20:28:31.338867Z"Zhou, Guanyu"https://zbmath.org/authors/?q=ai:zhou.guanyuSummary: We propose two linear schemes (1st- and 2nd-order) for the generalized Newtonian flow with the shear-dependent viscosity, which combine the decoupling techniques with the projection methods. The linear stabilization terms mimic \(-k\partial_t \Delta{\boldsymbol{u}}\) and \(-k\partial_{tt} \Delta{\boldsymbol{u}}\) from the PDE point of view. By our schemes, each velocity component can be computed in parallel efficiently using the same solver \((I-\alpha^{-1}k\Delta)^{-1}\) at every time level. We analyze the convergence rates of the (temporally) semi- and the fully-discrete schemes. The theoretical results are testified by the numerical experiments.Quadratic spline function for the approximate solution of an intermediate space-fractional advection diffusion equationhttps://zbmath.org/1491.651082022-09-13T20:28:31.338867Z"Abdel-Rehi, E. A."https://zbmath.org/authors/?q=ai:abdel-rehi.e-a"Brikaa, M. G."https://zbmath.org/authors/?q=ai:brikaa.m-gSummary: The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.A new Lagrange multiplier approach for constructing structure preserving schemes. II: Bound preservinghttps://zbmath.org/1491.651102022-09-13T20:28:31.338867Z"Cheng, Qing"https://zbmath.org/authors/?q=ai:cheng.qing"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jieIn this paper, positivity preserving schemes using Lagrange multiplier approach in Part I [\textit{Q. Cheng} and \textit{J. Shen}, Comput. Methods Appl. Mech. Eng. 391, Article ID 114585, 25 p. (2022; Zbl 07487691)] are extended to construct bound preserving schemes for a class of nonlinear PDEs in the following form:
\[
u_t + {\mathcal L }u + {\mathcal N }(u) = 0
\]
with suitable initial and boundary conditions, where \({\mathcal L}\) is a linear or nonlinear nonnegative operator and \({\mathcal N}(u)\) is a semilinear or quasi-linear operator.
For problems which also conserve mass, these bound preserving schemes are modified to also conserve mass. For second-order parabolic-type equations stability results are established with a second-order scheme with mass conservation. A hybrid spectral method is considered for error analysis for a fully discretized second-order scheme. These schemes are applied to several typical PDEs with bound and/or mass preserving properties and are validated for a variety of problems with bound preserving solutions, including the Allen-Cahn and Cahn-Hilliard equations and a class of Fokker-Planck equations. The schemes constructed in this paper include the cutoff approach [\textit{C. Lu} et al., J. Computat. Phys., 242, 24--36 (2013; Zbl 1297.65097)] as a special case, so that they provide an alternative interpretation of the cutoff approach and allows us to construct new cutoff implicit-explicit (IMEX) schemes with mass conservation.
Reviewer: Bülent Karasözen (Ankara)Second-order SAV schemes for the nonlinear Schrödinger equation and their error analysishttps://zbmath.org/1491.651112022-09-13T20:28:31.338867Z"Deng, Beichuan"https://zbmath.org/authors/?q=ai:deng.beichuan"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie"Zhuang, Qingqu"https://zbmath.org/authors/?q=ai:zhuang.qingquSummary: We consider a second-order SAV scheme for the nonlinear Schrödinger equation in the whole space with typical generalized nonlinearities, and carry out a rigorous error analysis. We also develop a fully discretized SAV scheme with Hermite-Galerkin approximation for the space variables, and present numerical experiments to validate our theoretical results.A novel discrete fractional Grönwall-type inequality and its application in pointwise-in-time error estimateshttps://zbmath.org/1491.651122022-09-13T20:28:31.338867Z"Li, Dongfang"https://zbmath.org/authors/?q=ai:li.dongfang"She, Mianfu"https://zbmath.org/authors/?q=ai:she.mianfu"Sun, Hai-wei"https://zbmath.org/authors/?q=ai:sun.haiwei"Yan, Xiaoqiang"https://zbmath.org/authors/?q=ai:yan.xiaoqiangSummary: We present a family of fully-discrete schemes for numerically solving nonlinear sub-diffusion equations, taking the weak regularity of the exact solutions into account. Using a novel discrete fractional Grönwall inequality, we obtain pointwise-in-time error estimates of the time-stepping methods. It is proved that as \(t\rightarrow 0\), the convergence orders can be \(\sigma_k\), where \(\sigma_k\) is the regularity parameter. The initial convergence results are sharp. As \(t\) is far away from 0, the schemes give a better convergence results. Numerical experiments are given to confirm the theoretical results.The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applicationshttps://zbmath.org/1491.651132022-09-13T20:28:31.338867Z"Pourbabaee, Marzieh"https://zbmath.org/authors/?q=ai:pourbabaee.marzieh"Saadatmandi, Abbas"https://zbmath.org/authors/?q=ai:saadatmandi.abbasSummary: In this paper, the properties of Chebyshev polynomials and the Gauss-Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given.Analysis of the Fitzhugh Nagumo model with a new numerical schemehttps://zbmath.org/1491.651142022-09-13T20:28:31.338867Z"Mishra, Jyoti"https://zbmath.org/authors/?q=ai:mishra.jyotiSummary: The model describing a prototype of an excitable system was extended using the newly established concept of fractional differential operators with non-local and non-singular kernel in this paper. We presented a detailed discussion underpinning the well-poseness of the extended model. Due to the non-linearity of the modified model, we solved it using a newly established numerical scheme for partial differential equations that combines the fundamental theorem of fractional calculus, the Laplace transform and the Lagrange interpolation approximation. We presented some numerical simulations that, of course reflect asymptotically the real world observed behaviors.A highly accurate difference method for solving the Dirichlet problem for Laplace's equation on a rectanglehttps://zbmath.org/1491.651152022-09-13T20:28:31.338867Z"Dosiyev, Adiguzel A."https://zbmath.org/authors/?q=ai:dosiyev.adiguzel-a"Sarikaya, Hediye"https://zbmath.org/authors/?q=ai:sarikaya.hediyeSummary: \(O(h^8)\) order \((h\) is the mesh size) of accurate three-stage difference method on a square grid for the approximate solution of the Dirichlet problem for Laplace's equation on a rectangle is proposed and justified without taking more than 9 nodes of the grid. At the first stage, by using the 9-point scheme the sum of the pure fourth derivatives of the desired solution is approximated of order \(O(h^6)\). At the second stage, approximate values of the sum of the pure eighth derivatives is approximated of order \(O(h^2)\) by the 5-point scheme. At the final third stage, the system of simplest 5-point difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first and second stages. Numerical experiment is illustrated to support the analysis made.
For the entire collection see [Zbl 1436.46003].A numerical algorithm to computationally solve the Hemker problem using Shishkin mesheshttps://zbmath.org/1491.651162022-09-13T20:28:31.338867Z"Hegarty, A. F."https://zbmath.org/authors/?q=ai:hegarty.alan-f"O'Riordan, E."https://zbmath.org/authors/?q=ai:oriordan.eugeneA numerical algorithm is presented to solve a benchmark problem proposed by \textit{P. W. Hemker} [J. Comput. Appl. Math. 76, No. 1--2, 277--285 (1996; Zbl 0870.35020)]. The numerical algorithm is composed of several different Shishkin meshes defined across different coordinate systems aligned to three overlapping subdomains. It is constructed based on parameter explicit pointwise bounds on how the continuous solutions decay away from the circle. Using upwinding in all co-ordinate directions, numerical solutions exhibit no spurious oscillations. Several layer-adapted Shishkin meshes are utilized and these grids are aligned both to the geometry of the domain and to the dominant direction of decay within the boundary/interior layer functions. Numerical experiments indicate that the method is producing accurate approximations over an extensive range of the singular perturbation parameter. The numerical approximations are converging to the continuous solution for each value of the parameter, but not uniformly in the singular perturbation parameter
Reviewer: Bülent Karasözen (Ankara)Convergence rate analysis for deep Ritz methodhttps://zbmath.org/1491.651172022-09-13T20:28:31.338867Z"Duan, Chenguang"https://zbmath.org/authors/?q=ai:duan.chenguang"Jiao, Yuling"https://zbmath.org/authors/?q=ai:jiao.yuling"Lai, Yanming"https://zbmath.org/authors/?q=ai:lai.yanming"Li, Dingwei"https://zbmath.org/authors/?q=ai:li.dingwei"Lu, Xiliang"https://zbmath.org/authors/?q=ai:lu.xiliang"Yang, Jerry Zhijian"https://zbmath.org/authors/?q=ai:yang.jerry-zhijianSummary: Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [\textit{W. E} and \textit{B. Yu}, Commun. Math. Stat. 6, No. 1, 1--12 (2018; Zbl 1392.35306)] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in \(H^1\) norm for DRM using deep networks with \(\mathrm{ReLU}^2\) activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep \(\mathrm{ReLU}^2\) network in \(C^1\) norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and \(\mathrm{ReLU}^2\) network, both of which are of independent interest.A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaceshttps://zbmath.org/1491.651212022-09-13T20:28:31.338867Z"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Liu, Xiaoli"https://zbmath.org/authors/?q=ai:liu.xiaoli"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.bo"Zhang, Haiwen"https://zbmath.org/authors/?q=ai:zhang.haiwenUnique continuation on quadratic curves for harmonic functionshttps://zbmath.org/1491.651222022-09-13T20:28:31.338867Z"Ke, Yufei"https://zbmath.org/authors/?q=ai:ke.yufei"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.7|chen.yu.2|chen.yu.5|chen.yu.8|chen.yuqun|chen.yu.4|chen.yu.1|chen.yu.3|chen.yu.6Summary: The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated. The numerical algorithm is provided based on collocation method and Tikhonov regularization. The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.An efficient DWR-type a posteriori error bound of SDFEM for singularly perturbed convection-diffusion PDEshttps://zbmath.org/1491.651232022-09-13T20:28:31.338867Z"Avijit, D."https://zbmath.org/authors/?q=ai:avijit.d"Natesan, S."https://zbmath.org/authors/?q=ai:natesan.srinivasanSummary: This article deals with the residual-based a posteriori error estimation in the standard energy norm for the streamline-diffusion finite element method (SDFEM) for singularly perturbed convection-diffusion equations. The well-known dual-weighted residual (DWR) technique has been adopted to elevate the accuracy of the error estimator. Our main contribution is finding an efficient computable DWR-type robust residual-based a posteriori error bound for the SDFEM. The local lower error bound has also been provided. An adaptive mesh refinement algorithm has been addressed and lastly, some numerical experiments are carried out to justify the theoretical proofs.A polygonal discontinuous Galerkin method with minus one stabilizationhttps://zbmath.org/1491.651252022-09-13T20:28:31.338867Z"Bertoluzza, Silvia"https://zbmath.org/authors/?q=ai:bertoluzza.silvia"Prada, Daniele"https://zbmath.org/authors/?q=ai:prada.danieleSummary: We propose a discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element \(K\), a residual term involving the fluxes, measured in the norm of the dual of \(H^1 (K)\). The scalar product corresponding to such a norm is numerically realized \textit{via} the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the broken \(H^1 \) norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flowhttps://zbmath.org/1491.651262022-09-13T20:28:31.338867Z"Burman, Erik"https://zbmath.org/authors/?q=ai:burman.erik"Puppi, Riccardo"https://zbmath.org/authors/?q=ai:puppi.riccardoSummary: We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as \(\mathcal{O} (h^{-1})\), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as \(\mathcal{O} (h^{- k -1})\), \(k\) being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual \(L^2\)-norm. However, we are still able to recover the optimal a priori \(L^2\)-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEshttps://zbmath.org/1491.651282022-09-13T20:28:31.338867Z"Carstensen, C."https://zbmath.org/authors/?q=ai:carstensen.carsten"Nataraj, Neela"https://zbmath.org/authors/?q=ai:nataraj.neela"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients \({\mathbf{A}}, {\mathbf{b}},\gamma\) in \(L^\infty\) and symmetric and uniformly positive definite coefficient matrix \({\mathbf{A}} \), this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in \(H(\mathrm{div})\times L^2\) as well as in in \(L^2\times L^2\) up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to \(L^\infty\) coefficients. The compactness argument of \textit{A. H. Schatz} and \textit{J. Wang} [Math. Comput. 65, No. 213, 19--27 (1996; Zbl 0856.65129)] for the displacement-oriented problem does \textit{not} apply immediately to the mixed formulation in \(H(\mathrm{div})\times L^2\). But it allows the uniform approximation of some \(L^2\) contributions and can be combined with a recent \(L^2\) best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equationshttps://zbmath.org/1491.651292022-09-13T20:28:31.338867Z"Caucao, Sergio"https://zbmath.org/authors/?q=ai:caucao.sergio"Gatica, Gabriel N."https://zbmath.org/authors/?q=ai:gatica.gabriel-n"Ortega, Juan P."https://zbmath.org/authors/?q=ai:ortega.juan-pabloSummary: We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman-Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška-Brezzi's theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimal \textit{a priori} error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.A reduced basis method for fractional diffusion operators. IIhttps://zbmath.org/1491.651312022-09-13T20:28:31.338867Z"Danczul, Tobias"https://zbmath.org/authors/?q=ai:danczul.tobias"Schöberl, Joachim"https://zbmath.org/authors/?q=ai:schoberl.joachimSummary: We present a novel numerical scheme to approximate the solution map \(s \mapsto u(s) := \mathcal{L}^{-s} f\) to fractional PDEs involving elliptic operators. Reinterpreting \(\mathcal{L}^{-s}\) as an interpolation operator allows us to write \(u(s)\) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation \(L\) of the operator whose inverse is projected to the \(s\)-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously.
A second algorithm is presented to avoid inversion of \(L\). Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
For Part I, see [\textit{T. Danczul} and \textit{J. Schöberl}, Numer. Math. 151, No. 2, 369--404 (2022; Zbl 07536676)].New \(H(\mathrm{div})\)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal mesheshttps://zbmath.org/1491.651322022-09-13T20:28:31.338867Z"Devloo, Philippe R. B."https://zbmath.org/authors/?q=ai:devloo.philippe-remy-bernard"Farias, Agnaldo M."https://zbmath.org/authors/?q=ai:farias.agnaldo-m"Gomes, Sônia M."https://zbmath.org/authors/?q=ai:gomes.sonia-maria"Pereira, Weslley"https://zbmath.org/authors/?q=ai:pereira.weslley-s"Dos Santos, Antonio J. B."https://zbmath.org/authors/?q=ai:dos-santos.antonio-j-b"Valentin, Frédéric"https://zbmath.org/authors/?q=ai:valentin.fredericSummary: This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforced \textit{via} the rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in the \(H(\mathrm{div})\) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the \(L^2\)-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.Non-symmetric isogeometric FEM-BEM couplingshttps://zbmath.org/1491.651332022-09-13T20:28:31.338867Z"Elasmi, Mehdi"https://zbmath.org/authors/?q=ai:elasmi.mehdi"Erath, Christoph"https://zbmath.org/authors/?q=ai:erath.christoph"Kurz, Stefan"https://zbmath.org/authors/?q=ai:kurz.stefanSummary: We present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform \(h\)- and \(p\)-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.Singularly perturbed reaction-diffusion problems as first order systemshttps://zbmath.org/1491.651352022-09-13T20:28:31.338867Z"Franz, Sebastian"https://zbmath.org/authors/?q=ai:franz.sebastianSummary: We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and \(H_{\mathrm{div}}\)-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced \(H^2\)-norm for the second order formulation.MINRES for second-order PDEs with singular datahttps://zbmath.org/1491.651362022-09-13T20:28:31.338867Z"Führer, Thomas"https://zbmath.org/authors/?q=ai:fuhrer.thomas"Heuer, Norbert"https://zbmath.org/authors/?q=ai:heuer.norbert"Karkulik, Michael"https://zbmath.org/authors/?q=ai:karkulik.michaelIn this work, the minimum residual finite element methods (MINRES FEM) are analyzed with source functionals in \(H^{-1}(\Omega )\), the dual space of \(H^1_0 (\Omega )\) with \(\Omega \subset \mathbb{R}^d\) (\(d = 2, 3\)) a polytopal domain, and point sources by focusing on the first-order reformulation of the Poisson problem. The discontinuous Petrov-Galerkin (DPG) method with optimal test functions is also studied. They are MINRES methods that minimize a functional in the dual norm of broken test spaces. The results of the paper are also extended and analyzed for the DPG method for such data. The authors show that the MINRES FEM for the Poisson problem on Lipschitz domains can be modified to handle \(H^{-1}\) loads and lead to optimal convergence rates. The regularization approaches are extended that allow the use of \(H^{-1}\) loads and to the case of point loads. Appropriate convergence orders are proven for all cases. Several numerical experiments are presented that confirm the theoretical results. The approach can also be extended to general well-posed second-order problems.
Reviewer: Bülent Karasözen (Ankara)An embedded discontinuous Galerkin method for the Oseen equationshttps://zbmath.org/1491.651372022-09-13T20:28:31.338867Z"Han, Yongbin"https://zbmath.org/authors/?q=ai:han.yongbin"Hou, Yanren"https://zbmath.org/authors/?q=ai:hou.yanrenSummary: In this paper, the \textit{a priori} error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in the \(L^2 (\Omega)\) norm, has an optimal error bound with convergence order \(k+1\), where the constants are dependent on the Reynolds number (or \(\nu^{-1})\), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence order \(k+1/2\). Here, \(k\) is the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results.Numerical upscaling for heterogeneous materials in fractured domainshttps://zbmath.org/1491.651382022-09-13T20:28:31.338867Z"Hellman, Fredrik"https://zbmath.org/authors/?q=ai:hellman.fredrik"Målqvist, Axel"https://zbmath.org/authors/?q=ai:malqvist.axel"Wang, Siyang"https://zbmath.org/authors/?q=ai:wang.siyangSummary: We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise \textit{e.g.} in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An \textit{a priori} error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problemshttps://zbmath.org/1491.651402022-09-13T20:28:31.338867Z"He, Xiaoxiao"https://zbmath.org/authors/?q=ai:he.xiaoxiao"Song, Fei"https://zbmath.org/authors/?q=ai:song.fei"Deng, Weibing"https://zbmath.org/authors/?q=ai:deng.weibingThe paper deals with the numerical solution of the Stokes interface problem by a stabilized extended finite element method on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming \(P_1\) velocity and elementwise \(P_0\) pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size \(h\), the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in \(L_2\) norm for pressure are uniform to the viscosity and the location of the interface. Two numerical examples are presented to support the theoretical analysis.
Reviewer: Vit Dolejsi (Praha)Error analysis of higher order trace finite element methods for the surface Stokes equationhttps://zbmath.org/1491.651422022-09-13T20:28:31.338867Z"Jankuhn, Thomas"https://zbmath.org/authors/?q=ai:jankuhn.thomas"Olshanskii, Maxim A."https://zbmath.org/authors/?q=ai:olshanskii.maxim-a"Reusken, Arnold"https://zbmath.org/authors/?q=ai:reusken.arnold"Zhiliakov, Alexander"https://zbmath.org/authors/?q=ai:zhiliakov.alexanderSummary: The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in \(\mathbb{R}^3\). The method employs parametric \(\mathbf{P}_k\)-\(P_{k-1}\) finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin-Helmholtz instability problem on the unit sphere.An immersed hybrid difference method for the elliptic interface equationhttps://zbmath.org/1491.651432022-09-13T20:28:31.338867Z"Jeon, Youngmok"https://zbmath.org/authors/?q=ai:jeon.youngmokThe author proposes an immersed hybrid difference method for elliptic interface problems. The method can be extended to three-dimensional problems easily. Numerical analysis should give more details in the future.
Reviewer: Zhen Chao (Milwaukee)Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined meshhttps://zbmath.org/1491.651442022-09-13T20:28:31.338867Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyangSummary: The Galerkin finite element solution \(u_h\) of the Poisson equation \(-\Delta u=f\) under the Neumann boundary condition in a possibly nonconvex polygon \(\varOmega\), with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:
\[
\|u_h\|_{L^{\infty}(\varOmega)} \le C\ell_h\|u\|_{L^{\infty}(\varOmega)},
\]
where \(\ell_h = \ln (2+1/h)\) for piecewise linear elements and \(\ell_h=1\) for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:
\[
\|u-u_h\|_{L^{\infty}(\varOmega)} \le C\ell_h\|u-I_hu\|_{L^{\infty}(\varOmega)},
\]
where \(I_h\) denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm:
\[
\|u-u_h\|_{L^\infty (\varOmega)} \le
\begin{cases}
C\ell_h h^{k+2-\frac{2}{p}} \|f\|_{W^{k,p}(\varOmega)} &\text{if } r\ge k+1,\\
C\ell_h h^{k+1} \|f\|_{H^k(\varOmega)} &\text{if } r=k,
\end{cases}
\]
where \(r\ge 1\) is the degree of finite elements, \(k\) is any nonnegative integer no larger than \(r\), and \(p\in [2,\infty)\) can be arbitrarily large.High order unconditionally energy stable RKDG schemes for the Swift-Hohenberg equationhttps://zbmath.org/1491.651452022-09-13T20:28:31.338867Z"Liu, Hailiang"https://zbmath.org/authors/?q=ai:liu.hailiang"Yin, Peimeng"https://zbmath.org/authors/?q=ai:yin.peimengSummary: We propose unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows including the Swift-Hohenberg equation. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved for arbitrary time steps and spatial meshes. The method integrates a penalty free DG method for spatial discretization with a multi-stage algebraically stable RK method for temporal discretization by the energy quadratiztion (EQ) strategy. The resulting fully discrete DG method is proven to be unconditionally energy stable. By numerical tests on several benchmark problems we demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticityhttps://zbmath.org/1491.651462022-09-13T20:28:31.338867Z"Li, Yuwen"https://zbmath.org/authors/?q=ai:li.yuwenSummary: For the planar Navier-Lamé equation in mixed form with symmetric stress tensors, we prove the uniform quasi-optimal convergence of an adaptive method based on the hybridized mixed finite element proposed in [\textit{S. Gong} et al., Numer. Math. 141, No. 2, 569--604 (2019; Zbl 1412.65211)]. The main ingredients in the analysis consist of a discrete \textit{a posteriori} upper bound and a quasi-orthogonality result for the stress field under the mixed boundary condition. Compared with existing adaptive methods, the proposed adaptive algorithm could be directly applied to the traction boundary condition and be easily implemented.Multiscale scattering in nonlinear Kerr-type mediahttps://zbmath.org/1491.651482022-09-13T20:28:31.338867Z"Maier, Roland"https://zbmath.org/authors/?q=ai:maier.roland|maier.roland.1"Verfürth, Barbara"https://zbmath.org/authors/?q=ai:verfurth.barbaraSummary: We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.Projection in negative norms and the regularization of rough linear functionalshttps://zbmath.org/1491.651492022-09-13T20:28:31.338867Z"Millar, F."https://zbmath.org/authors/?q=ai:millar.felipe"Muga, I."https://zbmath.org/authors/?q=ai:muga.ignacio"Rojas, S."https://zbmath.org/authors/?q=ai:rojas.sergio"van der Zee, K. G."https://zbmath.org/authors/?q=ai:van-der-zee.kristoffer-georgeSummary: In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as \(W_0^{1,q}(\varOmega)\), where \(1<q<\infty\) and \(\varOmega\) is a Lipschitz domain, we propose a projection method in negative Sobolev spaces \(W^{-1,p}(\varOmega), p\) being the conjugate exponent satisfying \(p^{-1} + q^{-1} = 1\). Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of \(W^{-1,p}(\varOmega)\), though not of \(L^1(\varOmega)\), but one strives for a regular approximation in \(L^1(\varOmega)\). We focus on projections onto discrete finite element spaces \(G_n\), and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace \(V_m\). We show that this idea leads to a fully discrete method given by a mixed problem on \(V_m\times G_n\). We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods. We present numerical experiments that compute finite element approximations to Dirac delta's and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scatteringhttps://zbmath.org/1491.651502022-09-13T20:28:31.338867Z"Sheng, Qiwei"https://zbmath.org/authors/?q=ai:sheng.qiwei"Hauck, Cory D."https://zbmath.org/authors/?q=ai:hauck.cory-dSummary: We present an error analysis for the discontinuous Galerkin (DG) method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation with isotropic scattering. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter \(\varepsilon\) which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.The role of mesh quality and mesh quality indicators in the virtual element methodhttps://zbmath.org/1491.651512022-09-13T20:28:31.338867Z"Sorgente, T."https://zbmath.org/authors/?q=ai:sorgente.tommaso"Biasotti, S."https://zbmath.org/authors/?q=ai:biasotti.silvia"Manzini, G."https://zbmath.org/authors/?q=ai:manzini.gianmarco"Spagnuolo, M."https://zbmath.org/authors/?q=ai:spagnuolo.michelaSummary: Since its introduction, the virtual element method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the finite element method (FEM) analysis. In this work, we experimentally show that the VEM still converges, with almost optimal rates and low errors in the \(L^2, H^1\) and \(L^{\infty}\) norms, even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a mesh is potentially critical for VEM.Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2Dhttps://zbmath.org/1491.651532022-09-13T20:28:31.338867Z"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.3|zhang.jin|zhang.jin.1|zhang.jin.2"Lv, Yanhui"https://zbmath.org/authors/?q=ai:lv.yanhuiSummary: On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.Coupled iterative analysis for stationary inductionless magnetohydrodynamic system based on charge-conservative finite element methodhttps://zbmath.org/1491.651542022-09-13T20:28:31.338867Z"Zhang, Xiaodi"https://zbmath.org/authors/?q=ai:zhang.xiaodi"Ding, Qianqian"https://zbmath.org/authors/?q=ai:ding.qianqianSummary: This paper considers charge-conservative finite element approximation and three coupled iterations of stationary inductionless magnetohydrodynamics equations in Lipschitz domain. Using a mixed finite element method, we discretize the hydrodynamic unknowns by stable velocity-pressure finite element pairs, discretize the current density and electric potential by \(\boldsymbol{H}(\operatorname{div},\varOmega)\times L^2(\varOmega)\)-comforming finite element pairs. The well-posedness of this formula and the optimal error estimate are provided. In particular, we show that the error estimates for the velocity, the current density and the pressure are independent of the electric potential. With this, we propose three coupled iterative methods: Stokes, Newton and Oseen iterations. Rigorous analysis of convergence and stability for different iterative schemes are provided, in which we improve the stability conditions for both Stokes and Newton iterative method. Numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed methods.Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domainshttps://zbmath.org/1491.651552022-09-13T20:28:31.338867Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Ochiai, Hiroyuki"https://zbmath.org/authors/?q=ai:ochiai.hiroyuki"Tanaka, Yoshitaro"https://zbmath.org/authors/?q=ai:tanaka.yoshitaroSummary: The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, \(N^2a^N\) order, where \(a\) is a positive constant less than one and \(N\) is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points \(N\).A new patch up technique for elliptic partial differential equation with irregularitieshttps://zbmath.org/1491.651562022-09-13T20:28:31.338867Z"Singh, Swarn"https://zbmath.org/authors/?q=ai:singh.swarn"Singh, Suruchi"https://zbmath.org/authors/?q=ai:singh.suruchi"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilinSummary: This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution \(u\) and the derivatives \(u_x,u_{xx}\) and the mixed derivative \(u_{xy}\). The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme.Convergence analysis of the continuous and discrete non-overlapping double sweep domain decomposition method based on PMLs for the Helmholtz equationhttps://zbmath.org/1491.651572022-09-13T20:28:31.338867Z"Kim, Seungil"https://zbmath.org/authors/?q=ai:kim.seungil"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.2|zhang.hui.8|zhang.hui.1|zhang.hui.7|zhang.hui.9|zhang.hui.5|zhang.hui.3|zhang.hui.11|zhang.hui.10|zhang.hui.6|zhang.hui.4|zhang.huiSummary: In this paper we will analyze the convergence of the non-overlapping double sweep domain decomposition method (DDM) with transmission conditions based on PMLs for the Helmholtz equation. The main goal is to establish the convergence of the double sweep DDM of both the continuous level problem and the corresponding finite element problem. We show that the double sweep process can be viewed as a contraction mapping of boundary data used for local subdomain problems not only in the continuous level and but also in the discrete level. It turns out that the contraction factor of the contraction mapping of the continuous level problem is given by an exponentially small factor determined by PML strength and PML width, whereas the counterpart of the discrete level problem is governed by the dominant term between the contraction factor similar to that of the continuous level problem and the maximal discrete reflection coefficient resulting from fast decaying evanescent modes. Based on this analysis we prove the convergence of approximate solutions in the \(H^1\)-norm. We also analyze how the discrete double sweep DDM depends on the number of subdomains and the PML parameters as the finite element discretization resolves sufficiently the Helmholtz and PML equations. Our theoretical results suggest that the contraction factor for the propagating modes depends linearly on the number of subdomains. To ensure the convergence, it is sufficient to have the PML width growing logarithmically with the number of subdomains. In the end, numerical experiments illustrating the convergence will be presented as well.A mesh-free method using piecewise deep neural network for elliptic interface problemshttps://zbmath.org/1491.651582022-09-13T20:28:31.338867Z"He, Cuiyu"https://zbmath.org/authors/?q=ai:he.cuiyu"Hu, Xiaozhe"https://zbmath.org/authors/?q=ai:hu.xiaozhe"Mu, Lin"https://zbmath.org/authors/?q=ai:mu.linThe aim in this paper is to investigate the use of deep learning methods to solve interface problems. One focuses on the second-order elliptic interface problem and the research follows the following scheme: the interface problem is reformulated by means of a least-square (LS) approach as a minimization problem and two deep neural network structures (DNN) are built to approximate the solution on sub-domains. The following second-order scalar elliptic interface problem is considered:
\[
-\nabla.(\beta(\mathcal{X})\nabla u)=f,\text{ in }\Omega_1\cup\Omega_2,
\]
\[
[[u]]=g_i\text{ on }\Gamma,
\]
\[
[[\beta(\mathcal{X})\nabla u.\mathcal N]]=g_f\text{ on }\Gamma,
\]
\[
u=g_D,\text{ on }\partial\Omega
\]
where the interface \(\Gamma\) is closed and divides domain \(\Omega\) in two disjoint sub-domains \(\Omega _1\) (the interior) and \(\Omega _2\) (the exterior). One defines the (LS) functional incorporating the interface as well as boundary conditions \(\mathcal {J}\)(v;\(g_j\),\(g_f\),\(g_D\),f) and one looks for its minimum for v\(\in\) \(H^1\)(\(\Omega\)). The construction of the two neural networks to approximate the solution on both sub-domains \(\Omega_1\) and \(\Omega _2\) is discussed in the third section of the article. One develops an adaptive deep LS algorithm. Results on numerical experiments as for instance on Sunflower Shape Interface, Sphere Shape Interface, Heart Shape Interface, Circle Interface with High Contrast Coefficients, Flower Shape Interface are presented in the fourth section of the paper.
Reviewer: Claudia Simionescu-Badea (Wien)Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equationhttps://zbmath.org/1491.651592022-09-13T20:28:31.338867Z"Cai, Jiaxiang"https://zbmath.org/authors/?q=ai:cai.jiaxiang"Chen, Juan"https://zbmath.org/authors/?q=ai:chen.juan"Chen, Min"https://zbmath.org/authors/?q=ai:chen.min.2|chen.min.1|chen.min|chen.min.3Summary: A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.An integral equation formulation of the \(N\)-body dielectric spheres problem. I: Numerical analysishttps://zbmath.org/1491.651702022-09-13T20:28:31.338867Z"Hassan, Muhammad"https://zbmath.org/authors/?q=ai:hassan.muhammad"Stamm, Benjamin"https://zbmath.org/authors/?q=ai:stamm.benjaminSummary: In this article, we analyse an integral equation of the second kind that represents the solution of \(N\) interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants -- and thus the approximation error -- with respect to the number \(N\) of involved dielectric spheres. We develop a new \textit{a priori} error analysis that demonstrates \(N\)-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of \(N\).
For Part II, see [\textit{B. Bramas} et al., ESAIM, Math. Model. Numer. Anal. 55, 625--651 (2021; Zbl 1491.65165)].Calculation of the gradient of Tikhonov's functional in solving coefficient inverse problems for linear partial differential equationshttps://zbmath.org/1491.651722022-09-13T20:28:31.338867Z"Leonov, Alexander S."https://zbmath.org/authors/?q=ai:leonov.alexander-s"Sharov, Alexander N."https://zbmath.org/authors/?q=ai:sharov.alexander-n"Yagola, Anatoly G."https://zbmath.org/authors/?q=ai:yagola.anatolii-grigorevichSummary: A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young's modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.Group analysis of the equations of ideal plasticityhttps://zbmath.org/1491.740112022-09-13T20:28:31.338867Z"Senashov, T. I."https://zbmath.org/authors/?q=ai:senashov.t-i"Gomonova, O. V."https://zbmath.org/authors/?q=ai:gomonova.olga-v"Cherepanova, O. N."https://zbmath.org/authors/?q=ai:cherepanova.olga-nSummary: This paper deals with the problem of constructing exact solutions of the von Mises three-dimensional equations of plasticity based on the group of continuous transformations admitted by the system (B. D. Annin's problem). New classes of solutions of the three-dimensional equations of plasticity are given. The problem of compression of an elastoplastic material layer by rigid plates is solved. In this case, the material obeys the exponential plasticity condition proposed by \textit{B. D. Annin} [``One plane elastoplastic problem with an exponential yield condition'' (Russian), Inzh. Zhurn. Mekh. Tverd. Tela 3, 122--123 (1966)].Structural acoustic problem and dynamic nonlinear plate equationshttps://zbmath.org/1491.740212022-09-13T20:28:31.338867Z"Oudaani, Jaouad"https://zbmath.org/authors/?q=ai:oudaani.jaouad"Raïssouli, Mustapha"https://zbmath.org/authors/?q=ai:raissouli.mustapha"El Mouatasim, Abdelkrim"https://zbmath.org/authors/?q=ai:el-mouatasim.abdelkrimSummary: The purpose of this paper is to investigate a structural interaction model coupled with dynamic von Karman equations, without neither rotational inertia nor clamped boundary conditions. Our fundamental goal is to establish the existence and the uniqueness of the weak solution for the so-called global functional energy. The stability is also discussed. At the end, a numerical study based on the finite difference method is presented as well.On the existence of rotationally symmetric solution of a constrained minimization problem of elasticityhttps://zbmath.org/1491.740322022-09-13T20:28:31.338867Z"Aguiar, Adair R."https://zbmath.org/authors/?q=ai:aguiar.adair-roberto"Rocha, Lucas A."https://zbmath.org/authors/?q=ai:rocha.lucas-aIn this paper the authors consider the equilibrium problem, with no body force, of a cylindrically orthotropic disk subject to a prescribed displacement along its boundary. As the classic theory of elasticity allows for solutions with material overlapping, which is not physical, one way to prevent this anomalous behavior is to consider the minimization of the total potential energy of classical linear elasticity subject to the local injectivity constraint. In this context bifurcation is expected to occur from a radially symmetric solution to a secondary one. The authors present analytical and computational results indicating that this secondary solution is rotationally symmetric.
The problem of minimization of the total potential energy of classical linear elasticity is defined by
\begin{align*}
\min_{ u\in A_\varepsilon } \mathcal{E}(u),\qquad
\mathcal{E}(u):= \frac{1}{2}\int_{ \mathcal{B} } \mathbf{T} \cdot \mathbf{E}d X
-\int_{ \mathcal{B} } \mathbf{b} \cdot \mathbf{u}d X-
\int_{ \partial_2\mathcal{B} } \bar{ \mathbf{t} } \cdot \mathbf{u}d X.
\end{align*}
Here \(\mathcal{B}\subset\mathbb{R}^2\) is the undistorted natural reference configuration of a linearly elastic solid, \(\mathbf{u}\) is the displacement of \(X\), \(\bar{ \mathbf{t} }\) is a prescribed dead load traction field, and the stress tensor \(\mathbf{T}\) is related to \(\mathbf{E}\) by the generalized Hooke's law \[\mathbf{T} = \mathbb{C}[\mathbf{E}],\]
with \(\mathbb{C}\) being the stress tensor and \(\mathbf{E}\) being the infinitesimal strain tensor defined
by
\[\mathbf{E} := \frac{1}{2} ( \nabla u+ \nabla u^T ). \]
Finally,
\[ A_\varepsilon :=\{ u\in H^1( \mathcal{B} )\longrightarrow \mathbb{R}^2: \text{det}(1+\nabla u)\ge \varepsilon>0,
u=\tilde{u} \text{ on } \partial_1\mathcal{B} \} \]
is the set of kinematically admissible displacements.
The authors' approach mainly consists of three steps:
\begin{itemize}
\item[1.] First, assuming that the solution is rotationally symmetric, they solve the Euler-Lagrange equations of the corresponding minimization problem in the region where the local injectivity constraint is not active, and obtain a solution that depends on constants of integration. In the region where the constraint is active, they then find a nonlinear
relation between the radial and tangential displacements, which contains a constant of
integration that is also determined numerically.
\item[2.] Second, still assuming rotational symmetry, the authors use an interior penalty formulation together with a standard finite element method to obtain sequences of numerical solutions that converge to a limit function that is in very good agreement with analytical results in the non active region.
\item[3.] Finally, the authors investigate numerically the influence of both the shear modulus and the boundary condition on the existence of the rotationally symmetric solution. For a given mesh, there are thresholds outside which this solution is not possible.
\end{itemize}
Reviewer: Xin Yang Lu (Thunder Bay)Equations of motion for cracked beams and shallow archeshttps://zbmath.org/1491.740642022-09-13T20:28:31.338867Z"Gutman, Semion"https://zbmath.org/authors/?q=ai:gutman.semion"Ha, Junhong"https://zbmath.org/authors/?q=ai:ha.junhong"Shon, Sudeok"https://zbmath.org/authors/?q=ai:shon.sudeokSummary: Cracks in beams and shallow arches are modeled by massless rotational springs. First, we introduce a specially designed linear operator that ``absorbs'' the boundary conditions at the cracks. Then the equations of motion are derived from the first principles using the Extended Hamilton's Principle, accounting for non-conservative forces. The variational formulation of the equations is stated in terms of the subdifferentials of the bending and axial potential energies. The equations are given in their abstract (weak), as well as in classical forms.About well-posedness and lack of exponential stability of Shear beam modelshttps://zbmath.org/1491.740652022-09-13T20:28:31.338867Z"Ramos, A. J. A."https://zbmath.org/authors/?q=ai:ramos.anderson-j-a"Almeida Júnior, D. S."https://zbmath.org/authors/?q=ai:almeida.dilberto-s-jun|almeida-junior.dilberto-da-silva"Freitas, M. M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Veras, L. S."https://zbmath.org/authors/?q=ai:veras.l-sSummary: In this paper, we consider the Shear beam model (no rotary inertia) and we stablished a decay result of the total energy of solutions by taking a feedback law acting on angle rotation. Unlike the dissipative Shear beam model with damping effect acting on vertical displacement, where the exponential decay holds irrespective any relationship between coefficients of the system, here we prove that system is non-exponential stability by using semigroup techniques. Also, the well-posedness is achieved by using semigroup theory.A differential variational inequality in the study of contact problems with wearhttps://zbmath.org/1491.740752022-09-13T20:28:31.338867Z"Chen, Tao"https://zbmath.org/authors/?q=ai:chen.tao|chen.tao.1"Huang, Nan-Jing"https://zbmath.org/authors/?q=ai:huang.nan-jing"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mircea|sofonea.mircea-tSummary: We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.Variational analysis of unilateral contact problem for thermo-piezoelectric materials with frictionhttps://zbmath.org/1491.740762022-09-13T20:28:31.338867Z"Hachlaf, A."https://zbmath.org/authors/?q=ai:hachlaf.abdelhadi"Benaissa, H."https://zbmath.org/authors/?q=ai:benaissa.hicham"Benkhira, EL-H."https://zbmath.org/authors/?q=ai:benkhira.el-hassan"Fakhar, R."https://zbmath.org/authors/?q=ai:fakhar.rachidSummary: This paper deals with the mathematical analysis of quasi-static unilateral contact problem with friction between a thermo-piezoelectric body and a conductive foundation. The material is assumed to have thermo-electro-elastic behavior and the contact is modeled by the Signorini's law, the condition of dry friction and a regularized electrical conductivity condition. The effects of frictional heating and thermal conductivity on the mechanisms of material are taken into account. To prove the existence of a weak solution to the problem, an incremental formulation obtained by using an implicit time scheme is studied. Several estimates on the incremental solutions are given, which allow us to pass to the limit by using compactness results.Modelling of frictionless Signorini problem for a linear elastic membrane shellhttps://zbmath.org/1491.740772022-09-13T20:28:31.338867Z"Mezabia, M. E."https://zbmath.org/authors/?q=ai:mezabia.m-e"Chacha, D. A."https://zbmath.org/authors/?q=ai:chacha.djamel-ahmed|chacha.djamal-ahmed"Bensayah, A."https://zbmath.org/authors/?q=ai:bensayah.abdallahSummary: The purpose of this paper is the study of the asymptotic modeling of a linearly thin elastic isotropic membrane shell which is subjected to frictionless contact with a rigid obstacle on part of the boundary. Using the formal asymptotic expansions method, we obtain a two dimensional Signorini membrane shell model, then we justify the result by convergence theorem.Sweeping process arguments in the analysis and control of a contact problemhttps://zbmath.org/1491.740792022-09-13T20:28:31.338867Z"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mircea"Xiao, Yi-Bin"https://zbmath.org/authors/?q=ai:xiao.yibinSummary: This chapter presents some notation and preliminary material which are needed. It describes the mechanical assumptions they consider and the resulting mathematical model of contact and also presents a variational formulation of the problem and prove an existence and uniqueness result. The chapter provides three examples which differ by the choice of the control and cost function, together with the corresponding mechanical interpretation. The existence of the optimal pairs for the optimal control we consider is based on a continuous dependence result which has some interest in its own. The literature concerning optimal control problems in the study of mathematical models of contact is quite limited. The reason is the strong nonlinearities which arise in the boundary conditions involved in such models. Contact phenomena between deformable bodies arise in industry and everyday life.
For the entire collection see [Zbl 1483.93002].Energetic solutions for the coupling of associative plasticity with damage in geomaterialshttps://zbmath.org/1491.740912022-09-13T20:28:31.338867Z"Crismale, Vito"https://zbmath.org/authors/?q=ai:crismale.vitoSummary: We prove existence of globally stable quasistatic evolutions, referred to as energetic solutions, for a model proposed by \textit{J.-J. Marigo} and \textit{K. Kazymyrenko} in [``A micromechanical inspired model for the coupled to damage elasto-plastic behavior of geomaterials under compression'', Mech. Indust. 20, No. 1, Article No. 105, 16 p. (2019; \url{doi:10.1051/meca/2018043 })]. The behaviour of geomaterials under compression is studied through the coupling of Drucker-Prager plasticity model with a damage term tuning kinematical hardening. This provides a new approach to the modelling of geomaterials, for which non associative plasticity is usually employed. The kinematical hardening is null where the damage is complete, so there the behaviour is perfectly plastic. We analyse the model combining tools from the theory of capacity and from the treatment of linearly elastic materials with cracks.Global well posedness for a Q-tensor model of nematic liquid crystalshttps://zbmath.org/1491.760062022-09-13T20:28:31.338867Z"Murata, Miho"https://zbmath.org/authors/?q=ai:murata.miho"Shibata, Yoshihiro"https://zbmath.org/authors/?q=ai:shibata.yoshihiroSummary: In this paper, we prove the global well posedness and the decay estimates for a \(\mathbb{Q}\)-tensor model of nematic liquid crystals in \(\mathbb{R}^N\), \(N \ge 3\). This system is a coupled system by the Navier-Stokes equations with a parabolic-type equation describing the evolution of the director fields \(\mathbb{Q}\). The proof is based on the maximal \(L_p\)-\(L_q\) regularity and the \(L_p\)-\(L_q\) decay estimates to the linearized problem.Wave breaking in undular bores with shear flowshttps://zbmath.org/1491.760122022-09-13T20:28:31.338867Z"Bjørnestad, Maria"https://zbmath.org/authors/?q=ai:bjornestad.maria"Kalisch, Henrik"https://zbmath.org/authors/?q=ai:kalisch.henrik"Abid, Malek"https://zbmath.org/authors/?q=ai:abid.malek"Kharif, Christian"https://zbmath.org/authors/?q=ai:kharif.christian"Brun, Mats"https://zbmath.org/authors/?q=ai:brun.mats-kirkesaetherSummary: It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of \textit{H. Favre} [Ondes de translation. Paris: Dunod (1935)] that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg-de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.Vortex collapses for the Euler and quasi-geostrophic modelshttps://zbmath.org/1491.760182022-09-13T20:28:31.338867Z"Godard-Cadillac, Ludovic"https://zbmath.org/authors/?q=ai:godard-cadillac.ludovicSummary: This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by \textit{C. Marchioro} and \textit{M. Pulvirenti} [Mathematical theory of incompressible nonviscous fluids. New York, NY: Springer-Verlag (1994; Zbl 0789.76002)] concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.\(L^p\)-strong solution for the stationary exterior Stokes equations with Navier boundary conditionhttps://zbmath.org/1491.760212022-09-13T20:28:31.338867Z"Dhifaoui, Anis"https://zbmath.org/authors/?q=ai:dhifaoui.anisLet \(\Omega \subset \mathbb{R}^3\) be an unbounded domain with compact boundary of class \(C^{2,1}\) such that \(\mathbb{R}^3\setminus \overline \Omega \) is connected. The paper studies the Stokes system with Navier boundary condition \( -\Delta u+\nabla p=f\), \( \nabla \cdot u=0 \) in \( \Omega \), \( u_n=g\), \( [T(u,p)n^\Omega +\alpha u]_\tau = h \) on \( \partial \Omega \). A solution \( (u,p)\) is from the weighted Sobolev spaces \( W^{2,q}_{k+1}(\Omega )\times W^{1,q}_{k+1}(\Omega )\).
Reviewer: Dagmar Medková (Praha)Breathers, cascading instabilities and Fermi-Pasta-Ulam-Tsingou recurrence of the derivative nonlinear Schrödinger equation: effects of `self-steepening' nonlinearityhttps://zbmath.org/1491.760342022-09-13T20:28:31.338867Z"Yin, H. M."https://zbmath.org/authors/?q=ai:yin.khin-m|yin.huiming|yin.hui-min"Chow, K. W."https://zbmath.org/authors/?q=ai:chow.kwok-wing|chow.ka-wing|chow.kong-wingSummary: Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrödinger equation, which incorporates second order dispersion, cubic nonlinearity and self-steepening effect. By insisting on periodic boundary conditions, a cascading process will occur where initially small higher order Fourier modes can grow alongside with lower order modes. Typically a breather is first observed when all modes attain roughly the same order of magnitude. Beyond the formation of the first breather, analytical formula of spatially periodic but temporally localized breather ceases to be a valid indicator. However, numerical simulations display Fermi-Pasta-Ulam-Tsingou type recurrence. Self-steepening effect plays a crucial role in the dynamics, as it induces motion of the breather and generates chaotic behavior of the Fourier coefficients. Theoretically, correlation between breather motion and the Lax pair formulation is made. Physically, quantitative assessments of wave profile evolution are made for different initial conditions, e.g. random noise versus modulation instability mode of maximum growth rate. Potential application to fluid mechanics is discussed.Theoretical and numerical analysis of a simple model derived from compressible turbulencehttps://zbmath.org/1491.760372022-09-13T20:28:31.338867Z"Gavrilyuk, Sergey"https://zbmath.org/authors/?q=ai:gavrilyuk.sergey-l"Hérard, Jean-Marc"https://zbmath.org/authors/?q=ai:herard.jean-marc"Hurisse, Olivier"https://zbmath.org/authors/?q=ai:hurisse.olivier"Toufaili, Ali"https://zbmath.org/authors/?q=ai:toufaili.aliSummary: Turbulent compressible flows are encountered in many industrial applications, for instance when dealing with combustion or aerodynamics. This paper is dedicated to the study of a simple turbulent model for compressible flows. It is based on the Euler system with an energy equation and turbulence is accounted for with the help of an algebraic closure that impacts the thermodynamical behavior. Thereby, no additional PDE is introduced in the Euler system. First, a detailed study of the model is proposed: hyperbolicity, structure of the waves, nature of the fields, existence and uniqueness of the Riemann problem. Then, numerical simulations are proposed on the basis of existing finite-volume schemes. These simulations allow to perform verification test cases and more realistic explosion-like test cases with regards to the turbulence level.Maximal regularity for compressible two-fluid systemhttps://zbmath.org/1491.760592022-09-13T20:28:31.338867Z"Piasecki, Tomasz"https://zbmath.org/authors/?q=ai:piasecki.tomasz"Zatorska, Ewelina"https://zbmath.org/authors/?q=ai:zatorska.ewelinaThe authors studied a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure. They showed that regular solutions in a \(L^p-L^q\) maximal regularity setting exist both locally and globally in time, under additional smallness assumptions on the initial data. The interesting proof relies on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated linear problem.
Reviewer: Teng Wang (Beijing)Global strong solution to the Cauchy problem of 1D viscous two-fluid model without any domination conditionhttps://zbmath.org/1491.760602022-09-13T20:28:31.338867Z"Gao, Xiaona"https://zbmath.org/authors/?q=ai:gao.xiaona"Guo, Zhenhua"https://zbmath.org/authors/?q=ai:guo.zhenhua"Li, Zilai"https://zbmath.org/authors/?q=ai:li.zilaiSummary: In this paper, we consider the Cauchy problem to the compressible two-fluid Navier-Stokes equations in one-dimensional space allowing vacuum. It is shown that the compressible two-fluid Navier-Stokes equations admit global strong solution with the large initial value and no the domination condition1 which was posed in [\textit{A. Vasseur} et al., J. Math. Pures Appl. (9) 125, 247--282 (2019; Zbl 1450.76033)], when the initial vacuum can be permitted inside the region. Note that this result is proved without any smallness conditions on the initial value.Homogenization of the evolutionary compressible Navier-Stokes-Fourier system in domains with tiny holeshttps://zbmath.org/1491.760612022-09-13T20:28:31.338867Z"Pokorný, Milan"https://zbmath.org/authors/?q=ai:pokorny.milan"Skříšovský, Emil"https://zbmath.org/authors/?q=ai:skrisovsky.emilSummary: We study the homogenization of the evolutionary compressible Navier-Stokes-Fourier system in a bounded three-dimensional domain perforated with a large number of very tiny holes. We show that under suitable assumptions on the smallness and distribution of the holes the limit system remains the same in the unperforated domain. One of the main novelty in the paper consists in the treatment of the entropy inequality and thus the paper also improves the related result in the steady case from \textit{Y. Lu} and \textit{M. Pokorný} [J. Differ. Equations 278, 463--492 (2021; Zbl 1458.35043)].Accuracy of a low Mach number model for time-harmonic acousticshttps://zbmath.org/1491.760642022-09-13T20:28:31.338867Z"Mercier, J.-F."https://zbmath.org/authors/?q=ai:mercier.jean-francoisReacting multi-component fluids: regular solutions in Lorentz spaceshttps://zbmath.org/1491.760912022-09-13T20:28:31.338867Z"Mucha, Piotr Bogusław"https://zbmath.org/authors/?q=ai:mucha.piotr-boguslaw"Piasecki, Tomasz"https://zbmath.org/authors/?q=ai:piasecki.tomaszSummary: The paper deals with the analysis of a model of a multi-component fluid admitting chemical reactions. The flow is considered in the incompressible regime. The main result shows the global existence of regular solutions under the assumption of suitable smallness conditions. In order to control the solutions a special structure condition on the derivatives of chemical production functions determining the reactions is required. The existence is shown in a new critical functional framework of Lorentz spaces of type \(L_{p, r}(0, T; L_q)\), which allows to control the integral \(\int_0^\infty \|\nabla u(t)\|_\infty dt\).Asymptotic shallow models arising in magnetohydrodynamicshttps://zbmath.org/1491.760922022-09-13T20:28:31.338867Z"Alonso-Orán, Diego"https://zbmath.org/authors/?q=ai:alonso-oran.diegoSummary: In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in [\textit{A. Castro} and \textit{D. Lannes}, J. Fluid Mech. 759, 642--675 (2014; Zbl 1446.76077)] to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model.Two-dimensional resistive-wall impedance with finite thickness: its mathematical structures and their physical meaningshttps://zbmath.org/1491.780042022-09-13T20:28:31.338867Z"Shobuda, Yoshihiro"https://zbmath.org/authors/?q=ai:shobuda.yoshihiroSummary: When the skin depth is greater than the chamber thickness for relativistic beams, the two-dimensional longitudinal resistive-wall impedance of a cylindrical chamber with a finite thickness decreases proportionally to the frequency. The phenomenon is commonly interpreted as electromagnetic fields leaking out of the chamber over a frequency range. However, the relationship between the wall current on the chamber and the leakage fields from the chamber is unclear because the naive resistive-wall impedance formula does not dynamically express how the wall current converts to the leakage fields when the skin depth exceeds the chamber thickness. A prestigious textbook[1] re-expressed the resistive-wall impedance via a parallel circuit model with the resistive-wall and inductive terms to provide a dynamic picture of the phenomenon. However, there are some flaws in the formula. This study highlights them from a fundamental standpoint, and provides a more appropriate and rigorous picture of the longitudinal resistive-wall impedance with finite thickness. To demonstrate their physical meaning, we re-express the longitudinal impedance for non-relativistic beams, as well as the transverse resistive-wall impedance including space charge impedance based on a parallel circuit model.Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying indexhttps://zbmath.org/1491.780082022-09-13T20:28:31.338867Z"Balac, Stéphane"https://zbmath.org/authors/?q=ai:balac.stephane"Dauge, Monique"https://zbmath.org/authors/?q=ai:dauge.monique"Moitier, Zoïs"https://zbmath.org/authors/?q=ai:moitier.zoisSummary: Whispering gallery modes [WGM] are resonant modes displaying special features: they concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider 2D Helmholtz equations governing transverse electric or magnetic modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency \(m\to\infty\) for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy, we highlight three typical behaviors in such optical micro-disks, depending on the sign of an `effective curvature' that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual \(\mathscr{O}(m^{-\infty})\). We show using the black box scattering approach that quasi-resonances are \(\mathscr{O}(m^{-\infty})\) close to true resonances.Hybridization of the rigorous coupled-wave approach with transformation optics for electromagnetic scattering by a surface-relief gratinghttps://zbmath.org/1491.780092022-09-13T20:28:31.338867Z"Civiletti, B. J."https://zbmath.org/authors/?q=ai:civiletti.benjamin-j"Lakhtakia, A."https://zbmath.org/authors/?q=ai:lakhtakia.akhlesh"Monk, P. B."https://zbmath.org/authors/?q=ai:monk.peter-bThe authors combine transformation optics with the rigorous coupled-wave approach, in view to study the time-harmonic Maxwell equations in a spatial domain that contains a grating, being invariant in one dimension (and so that the chosen constitutive properties allow the reduction of the full Maxwell system to a 2D Helmholtz equation for each linear polarization state). The existence of solution to the original scattering problem is obtained. A convergence analysis was included for a discretized form of the transformed problem (with respect to two different parameters), and the uniqueness of solution of this discretized problem is obtained. A numerical example was presented as a test of the convergence theory, allowing also some comparison with other known methods.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Floquet engineering of electric polarization with two-frequency drivehttps://zbmath.org/1491.780132022-09-13T20:28:31.338867Z"Ikeda, Yuya"https://zbmath.org/authors/?q=ai:ikeda.yuya"Kitamura, Sota"https://zbmath.org/authors/?q=ai:kitamura.sota"Morimoto, Takahiro"https://zbmath.org/authors/?q=ai:morimoto.takahiroSummary: Electric polarization is a geometric phenomenon in solids and has a close relationship to the symmetry of the system. Here we propose a mechanism to dynamically induce and manipulate electric polarization by using an external light field. Specifically, we show that application of bicircular lights controls the rotational symmetry of the system and can generate electric polarization. To this end, we use Floquet theory to study a system subjected to a two-frequency drive. We derive an effective Hamiltonian with high-frequency expansions, for which the electric polarization is computed with the Berry phase formula. We demonstrate the dynamical control of polarization for a one-dimensional Su-Shrieffer-Heeger chain, a square lattice model, and a honeycomb lattice model.Nonlinear propagation of coupled surface TE and leaky TM electromagnetic waveshttps://zbmath.org/1491.780142022-09-13T20:28:31.338867Z"Smirnov, Yury"https://zbmath.org/authors/?q=ai:smirnov.yury-g"Smolkin, Eugene"https://zbmath.org/authors/?q=ai:smolkin.eugeneSummary: Propagation of the coupled electromagnetic wave, which is a superposition of TE surface and TM leaky waves, in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Nonlinear coupled TE-TM wave is characterised by two (independent) frequencies and two (coupled) propagation constants (propagation constants). The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell's equations. Existence of coupled TE-TM waves is proved. Intervals of localisation of propagation constants are found.Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiberhttps://zbmath.org/1491.780152022-09-13T20:28:31.338867Z"Yang, Dan-Yu"https://zbmath.org/authors/?q=ai:yang.danyu"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Qu, Qi-Xing"https://zbmath.org/authors/?q=ai:qu.qixing"Zhang, Chen-Rong"https://zbmath.org/authors/?q=ai:zhang.chen-rong"Chen, Su-Su"https://zbmath.org/authors/?q=ai:chen.su-su"Wei, Cheng-Cheng"https://zbmath.org/authors/?q=ai:wei.cheng-chengSummary: Optical fiber communication plays an important role in the modern communication. In this paper, we investigate a variable-coefficient coupled Hirota system which describes the vector optical pulses in an inhomogeneous optical fiber. With respect to the complex wave envelopes, we construct a Lax pair and a Darboux transformation, both different from the existing ones. Infinitely-many conservation laws are derived based on our Lax pair. We obtain the one/two-fold bright-bright soliton solutions, one/two-fold bright-dark soliton solutions and one/two-fold breather solutions via our Darboux transformation. When \(\alpha(t)\), \(\beta(t)\) and \(\delta(t)\) are the trigonometric functions, we present the bright-bright soliton, bright-dark soliton and breather which are all periodic along the propagation direction, where \(\alpha(t)\), \(\beta(t)\) and \(\delta(t)\) represent the group velocity dispersion, third-order dispersion and nonlinear terms of the self-phase modulation and cross-phase modulation. Interactions between the two bright-bright soliton, two bright-dark solitons and two breathers are presented. Bound state of the two bright-bright solitons is formed. Widths and velocities of the two bright-bright solitons do not change but their amplitudes change after their interaction via the asymptotic analysis. Periods of the bright-dark solitons decrease when the periods of the trigonometric \(\alpha(t)\), \(\beta(t)\) and \(\delta(t)\) decrease.A port-Hamiltonian formulation of coupled heat transferhttps://zbmath.org/1491.800042022-09-13T20:28:31.338867Z"Jäschke, Jens"https://zbmath.org/authors/?q=ai:jaschke.jens"Ehrhardt, Matthias"https://zbmath.org/authors/?q=ai:ehrhardt.matthias-joachim|ehrhardt.matthias"Günther, Michael"https://zbmath.org/authors/?q=ai:gunther.michael"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgitThe authors apply a generalized port-Hamiltonian formalism to a conjugate heat transfer problem in gas turbine blades. Introducing some simplifying hypotheses and a rescaling process, they end with the system: \(\frac{\partial \vartheta _{m}}{\partial t}=\frac{k}{c_{m}l_{m}^{2}}\frac{\partial ^{2}\vartheta _{m}}{\partial \xi _{m}^{2}}\), \(0<\xi _{m}<1\), \(\frac{\partial \vartheta _{in}}{\partial t}=-\frac{\nu }{l_{in}}\frac{\partial ^{2}\vartheta _{in}}{\partial \xi _{in}^{2}}\), \(0<\xi _{in}<1\), \(\frac{ \partial \vartheta _{out}}{\partial t}=-\frac{\nu }{l_{out}}\frac{\partial ^{2}\vartheta _{out}}{\partial \xi _{out}^{2}}\), \(0<\xi _{out}<1\). The following boundary conditions are added: \(-\frac{k}{l_{m}}\frac{\partial \vartheta _{m}}{\partial \xi _{m}}(0,t)=h_{a}(T_{ext}(t)-\vartheta _{m}(0,t)) \), \(-\frac{k}{l_{m}}\frac{\partial \vartheta _{m}}{\partial \xi _{m}} (1,t)=h_{b}(\vartheta _{m}(1,t)-\vartheta _{in}(1,t))\), \(\vartheta _{in}(0,t)=T_{\inf low}(t)\), \(c_{c}\nu (\vartheta _{out}(0,t)-\vartheta _{in}(1,t))=h_{b}(\vartheta _{m}(1,t)-\vartheta _{in}(1,t))\), for different constants or known quantities. The authors introduce a generalized finite-dimensional port-Hamiltonian framework for distributed parameter systems. They first recall the notions of Dirac and Stokes-Dirac structures. Let \(P_{1}\in \mathbb{R}^{n\times n}\) be invertible and selfadjoint, \(P_{0}\in \mathbb{R}^{n\times n}\) be skew-adjoint, \(\mathcal{H} \in \mathbb{R}^{n\times n}\) be symmetric such that \(mI<\mathcal{H}<MI\) with constants \(m,M>0\), and \(X=L^{2}([a,b],\mathcal{R}^{n})\) be a Hilbert space. Then the differential equation \(\frac{\partial \Theta }{\partial t}(\xi ,t)=P_{1}\frac{\partial }{\partial \xi }(\mathcal{H}(\xi )\Theta (\xi ,t))+P_{0}\frac{\partial }{\partial \xi }(\mathcal{H}(\xi )\Theta (\xi ,t))\) is a linear first-order port-Hamiltonian system on a one-dimensional space with the associated Hamiltonian \(H(t)=\frac{1}{2}\int_{a}^{b}\Theta (\xi ,t)^{T}\mathcal{H}(\xi )\Theta (\xi ,t)d\xi \). In the present context, the authors introduce the Hamiltonian \(H_{m}=\frac{1}{2c_{m}}\int_{0}^{1} \vartheta _{m}(\xi ,t)^{2}d\xi \). They finally rewrite the boundary conditions and the transport equations within this framework.
Reviewer: Alain Brillard (Riedisheim)Interplay of quantum mechanics and nonlinearity. Understanding small-system dynamics of the discrete nonlinear Schrödinger equationhttps://zbmath.org/1491.810042022-09-13T20:28:31.338867Z"Kenkre, V. M. (Nitant)"https://zbmath.org/authors/?q=ai:kenkre.vasudev-mangeshPublisher's description: This book presents an in-depth study of the discrete nonlinear Schrödinger equation (DNLSE), with particular emphasis on spatially small systems that permit analytic solutions. In many quantum systems of contemporary interest, the DNLSE arises as a result of approximate descriptions despite the fundamental linearity of quantum mechanics. Such scenarios, exemplified by polaron physics and Bose-Einstein condensation, provide application areas for the theoretical tools developed in this text. The book begins with an introduction of the DNLSE illustrated with the dimer, development of fundamental analytic tools such as elliptic functions, and the resulting insights into experiment that they allow. Subsequently, the interplay of the initial quantum phase with nonlinearity is studied, leading to novel phenomena with observable implications in fields such as fluorescence depolarization of stick dimers, followed by analysis of more complex and/or larger systems. Specific examples analyzed in the book include the nondegenerate nonlinear dimer, nonlinear trapping, rotational polarons, and the nonadiabatic nonlinear dimer. Phenomena treated include strong carrier-phonon interactions and Bose-Einstein condensation. This book is aimed at researchers and advanced graduate students, with chapter summaries and problems to test the reader's understanding, along with an extensive bibliography. The book will be essential reading for researchers in condensed matter and low-temperature atomic physics, as well as any scientist who wants fascinating insights into the role of nonlinearity in quantum physics.Two-particle bound states at interfaces and cornershttps://zbmath.org/1491.810182022-09-13T20:28:31.338867Z"Roos, Barbara"https://zbmath.org/authors/?q=ai:roos.barbara"Seiringer, Robert"https://zbmath.org/authors/?q=ai:seiringer.robertSummary: We study two interacting quantum particles forming a bound state in \(d\)-dimensional free space, and constrain the particles in \(k\) directions to \(( 0 , \infty )^k \times \mathbb{R}^{d - k} \), with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from \(k\) to \(k + 1\). This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all \(k\) the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of \textit{S. Egger} et al. [J. Spectr. Theory 10, No. 4, 1413--1444 (2020; Zbl 1469.81021)] to dimensions \(d > 1\).FZZ-triality and large \(\mathcal{N} = 4\) super Liouville theoryhttps://zbmath.org/1491.810252022-09-13T20:28:31.338867Z"Creutzig, Thomas"https://zbmath.org/authors/?q=ai:creutzig.thomas"Hikida, Yasuaki"https://zbmath.org/authors/?q=ai:hikida.yasuakiSummary: We examine dualities of two dimensional conformal field theories by applying the methods developed by the authors. We first derive the duality between \(SL(2|1)_k/(SL(2)_k \otimes U(1))\) coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large \(\mathcal{N} = 4\) super Liouville theory and a coset of the form \(Y(k_1, k_2)/SL(2)_{k_1 + k_2}\), where \(Y(k_1, k_2)\) consists of two \(SL(2|1)_{k_i}\) and free bosons or equivalently two \(U(1)\) cosets of \(D(2, 1; k_i - 1)\) at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden \(SL(2)_{k^\prime}\) in \(SL(2|1)_k\) or \(D(2, 1; k - 1)_1\). The relation of levels is \(k^\prime - 1 = 1/(k-1)\). We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with \(SL(n|1)_k /(SL(n)_k \otimes U(1))\) for arbitrary \(n > 2\).Regularity of Boltzmann equation with Cercignani-Lampis boundary in convex domainhttps://zbmath.org/1491.820092022-09-13T20:28:31.338867Z"Chen, Hongxu"https://zbmath.org/authors/?q=ai:chen.hongxuApproximating the ground state eigenvalue via the effective potentialhttps://zbmath.org/1491.820112022-09-13T20:28:31.338867Z"Chenn, Ilias"https://zbmath.org/authors/?q=ai:chenn.ilias"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.41"Zhang, Shiwen"https://zbmath.org/authors/?q=ai:zhang.shiwenThe Parisi formula is a Hamilton-Jacobi equation in Wasserstein spacehttps://zbmath.org/1491.820122022-09-13T20:28:31.338867Z"Mourrat, Jean-Christophe"https://zbmath.org/authors/?q=ai:mourrat.jean-christopheSummary: The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line.Violation of the second fluctuation-dissipation relation and entropy production in nonequilibrium mediumhttps://zbmath.org/1491.820152022-09-13T20:28:31.338867Z"Tanogami, Tomohiro"https://zbmath.org/authors/?q=ai:tanogami.tomohiroSummary: We investigate a class of nonequilibrium media described by Langevin dynamics that satisfies the local detailed balance. For the effective dynamics of a probe immersed in the medium, we derive an inequality that bounds the violation of the second fluctuation-dissipation relation (FDR). We also discuss the validity of the effective dynamics. In particular, we show that the effective dynamics obtained from nonequilibrium linear response theory is consistent with that obtained from a singular perturbation method. As an example of these results, we propose a simple model for a nonequilibrium medium in which the particles are subjected to potentials that switch stochastically. For this model, we show that the second FDR is recovered in the fast switching limit, although the particles are out of equilibrium.On a class of Fokker-Planck equations with subcritical confinementhttps://zbmath.org/1491.820162022-09-13T20:28:31.338867Z"Toscani, Giuseppe"https://zbmath.org/authors/?q=ai:toscani.giuseppe"Zanella, Mattia"https://zbmath.org/authors/?q=ai:zanella.mattiaSummary: We study the relaxation to equilibrium for a class of linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density \(e(x)\), the diffusion coefficient can be built to have \(e(x)\) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density \(e(x)\), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.Gamma-convergence results for nematic elastomer bilayers: relaxation and actuationhttps://zbmath.org/1491.820212022-09-13T20:28:31.338867Z"Cesana, Pierluigi"https://zbmath.org/authors/?q=ai:cesana.pierluigi"León Baldelli, Andrés A."https://zbmath.org/authors/?q=ai:leon-baldelli.andres-aSummary: We compute effective energies of thin bilayer structures composed of soft nematic elastic liquid crystals in various geometrical regimes and functional configurations. Our focus is on elastic foundations composed of an isotropic layer attached to a nematic substrate where order-strain interaction results in complex opto-mechanical instabilities activated \textit{via} coupling through the common interface. Allowing out-of-plane displacements, we compute Gamma-limits for vanishing thickness which exhibit spontaneous stress relaxation and shape-morphing behaviour. This extends the plane strain modelling of the authors [Math. Models Methods Appl. Sci. 28, No. 14, 2863--2904 (2018; Zbl 1411.49008)], and shows the asymptotic emergence of fully coupled active macroscopic nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy non-convex and non-coercive. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields.Regular solution for the compressible Landau-Lifshitz-Bloch equation in a bounded domain of \(\mathbb{R}^3\)https://zbmath.org/1491.820222022-09-13T20:28:31.338867Z"Ayouch, C."https://zbmath.org/authors/?q=ai:ayouch.chahid"Benmouane, M."https://zbmath.org/authors/?q=ai:benmouane.m"Essoufi, E. H."https://zbmath.org/authors/?q=ai:essoufi.el-hassanSummary: In this paper, we prove a local in time existence of regular solution for the compressible Landau-Lifshitz-Bloch equation in a bounded domain of \(\mathbb{R}^3 \). The uniqueness of the solution is also established.Relativistic and non-Gaussianity contributions to the one-loop power spectrumhttps://zbmath.org/1491.830052022-09-13T20:28:31.338867Z"Martinez-Carrillo, Rebeca"https://zbmath.org/authors/?q=ai:martinez-carrillo.rebeca"De-Santiago, Josue"https://zbmath.org/authors/?q=ai:de-santiago.josue"Hidalgo, Juan Carlos"https://zbmath.org/authors/?q=ai:hidalgo.juan-carlos"Malik, Karim A."https://zbmath.org/authors/?q=ai:malik.karim-a(no abstract)Primordial black holes and gravitational waves from parametric amplification of curvature perturbationshttps://zbmath.org/1491.830092022-09-13T20:28:31.338867Z"Cai, Rong-Gen"https://zbmath.org/authors/?q=ai:cai.ronggen"Guo, Zong-Kuan"https://zbmath.org/authors/?q=ai:guo.zong-kuan"Liu, Jing"https://zbmath.org/authors/?q=ai:liu.jing.1|liu.jing"Liu, Lang"https://zbmath.org/authors/?q=ai:liu.lang"Yang, Xing-Yu"https://zbmath.org/authors/?q=ai:yang.xingyu(no abstract)Generation of primordial black holes and gravitational waves from dilaton-gauge field dynamicshttps://zbmath.org/1491.830112022-09-13T20:28:31.338867Z"Kawasaki, Masahiro"https://zbmath.org/authors/?q=ai:kawasaki.masahiro"Nakatsuka, Hiromasa"https://zbmath.org/authors/?q=ai:nakatsuka.hiromasa"Obata, Ippei"https://zbmath.org/authors/?q=ai:obata.ippei(no abstract)GRAMSES: a new route to general relativistic \(N\)-body simulations in cosmology. II: Initial conditionshttps://zbmath.org/1491.830172022-09-13T20:28:31.338867Z"Barrera-Hinojosa, Cristian"https://zbmath.org/authors/?q=ai:barrera-hinojosa.cristian"Li, Baojiu"https://zbmath.org/authors/?q=ai:li.baojiuExistence of new singularities in Einstein-aether theoryhttps://zbmath.org/1491.830182022-09-13T20:28:31.338867Z"Chan, R."https://zbmath.org/authors/?q=ai:chan.raymond-hon-fu|chan.roy|chan.ringo|chan.roberto|chan.roath|chan.ray"da Silva, M. F. A."https://zbmath.org/authors/?q=ai:da-silva.m-f-a"Satheeshkumar, V. H."https://zbmath.org/authors/?q=ai:satheeshkumar.v-h(no abstract)Parametrising non-linear dark energy perturbationshttps://zbmath.org/1491.830222022-09-13T20:28:31.338867Z"Hassani, Farbod"https://zbmath.org/authors/?q=ai:hassani.farbod"L'Huillier, Benjamin"https://zbmath.org/authors/?q=ai:lhuillier.benjamin"Shafieloo, Arman"https://zbmath.org/authors/?q=ai:shafieloo.arman"Kunz, Martin"https://zbmath.org/authors/?q=ai:kunz.martin"Adamek, Julian"https://zbmath.org/authors/?q=ai:adamek.julian(no abstract)Probing cosmological fields with gravitational wave oscillationshttps://zbmath.org/1491.830232022-09-13T20:28:31.338867Z"Jiménez, Jose Beltrán"https://zbmath.org/authors/?q=ai:jimenez.jose-beltran"Ezquiaga, Jose María"https://zbmath.org/authors/?q=ai:ezquiaga.jose-maria"Heisenberg, Lavinia"https://zbmath.org/authors/?q=ai:heisenberg.lavinia(no abstract)Perturbations in tachyon dark energy and their effect on matter clusteringhttps://zbmath.org/1491.830252022-09-13T20:28:31.338867Z"Singh, Avinash"https://zbmath.org/authors/?q=ai:singh.avinash-c|singh.avinash-k"Jassal, H. K."https://zbmath.org/authors/?q=ai:jassal.h-k"Sharma, Manabendra"https://zbmath.org/authors/?q=ai:sharma.manabendra(no abstract)Consistent Blandford-Znajek expansionhttps://zbmath.org/1491.830272022-09-13T20:28:31.338867Z"Armas, Jay"https://zbmath.org/authors/?q=ai:armas.jay"Cai, Yangyang"https://zbmath.org/authors/?q=ai:cai.yangyang"Compère, Geoffrey"https://zbmath.org/authors/?q=ai:compere.geoffrey"Garfinkle, David"https://zbmath.org/authors/?q=ai:garfinkle.david"Gralla, Samuel E."https://zbmath.org/authors/?q=ai:gralla.samuel-e(no abstract)The effect of nonlinear electrodynamics on Joule-Thomson expansion of a 5-dimensional charged AdS black hole in Einstein-Gauss-Bonnet gravityhttps://zbmath.org/1491.830282022-09-13T20:28:31.338867Z"Assrary, M."https://zbmath.org/authors/?q=ai:assrary.m"Sadeghi, J."https://zbmath.org/authors/?q=ai:sadeghi.jafar|sadeghi.jonathan|sadeghi.javad"Zomorrodian, M. E."https://zbmath.org/authors/?q=ai:zomorrodian.mohammad-ebrahimSummary: We have studied in this paper the Joule-Thomson expansion of a new charged AdS black hole with a nonlinear electrodynamics in framework of Einstein-Gauss-Bonnet gravity in AdS space. We investigated effects of mass (\(m\)), electric charge (\(q\)), GB coupling constant (\(\alpha\)) and nonlinear electrodynamics parameter (\(k\)) on Joule-Thomson expansion by depicting different graphs. The fact that inversion temperature tends to decrease by increasing \(k\), is in contrast to the effect of electric charge. The divergent point as well as the zero point of Joule-Thomson coefficient are also discussed. Results show that, this black hole exhibits a phase transition similar to that of van der Waals system. Furthermore, the isonthalpic curve is obtained and an interesting dependence of these curves on charge and nonlinear electrodynamics parameter is revealed. In \(T\)-\(P\) graphs, the cooling region shrinks with charge, while this region expands both with mass and with nonlinear electrodynamics parameter. Our study shows that nonlinear electrodynamics parameter plays an important role in Joule Thomson expansion.Primordial black holes from a tiny bump/dip in the inflaton potentialhttps://zbmath.org/1491.830352022-09-13T20:28:31.338867Z"Mishra, Swagat S."https://zbmath.org/authors/?q=ai:mishra.swagat-s"Sahni, Varun"https://zbmath.org/authors/?q=ai:sahni.varun(no abstract)Generalized uncertainty principle effects in the Hořava-Lifshitz quantum theory of gravityhttps://zbmath.org/1491.830372022-09-13T20:28:31.338867Z"García-Compeán, H."https://zbmath.org/authors/?q=ai:garcia-compean.hugo|garcia-compean.hector"Mata-Pacheco, D."https://zbmath.org/authors/?q=ai:mata-pacheco.dSummary: The Wheeler-DeWitt equation for a Kantowski-Sachs metric in Hořava-Lifshitz gravity with a set of coordinates in minisuperspace that obey a generalized uncertainty principle is studied. We first study the equation coming from a set of coordinates that obey the usual uncertainty principle and find analytic solutions in the infrared as well as a particular ultraviolet limit that allows us to find the solution found in Hořava-Lifshitz gravity with projectability and with detailed balance but now as an approximation of the theory without detailed balance. We then consider the coordinates that obey the generalized uncertainty principle by modifying the previous equation using the relations between both sets of coordinates. We describe two possible ways of obtaining the Wheeler-DeWitt equation. One of them is useful to present the general equation but it is found to be very difficult to solve. Then we use the other proposal to study the limiting cases considered before, that is, the infrared limit that can be compared to the equation obtained by using general relativity and the particular ultraviolet limit. For the second limit we use a ultraviolet approximation and then solve analytically the resulting equation. We find that and oscillatory behaviour is possible in this context but it is not a general feature for any values of the parameters involved.Multi-field inflation in high-slope potentialshttps://zbmath.org/1491.830422022-09-13T20:28:31.338867Z"Aragam, Vikas"https://zbmath.org/authors/?q=ai:aragam.vikas"Paban, Sonia"https://zbmath.org/authors/?q=ai:paban.sonia"Rosati, Robert"https://zbmath.org/authors/?q=ai:rosati.robert(no abstract)PBH in single field inflation: the effect of shape dispersion and non-Gaussianitieshttps://zbmath.org/1491.830442022-09-13T20:28:31.338867Z"Atal, Vicente"https://zbmath.org/authors/?q=ai:atal.vicente"Cid, Judith"https://zbmath.org/authors/?q=ai:cid.judith"Escrivà, Albert"https://zbmath.org/authors/?q=ai:escriva.albert"Garriga, Jaume"https://zbmath.org/authors/?q=ai:garriga.jaume(no abstract)Keeping an eye on DBI: power-counting for small-\(c_s\) cosmologyhttps://zbmath.org/1491.830462022-09-13T20:28:31.338867Z"Babic, Ivana"https://zbmath.org/authors/?q=ai:babic.ivana"Burgess, C. P."https://zbmath.org/authors/?q=ai:burgess.clifford-p|burgess.cliff-p"Geshnizjani, Ghazal"https://zbmath.org/authors/?q=ai:geshnizjani.ghazal(no abstract)On the slope of the curvature power spectrum in non-attractor inflationhttps://zbmath.org/1491.830532022-09-13T20:28:31.338867Z"Özsoy, Ogan"https://zbmath.org/authors/?q=ai:ozsoy.ogan"Tasinato, Gianmassimo"https://zbmath.org/authors/?q=ai:tasinato.gianmassimo(no abstract)Primordial black holes from metric preheating: mass fraction in the excursion-set approachhttps://zbmath.org/1491.830582022-09-13T20:28:31.338867Z"Auclair, Pierre"https://zbmath.org/authors/?q=ai:auclair.pierre"Vennin, Vincent"https://zbmath.org/authors/?q=ai:vennin.vincentCosmic microwave background anisotropy numerical solution (CMBAns). I: An introduction to \(C_l\) calculationhttps://zbmath.org/1491.830592022-09-13T20:28:31.338867Z"Das, Santanu"https://zbmath.org/authors/?q=ai:das.santanu-kumar"Phan, Anh"https://zbmath.org/authors/?q=ai:phan.anh-dung|phan.anh-vu|phan.anh-huy(no abstract)Perturbations in \(f(\mathbb{T})\) cosmology and the spin connectionhttps://zbmath.org/1491.830612022-09-13T20:28:31.338867Z"Golovnev, Alexey"https://zbmath.org/authors/?q=ai:golovnev.alexey-v(no abstract)On the stability of bimetric structure formationhttps://zbmath.org/1491.830622022-09-13T20:28:31.338867Z"Högås, Marcus"https://zbmath.org/authors/?q=ai:hogas.marcus"Torsello, Francesco"https://zbmath.org/authors/?q=ai:torsello.francesco"Mörtsell, Edvard"https://zbmath.org/authors/?q=ai:mortsell.edvard(no abstract)The EFT likelihood for large-scale structurehttps://zbmath.org/1491.850032022-09-13T20:28:31.338867Z"Cabass, Giovanni"https://zbmath.org/authors/?q=ai:cabass.giovanni"Schmidt, Fabian"https://zbmath.org/authors/?q=ai:schmidt.fabian(no abstract)The master equation in a bounded domain with Neumann conditionshttps://zbmath.org/1491.910212022-09-13T20:28:31.338867Z"Ricciardi, Michele"https://zbmath.org/authors/?q=ai:ricciardi.micheleSummary: In this article, we study the well-posedness of the master equation of mean field games in a framework of Neumann boundary condition. The definition of solution is closely related to the classical one of the mean field games system, but the boundary condition here leads to two Neumann conditions in the master equation formulation, for both space and measure. The global regularity of the linearized system, which is crucial in order to prove the existence of solutions, is obtained with a deep study of the boundary conditions and the global regularity at the boundary of a suitable class of parabolic equations.A probabilistic numerical method for a class of mean field gameshttps://zbmath.org/1491.910222022-09-13T20:28:31.338867Z"Sahar, Ben Aziza"https://zbmath.org/authors/?q=ai:sahar.ben-aziza"Salwa, Toumi"https://zbmath.org/authors/?q=ai:salwa.toumiExact solutions and numerical simulation for Bakstein-Howison modelhttps://zbmath.org/1491.911422022-09-13T20:28:31.338867Z"Dastranj, Elham"https://zbmath.org/authors/?q=ai:dastranj.elham"Fard, Hossein Sahebi"https://zbmath.org/authors/?q=ai:fard.hossein-sahebiSummary: In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact, stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally, European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.On approximation of transition densities in calibration of 1-dimensional stochastic models of asset priceshttps://zbmath.org/1491.911522022-09-13T20:28:31.338867Z"Merkin, L. A."https://zbmath.org/authors/?q=ai:merkin.l-a"Rezin, R. M."https://zbmath.org/authors/?q=ai:rezin.r-mSummary: For a 1-dimensional stochastic differential equation (SDE) of the type \(dS_t=\tilde{\mu}(S_t,t;\mathbf{q})dt+\tilde{\sigma}(S_t,t;\mathbf{q})dW_t \), where \(S_t\) is e.g. the price of a certain financial asset, \(W_t\) a Brownian motion and \(\mathbf{q}\) is a finite-dimensional vector of unknown model parameters, we provide a survey of methods for estimating the vector \(\mathbf{q}\) using the historical time series of \(S_t\). In quantitative finance terms, the SDEs under consideration are called parametric local volatility models, and the family of parameter estimation methods utilizing the historical time series is called ``real-measure model calibration methods'', as opposed to ``risk-neutral model calibration methods'' based on the prices of non-linear derivatives (options) with the underlying price \(S_t\). We are primarily focused on the classical maximum likelihood estimation (MLE) framework, and provide extensions of the previously-known MLE methods in two major respects. \textit{First,} we show that the likelihood values can be computed using the probability density functions (PDFs) for 3-variate distributions of open, high, low and close (OHLC) prices for each historical interval, instead of commonly-used 1-variate PDFs of open and close prices only. This allows us to take into account additional market information, i.e. \textit{ranges} of \(S_t\) values as opposed to just point-wise values. \textit{Second}, in order to construct the required 3-variate OHLC PDFs for general non-constant coefficients \(\tilde{\mu}\) and \(\tilde{\sigma} \), we apply results from the heat kernel theory. The constructed method may be applicable to a wide class of realistic SDEs in quantitative finance. In particular, this method can be applied to models of crypto-asset prices (e.g., for the purpose of price prediction in algorithmic trading), whereas risk-neutral (derivatives-based) methods would not be applicable due to insufficient liquidity of the crypto-options market. We consider an example of a reversion-based algorithmic trading strategy as an application of this method.Reaction-diffusion dynamics and biological pattern formationhttps://zbmath.org/1491.920062022-09-13T20:28:31.338867Z"Dutta, Kishore"https://zbmath.org/authors/?q=ai:dutta.kishoreSummary: The spontaneous formation of a wide variety of natural patterns with different shapes and symmetries in many physical and biological systems is one of the deep mysteries in science. This article describes the physical principles underlying the formation of various intriguing spatio-temporal patterns in Nature with special emphasis on some biological structures. We discuss how the spontaneous symmetry breaking due to diffusion driven instability in the reaction dynamics lead to the emergence of such complicated natural patterns. The mechanism of the formation of various animal coat patterns is explained via the Turing-type reaction-diffusion models.Spatial and color hallucinations in a mathematical model of primary visual cortexhttps://zbmath.org/1491.920102022-09-13T20:28:31.338867Z"Faugeras, Olivier D."https://zbmath.org/authors/?q=ai:faugeras.olivier-d"Song, Anna"https://zbmath.org/authors/?q=ai:song.anna"Veltz, Romain"https://zbmath.org/authors/?q=ai:veltz.romainThere is a study on chromatic aspects of visual perception by means of neural fields theory. The article considers a neural field model for color perceptions introduced by the authors in a previous article, [\textit{A. Song} et al., ``A neural field model for color perception unifying assimilation and contrast'', PLoS Comput. Biol. 15, No. 6, Article ID e1007050, 28 p. (2019; \url{doi:10.1371/journal.pcbi.1007050})]. The investigation here focuses on how this model can predict visual hallucinations. The model is briefly presented in the second section and is described by an initial value problem to the Hammerstein equation which is an integro-partial differential equation for the average membrane potential V (r, c, t). Under some assumptions and choice of an appropriate function space, one reminds that the existence of a unique solution to the Cauchy problem has been proved in an earlier paper from, [\textit{R. Veltz} and \textit{O. Faugeras}, SIAM J. Appl. Dyn. Syst. 9, No. 3, 954--998 (2010; Zbl 1194.92015)]. The aim further is to study the stationary solutions to the considered equations and their bifurcations which are interpreted as possible metaphors of visual hallucinations. In the next three sections one introduces the notion of stationary solutions, their bifurcations, one performs the computation of the spectrum of the linear operator in the neural model and one shows the symmetries of the model and equivariant bifurcations of the solutions. Examples of four types of planforms are presented in the sixth Section. A numerical bifurcation analysis is performed in the seventh Section. One describes the numerical experiments and presents some first, second and third bifurcation diagrams. Conclusions are provided in the eighth Section.
A very interesting research paper and good presentation.
Reviewer: Claudia Simionescu-Badea (Wien)A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signalhttps://zbmath.org/1491.920332022-09-13T20:28:31.338867Z"Wang, Yu Lan"https://zbmath.org/authors/?q=ai:wang.yulan"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michael"Xiang, Zhao Yin"https://zbmath.org/authors/?q=ai:xiang.zhaoyinSummary: The chemotaxis-Navier-Stokes system
\[
\begin{cases}
n_t + u \cdot \nabla n = \Delta n - \nabla \cdot (n\nabla c), \\
c_t + u \cdot \nabla c = \Delta c - nc, \\
u_t + (u \cdot \nabla) u = \Delta u + \nabla P + n\nabla \phi, \quad \nabla \cdot u = 0
\end{cases}
\]
is considered in a smoothly bounded planar domain \(\Omega\) under the boundary conditions
\[
(\nabla n - n\nabla c) \cdot \nu = 0,\quad c = c_{\star},\quad u = 0,\quad x \in \partial \Omega, t > 0,
\]
with a given nonnegative constant \(c_{\star}\). It is shown that if \((n_0, c_0, u_0)\) is sufficiently regular and such that the product \(\| n_0 \|_{L^1 (\Omega)} \| c_0\|_{L^{\infty} (\Omega)}^2\) is suitably small, an associated initial value problem possesses a bounded classical solution with \((n, c, u) |_{t=0} = (n_0, c_0, u_0)\).Predict blood pressure by photoplethysmogram with the fluid-structure interaction modelinghttps://zbmath.org/1491.920452022-09-13T20:28:31.338867Z"Chen, Jianhong"https://zbmath.org/authors/?q=ai:chen.jianhong"Hao, Wenrui"https://zbmath.org/authors/?q=ai:hao.wenrui"Sun, Pengtao"https://zbmath.org/authors/?q=ai:sun.pengtao"Zhang, Lian"https://zbmath.org/authors/?q=ai:zhang.lianSummary: Blood pressure (BP) has been identified as one of the main factors in cardiovascular disease and other related diseases. Then how to accurately and conveniently measure BP is important to monitor BP and to prevent hypertension. This paper proposes an efficient BP measurement model by integrating a fluid-structure interaction model with the photoplethysmogram (PPG) signal and developing a data-driven computational approach to fit two optimization parameters in the proposed model for each individual. The developed BP model has been validated on a public BP dataset and has shown that the average prediction errors among the root mean square error (RMSE), the mean absolute error (MAE), the systolic blood pressure (SBP) error, and the diastolic blood pressure (DBP) error are all below 5 mmHg for normal BP, stage I, and stage II hypertension groups, and, prediction accuracies of the SBP and the DBP are around 96\% among those three groups.A DNA-structured mathematical model of cell-cycle progression in cyclic hypoxiahttps://zbmath.org/1491.920472022-09-13T20:28:31.338867Z"Celora, Giulia L."https://zbmath.org/authors/?q=ai:celora.giulia-l"Bader, Samuel B."https://zbmath.org/authors/?q=ai:bader.samuel-b"Hammond, Ester M."https://zbmath.org/authors/?q=ai:hammond.ester-m"Maini, Philip K."https://zbmath.org/authors/?q=ai:maini.philip-k"Pitt-Francis, Joe M."https://zbmath.org/authors/?q=ai:pitt-francis.joe-m"Byrne, Helen M."https://zbmath.org/authors/?q=ai:byrne.helen-mSummary: New experimental data have shown how the periodic exposure of cells to low oxygen levels (\textit{i.e.}, cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced \textit{in vitro}, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with \textit{in vitro} experiments to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (\textit{i.e}, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodeshttps://zbmath.org/1491.920802022-09-13T20:28:31.338867Z"Harrach, Bastian"https://zbmath.org/authors/?q=ai:harrach.bastianAn invitation to stochastic mathematical biologyhttps://zbmath.org/1491.920872022-09-13T20:28:31.338867Z"Constable, George"https://zbmath.org/authors/?q=ai:constable.george-william-albert"Krumbeck, Yvonne"https://zbmath.org/authors/?q=ai:krumbeck.yvonne"Rogers, Tim"https://zbmath.org/authors/?q=ai:rogers.timThis paper represents an invitation to the use of stochastic models in mathematical biology, as motivated by the randomness and unpredictability of the natural world over a wide range of scales, from microscopic (transmission of genes and intracellular physiological processes) to macroscopic (species interaction and population dynamics). The authors make a convincing case that embracing stochastic thinking rather than relying on complete predictability can make a difference in the quality of predictions made and can handle the unpredictability of environmental factors and randomness in timing of individual events.
Several examples from a wide range of biological problems previously studied via deterministic approaches are showcased, along with the areas of mathematics that are susceptible to contribute. The paper is organized by mathematical discipline, from probability theory and statistical mechanics to stochastic differential systems and random matrix theory, open challenges being outlined along with their motivations, historical details and brief surveys of the state-of-art along with further references.
Reviewer: Paul Georgescu (Iaşi)Spatially nonhomogeneous periodic patterns in a delayed predator-prey model with predator-taxis diffusionhttps://zbmath.org/1491.920982022-09-13T20:28:31.338867Z"Shi, Qingyan"https://zbmath.org/authors/?q=ai:shi.qingyan"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongliSummary: In this paper, we study the effect of time delay on the dynamics of a diffusive predator-prey model with predator-taxis under Neumann boundary condition. The joint effect of predator-taxis and delay can lead to spatially nonhomogeneous periodic patterns via spatially nonhomogeneous Hopf bifurcations. It is also shown that there exist double Hopf bifurcations due to the interaction either between homogeneous and nonhomogeneous or between nonhomogeneous Hopf bifurcations with different modes, which cannot occur for the system with only either delay or predator-taxis diffusion.Spreading speeds and monostable waves in a reaction-diffusion model with nonlinear competitionhttps://zbmath.org/1491.921002022-09-13T20:28:31.338867Z"Zhang, Qiming"https://zbmath.org/authors/?q=ai:zhang.qiming"Han, Yazhou"https://zbmath.org/authors/?q=ai:han.yazhou"van Horssen, Wim T."https://zbmath.org/authors/?q=ai:van-horssen.wilhem-teunis"Ma, Manjun"https://zbmath.org/authors/?q=ai:ma.manjunThe authors study the cooperative Lotka-Volterra system with nonlinear competition
\begin{align*}
u_t&=u_{xx}+(1-u)(a_1v^2-u) \\
v_t&=dv_{xx}+rv(1-v-a_2(1-u)^2) \quad \text{in }\mathbb{R}\times (0,\infty),
\end{align*}
where \(r\), \(d\), \(a_1\), \(a_2\) are positive constants satisfying \(a_1>1\), \(a_2<\frac{1}{3}\). First, the minimal wave speed for traveling waves \(\overline{c}_+\) is calculated, which coincides with the single rightward spreading speed of the solution semiflow. Second, the linear speed \(c_0=2\sqrt{dr(1-a_2)}\) is detected from the linearized wave profile system. Then, selection mechanism of the minimal wave speed is defined, that is, linear and nonlinear selections according as the minimal wave speed is equal to and greater than \(c_0\), respectively. General criteria for these properties are derived, which prescribe the parameter region of this model for these selections. Finally, comparison to the Lotka-Volterra model with linear competition is discussed.
Reviewer: Takashi Suzuki (Osaka)Dynamics of antibody levels: asymptotic propertieshttps://zbmath.org/1491.921192022-09-13T20:28:31.338867Z"Pichór, Katarzyna"https://zbmath.org/authors/?q=ai:pichor.katarzyna"Rudnicki, Ryszard"https://zbmath.org/authors/?q=ai:rudnicki.ryszardSummary: We study properties of a piecewise deterministic Markov process modeling the changes in concentration of specific antibodies. The evolution of densities of the process is described by a stochastic semigroup. The long-time behavior of this semigroup is studied. In particular, we prove theorems on its asymptotic stability.Non-Markovian modelling highlights the importance of age structure on Covid-19 epidemiological dynamicshttps://zbmath.org/1491.921202022-09-13T20:28:31.338867Z"Reyné, Bastien"https://zbmath.org/authors/?q=ai:reyne.bastien"Richard, Quentin"https://zbmath.org/authors/?q=ai:richard.quentin"Selinger, Christian"https://zbmath.org/authors/?q=ai:selinger.christian"Sofonea, Mircea T."https://zbmath.org/authors/?q=ai:sofonea.mircea-t"Djidjou-Demasse, Ramsès"https://zbmath.org/authors/?q=ai:demasse.ramses-djidjou"Alizon, Samuel"https://zbmath.org/authors/?q=ai:alizon.samuelSummary: The Covid-19 pandemic outbreak was followed by a huge amount of modelling studies in order to rapidly gain insights to implement the best public health policies. Most of these compartmental models involved ordinary differential equations (ODEs) systems. Such a formalism implicitly assumes that the time spent in each compartment does not depend on the time already spent in it, which is at odds with the clinical data. To overcome this ``memoryless'' issue, a widely used solution is to increase and chain the number of compartments of a unique reality (\textit{e.g.} have infected individual move between several compartments). This allows for greater heterogeneity and thus be closer to the observed situation, but also tends to make the whole model more difficult to apprehend and parameterize. We develop a non-Markovian alternative formalism based on partial differential equations (PDEs) instead of ODEs, which, by construction, provides a memory structure for each compartment thereby allowing us to limit the number of compartments. We apply our model to the French 2021 SARS-CoV-2 epidemic and, while accounting for vaccine-induced and natural immunity, we analyse and determine the major components that contributed to the Covid-19 hospital admissions. The results indicate that the observed vaccination rate alone is not enough to control the epidemic, and a global sensitivity analysis highlights a huge uncertainty attributable to the age-structured contact matrix. Our study shows the flexibility and robustness of PDE formalism to capture national COVID-19 dynamics and opens perspectives to study medium or long-term scenarios involving immune waning or virus evolution.Global dynamics of an age-space structured HIV/AIDS model with viral load-dependent infection and conversion rateshttps://zbmath.org/1491.921262022-09-13T20:28:31.338867Z"Wu, Peng"https://zbmath.org/authors/?q=ai:wu.peng"Feng, Zhaosheng"https://zbmath.org/authors/?q=ai:feng.zhaosheng"Zhang, Xuebing"https://zbmath.org/authors/?q=ai:zhang.xuebingSummary: Distinguished from the existing HIV/AIDS epidemic models, we formulate a model by considering three-age-structured, spacial diffusion, viral load-dependent infection and conversion rates to study the global dynamical behaviors of HIV/AIDS transmission. The generation operator \(\mathscr{R}\) and the explicit expression of basic reproduction ratio \(R_0\) are introduced. The global stability of steady states and the uniform persistence of the disease are presented. Sensitivity analysis indicates that some parameters impact the value of \(R_0\) markedly and the intervening measure plays a crucial role in the intervention of HIV infection at population level. Furthermore, numerical simulations suggest that intervening measure at the individual and population levels is highly effective in controlling the transmission of the disease.Global controllability of the Navier-Stokes equations in the presence of curved boundary with no-slip conditionshttps://zbmath.org/1491.930172022-09-13T20:28:31.338867Z"Liao, Jiajiang"https://zbmath.org/authors/?q=ai:liao.jiajiang"Sueur, Franck"https://zbmath.org/authors/?q=ai:sueur.franck"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.3|zhang.ping.6|zhang.ping.2|zhang.ping|zhang.ping.1|zhang.ping.5Summary: We consider the issue of the small-time global exact null controllability of the axi-symmetric incompressible Navier-Stokes equation in a 3D finite vertical cylinder with circular section. We assume that we are able to act on the fluid flow on the top and on the bottom of the cylinder while no-slip conditions are prescribed on the boundary of the lateral section. We also make use of a distributed control, which can be chosen arbitrarily small for any Sobolev regularity in space. Our work improves earlier results in
[\textit{S. Guerrero} et al., C. R., Math., Acad. Sci. Paris 343, No. 9, 573--577 (2006; Zbl 1109.35087); J. Math. Pures Appl. (9) 98, No. 6, 689--709 (2012; Zbl 1253.35100)] where the distributed force is small only in a negative Sobolev space and the recent work [\textit{J.-M. Coron} et al., Ann. PDE 5, No. 2, Paper No. 17, 49 p. (2019; Zbl 1439.35367)]
where the case of the 2D incompressible Navier-Stokes equation in a rectangle was considered. Our analysis actually follows quite narrowly the one in [Coron et al., loc. cit.] by making use of Coron's return method, of the well-prepared dissipation method and of long-time nonlinear Cauchy-Kovalevskaya estimates. An extra difficulty here is the curvature of the uncontrolled part of the boundary which requires further analysis to apply the well-prepared dissipation method to lower order boundary layer terms.Output regulation for 1-D reaction-diffusion equation with a class of time-varying disturbances from exosystemhttps://zbmath.org/1491.930432022-09-13T20:28:31.338867Z"Wei, Jing"https://zbmath.org/authors/?q=ai:wei.jing"Guo, Bao-Zhu"https://zbmath.org/authors/?q=ai:guo.baozhuSummary: In this paper, we consider boundary output regulation for one-dimensional reaction-diffusion equation that has disturbances entering the system from in-domain and both boundaries. The reference signal and disturbances are generated from an exosystem with time varying coefficients. First, a feedforward control is designed on the basis of an infinite-dimensional regulator equation and a backstepping transformation. Second, with the measurement output, we design an observer to estimate both states of the plant and the external system. The output feedback boundary control is then designed by replacing the states with their estimates. As a result, we show that the output converges to the reference signal exponentially as time goes on.Exact control of a distributed system described by the wave equation with integral memoryhttps://zbmath.org/1491.930582022-09-13T20:28:31.338867Z"Romanov, I. V."https://zbmath.org/authors/?q=ai:romanov.ivan-v"Shamaev, A. S."https://zbmath.org/authors/?q=ai:shamaev.alexei-sSummary: We consider the distributed control problem for the wave equation with memory, where the kernel is the sum of decreasing exponential functions and the control is bounded in modulus. We prove that the oscillations of the system can be brought to the state of rest in a finite time.Passivity-based boundary control for delay reaction-diffusion systemshttps://zbmath.org/1491.930592022-09-13T20:28:31.338867Z"Wu, Kai-Ning"https://zbmath.org/authors/?q=ai:wu.kaining"Zhou, Wei-Jie"https://zbmath.org/authors/?q=ai:zhou.weijie"Liu, Xiao-Zhen"https://zbmath.org/authors/?q=ai:liu.xiaozhenSummary: Passivity-based boundary control is considered for time-varying delay reaction-diffusion systems (DRDSs) with boundary input-output. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are obtained for input strict passivity and output strict passivity of DRDSs, respectively. When the parameter uncertainties appear in DRDSs, sufficient conditions are presented to guarantee the robust passivity. Moreover, we apply our theoretical results to the synchronization problem of coupled delay reaction-diffusion systems and get the criterion to ensure the asymptotic synchronization. Finally, numerical simulations are provided to show the validity of our theoretical results.