Recent zbMATH articles in MSC 35Qhttps://zbmath.org/atom/cc/35Q2021-06-15T18:09:00+00:00WerkzeugMathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in 2 space dimensions.https://zbmath.org/1460.353442021-06-15T18:09:00+00:00"Xu, Xiangsheng"https://zbmath.org/authors/?q=ai:xu.xiangshengThe paper deals with the mathematical description of the evolution of a crystal surface. Mathematical model originates in the boundary value problem for nonlinear fourth-order partial differential equation in 2D. Physical background is discussed. However, the main attention is devoted to the weak solution of the stationary boundary value problem in 2D. The author proves the existence of weak solution by constructing it as a limit of a sequence of approximate solutions obtained by means of the backward difference method. One of the main tools here is the Leray-Schauder fixed point theorem. The paper also deals with stability of the weak solution.
Reviewer: Pavel Burda (Praha)A note on optimal \(H^1\)-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation.https://zbmath.org/1460.353262021-06-15T18:09:00+00:00"Henning, Patrick"https://zbmath.org/authors/?q=ai:henning.patrick"Wärnegård, Johan"https://zbmath.org/authors/?q=ai:warnegard.johanSummary: In this paper we consider a mass- and energy-conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal \(L^{\infty}(H^1)\)-error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments.Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density.https://zbmath.org/1460.353032021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We consider the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity on the whole space \(\mathbb{R}^2\). By weighted energy method, we show the local existence and uniqueness of strong solutions provided that the initial density decays not too slowly at infinity.Existence of global weak solutions for a 3D Navier-Stokes-Poisson-Korteweg equations.https://zbmath.org/1460.352992021-06-15T18:09:00+00:00"Yang, Jianwei"https://zbmath.org/authors/?q=ai:yang.jianwei"Wang, Zhengyan"https://zbmath.org/authors/?q=ai:wang.zhengyan"Ding, Fengxia"https://zbmath.org/authors/?q=ai:ding.fengxiaSummary: The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier-Stokes-Poisson-Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure.Sensitivity analysis of Burgers' equation with shocks.https://zbmath.org/1460.651292021-06-15T18:09:00+00:00"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Shu, Ruiwen"https://zbmath.org/authors/?q=ai:shu.ruiwenSensitivity analysis of Burgers' equation with shocks is discussed. It is known that the generalized polynomial chaos (gPC) method is used in many problems. For gPC to achieve high accuracy, PDE solutions need to have high regularity, but as usual, this is not true for hyperbolic type problems. In this paper, a counterargument is provided and is shown that even though the solution profile develops singularities, which destroys the spectral accuracy of gPC, the physical quantities are all smooth functions of the uncertainties coming from both initial data and the wave speed. The paper is organized as follows. Section 1 is an introduction. In Section 2, some notations are introduced and the precise quantitative versions of the main two theorems are stated. The deterministic case is focused on in Section 3 and some necessary tools for analyzing are prepared too. These tools are used in Section 4 and the abovementioned two main theorems are proved. The results to treat conservation laws with general convex fluxes are extended in Section 5. Conclusions are given in Section 6 and finally, some proofs not essential to the main context are given in the Appendix.
Reviewer: Temur A. Jangveladze (Tbilisi)A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source.https://zbmath.org/1460.353592021-06-15T18:09:00+00:00"Xie, Jianing"https://zbmath.org/authors/?q=ai:xie.jianingA version of the doubly parabolic Keller-Segel system with nonlinear (but nondegenerate) diffusion and logistic source term is studied in bounded domains of \(\mathbb R^N\), \(N\ge 2\). Using a new energy-type inequality global-in-time and uniform boundedness of solutions is studied under suitable assumptions on the diffusion term.
Reviewer: Piotr Biler (Wrocław)A network immuno-epidemiological HIV model.https://zbmath.org/1460.922002021-06-15T18:09:00+00:00"Gupta, Churni"https://zbmath.org/authors/?q=ai:gupta.churni"Tuncer, Necibe"https://zbmath.org/authors/?q=ai:tuncer.necibe"Martcheva, Maia"https://zbmath.org/authors/?q=ai:martcheva.maiaSummary: In this paper we formulate a multi-scale nested immuno-epidemiological model of HIV on complex networks. The system is described by ordinary differential equations coupled with a partial differential equation. First, we prove the existence and uniqueness of the immunological model and then establish the well-posedness of the multi-scale model. We derive an explicit expression of the basic reproduction number \({\mathscr{R}}_0\) of the immuno-epidemiological model. The system has a disease-free equilibrium and an endemic equilibrium. The disease-free equilibrium is globally stable when \({\mathscr{R}}_0<1\) and unstable when \({\mathscr{R}}_0 >1\). Numerical simulations suggest that \({\mathscr{R}}_0\) increases as the number of nodes in the network increases. Further, we find that for a scale-free network the number of infected individuals at equilibrium is a hump-like function of the within-host reproduction number; however, the dependence becomes monotone if the network has predominantly low connectivity nodes or high connectivity nodes.On a discrete scheme for time fractional fully nonlinear evolution equations.https://zbmath.org/1460.353632021-06-15T18:09:00+00:00"Giga, Yoshikazu"https://zbmath.org/authors/?q=ai:giga.yoshikazu"Liu, Qing"https://zbmath.org/authors/?q=ai:liu.qing.1"Mitake, Hiroyoshi"https://zbmath.org/authors/?q=ai:mitake.hiroyoshiSummary: We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.New general decay of solutions in a porous-thermoelastic system with infinite memory.https://zbmath.org/1460.740192021-06-15T18:09:00+00:00"Al-Mahdi, Adel M."https://zbmath.org/authors/?q=ai:al-mahdi.adel-m"Al-Gharabli, Mohammad M."https://zbmath.org/authors/?q=ai:algharabli.mohammad-m"Messaoudi, Salim A."https://zbmath.org/authors/?q=ai:messaoudi.salim-aSummary: This work is concerned with a one-dimensional thermoelastic porous system with infinite memory effect. We show that the stability of the system holds for a much larger class of kernels than the ones considered in the literature such as the one in [the last author and \textit{A. Fareh}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 6895--6906 (2011; Zbl 1228.35055); Discrete Contin. Dyn. Syst., Ser. B 20, No. 2, 599--612 (2015; Zbl 1304.35676)]. More precisely, we consider the kernel \(g : [0, + \infty) \to (0, + \infty)\) satisfying
\[
g^\prime(t) \leq - \gamma(t) G(g(t)),
\]
where \(\gamma\) and \(G\) are functions satisfying some specific properties. Under this very general assumption on the behavior of \(g\) at infinity, we establish a relation between the decay rate of the solutions and the growth of \(g\) at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data.Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains.https://zbmath.org/1460.353432021-06-15T18:09:00+00:00"Mohammed, Mogtaba"https://zbmath.org/authors/?q=ai:mohammed.mogtaba-a-y"Ahmed, Noor"https://zbmath.org/authors/?q=ai:ahmed.noor-aSummary: In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [\textit{D. Cioranescu} et al., Port. Math. (N.S.) 63, No. 4, 467--496 (2006; Zbl 1119.49014)]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.Propagation of electric field generated by periodic pumping in a stable medium of two-level atoms of the Maxwell-Bloch model.https://zbmath.org/1460.780212021-06-15T18:09:00+00:00"Filipkovska, M. S."https://zbmath.org/authors/?q=ai:filipkovska.maria-s"Kotlyarov, V. P."https://zbmath.org/authors/?q=ai:kotlyarov.vladimir-p|kotlyarov.v-pThe authors use the inverse scattering transform and the Riemann-Hilbert formalism to study a mixed problem for the Maxwell-Bloch equations, which describe the propagation of an electromagnetic wave in a resonant medium with distributed two-level atoms. In particular, the authors consider the propagation of a single mode signal. It is shown that the Riemann-Hilbert problem can provide asymptotics of the transmitted signal. Asymptotics are obtained both for the Riemann-Hilbert problem for large times and for the Maxwell-Bloch equations.
Reviewer: Eric Stachura (Marietta)Identification of the minimum value of reservoir permeability in nonlinear single phase mud filtrate invasion model.https://zbmath.org/1460.490012021-06-15T18:09:00+00:00"Boaca, Tudor"https://zbmath.org/authors/?q=ai:boaca.tudorSummary: In this paper we study an identification problem related to a nonlinear parabolic system. This system, called the nonlinear single phase mud filtrate invasion model, arises in the study of the mud filtrate invasion phenomenon and is presented in Boaca and Boaca (2018). Our objective is to determine the minimal value of the oil reservoir permeability starting from the observed values of the mud filtrate concentration at some time \(T\). We reduce this identification problem to a nonlinear optimal control problem. We prove the existence of an optimal control and obtain the optimal condition. The influence of the noisy measurements and some numerical results are also presented.Coordination of local and long range signaling modulates developmental patterning.https://zbmath.org/1460.920272021-06-15T18:09:00+00:00"Williamson, Carly"https://zbmath.org/authors/?q=ai:williamson.carly"Chamberlin, Helen M."https://zbmath.org/authors/?q=ai:chamberlin.helen-m"Dawes, Adriana T."https://zbmath.org/authors/?q=ai:dawes.adriana-tSummary: The development of multicellular organisms relies on correct patterns of cell fates to produce functional tissues in the mature organism. A commonly observed developmental pattern consists of alternating cell fates, where neighboring cells take on distinct cell fates characterized by contrasting gene and protein expression levels, and this cell fate pattern repeats over two or more cells. The patterns produced by these fate decisions are regulated by a small number of highly conserved signaling networks, some of which are mediated by long range diffusible signals and others mediated by local contact-dependent signals. However, it is not completely understood how local and long range signals associated with these networks interact to produce fate patterns that are both robust and flexible. Here we analyze mathematical models to investigate the patterning of cell fates in an array of cells, focusing on a two cell repeating pattern. Bifurcation analysis of a multicellular ODE model, where we consider the cells as discrete compartments, suggests that cells must balance sensitivity to external signals with robustness to perturbations. To focus on the patterning dynamics close to the bifurcation point, we derive a continuum PDE model that integrates local and long range signaling. For those cells with dynamics close to the bifurcation point, sensitivity to long range signals determines how far a pattern extends in space, while the number of local signaling connections determines the type of pattern produced. This investigation provides a general framework for understanding developmental patterning, and how both long range and local signals play a role in generating features observed across biology, such as species differences in nematode vulval development and insect bristle patterning, as well as medically relevant processes such as control of stem cell fate in the intestinal crypt.Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains.https://zbmath.org/1460.352592021-06-15T18:09:00+00:00"Kreml, Ondřej"https://zbmath.org/authors/?q=ai:kreml.ondrej"Nečasová, Šárka"https://zbmath.org/authors/?q=ai:necasova.sarka"Piasecki, Tomasz"https://zbmath.org/authors/?q=ai:piasecki.tomaszA barotropic flow of compressible viscous fluid is considered in domains with prescribed motion of the walls with either no-slip or slip boundary conditions. Local-in-time existence of strong solutions is established using a tranformation to a problem in a fixed domain leading to the Navier-Stokes equations with lower order terms and perturbed boundary conditions. Moreover, a weak-strong uniqueness result is proved for the problem with slip boundary conditions.
Reviewer: Piotr Biler (Wrocław)Dynamics in a quasilinear parabolic-elliptic Keller-Segel system with generalized logistic source and nonlinear secretion.https://zbmath.org/1460.920332021-06-15T18:09:00+00:00"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.5|wang.xin.4|wang.xin.13|wang.xin.3|wang.xin.10|wang.xin.9|wang.xin|wang.xin.8|wang.xin.12|wang.xin.11|wang.xin.7|wang.xin.2|wang.xin.1|wang.xin.6"Xiang, Tian"https://zbmath.org/authors/?q=ai:xiang.tian"Zhang, Nina"https://zbmath.org/authors/?q=ai:zhang.ninaSummary: In this chapter, we study dynamical properties of nonnegative solutions for the following quasilinear parabolic-elliptic Keller-Segel chemotaxis system with generalized logistic source and nonlinear secretion:
\[
\left\{\begin{array}{ll}
u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot (S(u)\nabla v)+f(u),\quad & x\in \varOmega,\, t>0,\\
0=\Delta v-v+u^\kappa,\quad & x\in\varOmega,\, t>0,\\
u(x,0)=u_0(x),\quad & x\in \varOmega,
\end{array}\right.\tag{\(\ast\)}
\]
with homogeneous Neumann boundary conditions in a bounded domain \(\varOmega\subset\mathbf{R}^n(n\geq 2)\) with smooth boundary, where \(\kappa >0\) and the parameter functions Dand Sare smooth and, for some \(d,\chi >0\), \(\alpha,\beta\in\mathbf{R}\), \(D(u)\geq du^{-\alpha}\), \(S(u)\leq\chi u^\beta\) for all \(u>1\) and the logistic source f(u) fulfills \(f(0)\geq 0\) as well as \(f(u)\leq a_0-b u^\gamma\) with \(a_0\geq 0\), \(b>0\), \(\gamma>1\). We first establish a boundedness principle for the chemotaxis system \((*)\) asserting that blow-up of the solution is impossible if \(\Vert u(\cdot ,t)\Vert_{L^q(\varOmega)}\) is bounded for some \(q>\max\{\frac{n}{2}(\alpha+\beta+\kappa-1),0\}\). Then, with the aid of this criterion, we show the uniform-in-time \(L^\infty\)-boundedness of solutions under either one of the followings:
(B1) \(\beta+\kappa<\max\{\gamma,1+\frac{2}{n}-\alpha\}\),
(B2) \(\beta+\kappa=\gamma\) and \(b>b_*=\left\{\begin{array}{ll} \frac{[n(\alpha+\gamma-1)-2]}{n(\alpha+\gamma-1)+2(\beta-1)}\chi &\text{ for }\beta > 0,\\ \chi & \text{ for }\beta\leq 0,\end{array}\right.\)
(B3) \(\beta>0\), \(\beta+\kappa=\gamma\), \(b=b_*\) and either \(a_0=0\) or
\[
\left\{\begin{array}{ll}
\alpha\le1 1 & \text{ for }\gamma>1,\\
1<\alpha\leq\frac{1}{2}+\frac{2}{n} & \text{ for }1-\alpha+\frac{2}{n+2-n\alpha}<\gamma\leq 1-\alpha+\frac{4}{n},\\
\frac{1}{2}+\frac{2}{n}<\alpha<1+\frac{2}{n} & \text{ for }\gamma>1-\alpha+\frac{4}{n},
\end{array}\right.
\]
(B4) \(\beta=0\), \(\kappa=\gamma>1\), \(b=b_*=\chi\) and either \(a_0=0\) or \(\alpha<1+\frac{2}{n}\).
Our results capture the effects of the net proliferation rate (whether \(a_0=0\) or not) of cells and weak chemotaxis \((\beta\leq 0)\) and, they encompass and extend the existing boundedness results, and hence enlarge the parameter range of boundedness. Finally, for the prototypical choices \(D(u)=(u+1)^{-\alpha}\), \(S(u)=\chi u(u+1)^{\beta-1}\) for \(\beta<1\) or \(S(u)=\chi u^\beta\) for \(\beta\geq 1\) and \(f(u)=au-bu^\gamma\) for some \(a\in\mathbb{R}\), \(b>0\), the global stabilities of the equilibria \(((a/b)^{\frac{1}{\gamma -1}},(a/b)^{\frac{\kappa}{\gamma-1}})\) and (0, 0) are investigated in great detail and their respective convergence rates are explicitly calculated out. These stabilization results exhibit the effect of each ingredient in \((*)\) and, in particular, illustrate that no pattern formation can arise for small chemosensitivity \(\chi\) or large damping \(b\).
For the entire collection see [Zbl 1459.91003].The existence of full-dimensional invariant tori for an almost-periodically forced nonlinear beam equation.https://zbmath.org/1460.740522021-06-15T18:09:00+00:00"Liu, Shujuan"https://zbmath.org/authors/?q=ai:liu.shujuan"Shi, Guanghua"https://zbmath.org/authors/?q=ai:shi.guanghuaSummary: In this paper, we prove the existence of full-dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via Kolmogorov-Arnold-MoserAM theory.
{\copyright 2021 American Institute of Physics}Multiplicative control problems for nonlinear reaction-diffusion-convection model.https://zbmath.org/1460.352772021-06-15T18:09:00+00:00"Brizitskii, R. V."https://zbmath.org/authors/?q=ai:brizitskii.roman-victorovich|brizitskii.r-v|brizitskii.roman-viktorovich"Saritskaia, Zh. Yu."https://zbmath.org/authors/?q=ai:saritskaia.zh-yuSummary: Global solvability of a boundary value problem for a generalized Boussinesq model is proved in the case, when reaction coefficient depends nonlinearly on concentration of substance. Maximum principle is stated for substance's concentration. Solvability of control problem is proved, when the role of controls is played by diffusion and mass exchange coefficients from the equations and from the boundary conditions of the model. For a considered multiplicative control problem, optimality systems are obtained. On the base of the analysis of these systems for particular reaction coefficients and cost functionals, local stability estimates are deduced for optimal solutions.Control of tumor growth distributions through kinetic methods.https://zbmath.org/1460.920572021-06-15T18:09:00+00:00"Preziosi, Luigi"https://zbmath.org/authors/?q=ai:preziosi.luigi"Toscani, Giuseppe"https://zbmath.org/authors/?q=ai:toscani.giuseppe"Zanella, Mattia"https://zbmath.org/authors/?q=ai:zanella.mattiaSummary: The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker-Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.On the spectral stability of standing waves of nonlocal \(\mathcal{PT}\) symmetric systems.https://zbmath.org/1460.353242021-06-15T18:09:00+00:00"Feng, Wen"https://zbmath.org/authors/?q=ai:feng.wen"Stanislavova, Milena"https://zbmath.org/authors/?q=ai:stanislavova.milenaSummary: We consider standing wave solutions of the nonlocal NLS and the nonlocal Klein-Gordon Equations. Using a variety of different techniques such as energy estimates, direct spectral calculations and index count theorems, together with the spectral properties of operators, we prove spectral stability of these waves as solutions of five different nonlocal models.
For the entire collection see [Zbl 1457.35005].A geometrical demonstration for continuation of solutions of the generalised BBM equation.https://zbmath.org/1460.350602021-06-15T18:09:00+00:00"da Silva, Priscila Leal"https://zbmath.org/authors/?q=ai:da-silva.priscila-leal"Freire, Igor Leite"https://zbmath.org/authors/?q=ai:freire.igor-leiteSummary: A simple proof that if the generalised BBM equation has a solution vanishing on an open set of its domain then the solution is necessarily zero is given. In particular, the only compactly supported solution of the equation under consideration is the identically vanishing one.Stationary patterns of a predator-prey model with prey-stage structure and prey-taxis.https://zbmath.org/1460.921622021-06-15T18:09:00+00:00"Chen, Meijun"https://zbmath.org/authors/?q=ai:chen.meijun"Cao, Huaihuo"https://zbmath.org/authors/?q=ai:cao.huaihuo"Fu, Shengmao"https://zbmath.org/authors/?q=ai:fu.shengmaoLocal equi-attraction of pullback attractor sections.https://zbmath.org/1460.370132021-06-15T18:09:00+00:00"Li, Fuzhi"https://zbmath.org/authors/?q=ai:li.fuzhi"Xin, Jie"https://zbmath.org/authors/?q=ai:xin.jie"Cui, Hongyong"https://zbmath.org/authors/?q=ai:cui.hongyong"Kloeden, Peter E."https://zbmath.org/authors/?q=ai:kloeden.peter-erisSummary: In this paper, we study a local equi-attraction of pullback attractors for non-autonomous processes. By the local equi-attraction we mean that any local part of sections of a pullback attractor \(\mathfrak{A} = \{ A(\tau)\}_{\tau \in \mathbb{R}}\) are pullback attracting at the same rate, i.e., for any bounded (but arbitrarily large) interval \(I\),
\[
\lim_{t \to \infty}( \sup_{\tau \in I} \operatorname{dist}_X(U(t, \tau - t, B), A(\tau))) = 0,\quad \forall B \subset X \text{ bounded},
\]
where \(X\) is a metric space and \(U : \mathbb{R}^+ \times \mathbb{R} \times X \to X\) is a process in \(X\). Our technique makes use of the uniform attractor theory to consider in a dynamic way the time parameter \(\tau\) involved in the pullback attractor. The analysis shows that, roughly, when a system has a uniform attractor, then the pullback attractor can be locally equi-attracting. As an example, the pullback attractor of 2D Navier-Stokes equation is studied, where the joint continuity in initial time and initial data of the solutions plays a key role. In addition, the continuity of the set-valued mapping \(\tau \mapsto A(\tau)\) in more regular spaces is also studied.Exact steady solutions for a fifteen velocity model of gas.https://zbmath.org/1460.766782021-06-15T18:09:00+00:00"d'Ameida, Amah"https://zbmath.org/authors/?q=ai:dameida.amahSummary: Existence and boundedness of the solutions of the boundary value problem for a fifteen velocity tridimensional discrete model of gas is proved for bounded boundary conditions and exact analytic solutions are built. An application to the determination of the accommodation coefficients on the boundaries of a flow in a rectangular box is performed.
For the entire collection see [Zbl 1458.00035].Local existence and blowup criterion for the Euler equations in a function space with nondecaying derivatives.https://zbmath.org/1460.760732021-06-15T18:09:00+00:00"Hirata, Daisuke"https://zbmath.org/authors/?q=ai:hirata.daisukeSummary: We consider the Cauchy problem for the incompressible Euler equations on \(\mathbb{R}^d\) for \(d \geq 3\). Then we demonstrate the local-in-time solvability of classical solutions with the nondecaying derivatives and finite kinetic energy. Moreover, we establish the blowup criterion of such solutions in terms of the vorticity.Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces \(M_{p,q}^s(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\).https://zbmath.org/1460.353182021-06-15T18:09:00+00:00"Chaichenets, Leonid"https://zbmath.org/authors/?q=ai:chaichenets.leonid"Hundertmark, Dirk"https://zbmath.org/authors/?q=ai:hundertmark.dirk"Kunstmann, Peer Christian"https://zbmath.org/authors/?q=ai:kunstmann.peer-christian"Pattakos, Nikolaos"https://zbmath.org/authors/?q=ai:pattakos.nikolaosSummary: We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in [\textit{M. Sugimoto} et al., Integral Transforms Spec. Funct. 22, No. 4--5, 351--358 (2011; Zbl 1221.44007)], of the intersection \(M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\) for \(d\in\mathbb{N}\), \(p,q\in [1,\infty]\) and \(s\geq 0\). We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves a theorem by \textit{Á. Bényi} and \textit{K. A. Okoudjou} [Bull. Lond. Math. Soc. 41, No. 3, 549--558 (2009; Zbl 1173.35115)], where only the case \(q=1\) is considered, and closes a gap in the literature. If \(q>1\) and \(s>d\left(1-\frac{1}{q}\right)\) or if \(q=1\) and \(s\geq 0\) then \(M^s_{p,q}(\mathbb{R}^d)\hookrightarrow M_{\infty,1}(\mathbb{R}^d)\) and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.
For the entire collection see [Zbl 1457.35005].Stability of smooth solutions for the compressible Euler equations with time-dependent damping and one-side physical vacuum.https://zbmath.org/1460.350312021-06-15T18:09:00+00:00"Pan, Xinghong"https://zbmath.org/authors/?q=ai:pan.xinghongThe author considers the 1D compressible Euler equations with time-dependent damping and focuses on stability of the one-side vacuum solutions. The gas is isentropic and adiabatic. The domain is time-dependent, with gas expanding from the initial compact domain. The author proves the global existence and stability of smooth solutions. Actually, the solution is obtained in the explicit form and its stability is established.
Reviewer: Ilya A. Chernov (Petrozavodsk)Central-upwind scheme for a non-hydrostatic Saint-Venant system.https://zbmath.org/1460.651052021-06-15T18:09:00+00:00"Chertock, Alina"https://zbmath.org/authors/?q=ai:chertock.alina-e"Kurganov, Alexander"https://zbmath.org/authors/?q=ai:kurganov.alexander"Miller, Jason"https://zbmath.org/authors/?q=ai:miller.jason"Yan, Jun"https://zbmath.org/authors/?q=ai:yan.junThe authors develop a second-order central-upwind scheme for a nonhydrostatic version of the Saint-Venant system. This scheme is well-balanced and positivity preserving. The scheme is used to study ability of the non-hydrostatic Saint-Venant system to model long-time propagation and on-shore arrival of the tsunami-type waves. It is remarked that for a certain range of the dispersive coefficients, both the shape and amplitude of the waves are preserved even when the computational grid is coarse. The importance of the dispersive terms in the description of on-shore arrival is shown.
For the entire collection see [Zbl 1453.35003].
Reviewer: Abdallah Bradji (Annaba)Global diffeomorphism of the Lagrangian flow-map for a Pollard-like internal water wave.https://zbmath.org/1460.353472021-06-15T18:09:00+00:00"Kluczek, Mateusz"https://zbmath.org/authors/?q=ai:kluczek.mateusz"Rodríguez-Sanjurjo, Adrián"https://zbmath.org/authors/?q=ai:rodriguez-sanjurjo.adrianSummary: In this article we provide an overview of a rigorous justification of the global validity of the fluid motion described by a new exact and explicit solution prescribed in terms of Lagrangian variables of the nonlinear geophysical equations. More precisely, the three-dimensional Lagrangian flow-map describing this exact and explicit solution is proven to be a global diffeomorphism from the labelling domain into the fluid domain. Then, the flow motion is shown to be dynamically possible.
For the entire collection see [Zbl 1432.35003].Multiple solutions for a class of \(p(x)\)-curl systems arising in electromagnetism.https://zbmath.org/1460.780222021-06-15T18:09:00+00:00"Nguyen Thanh Chung"https://zbmath.org/authors/?q=ai:nguyen-thanh-chung.Summary: In this paper, we study the existence of solutions for a class of of \(p(x)\)-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities, we obtain at least two non-trivial solutions for the problem by using the mountain pass theorem combined with the Ekeland variational principle. Our main result in this paper complements and extends some earlier ones concerning the \(p(x)\)-curl operator in [\textit{A. Bahrouni} and \textit{D. Repovš}, Complex Var. Elliptic Equ. 63, No. 2, 292--301 (2018; Zbl 1423.35124); \textit{D. Medková}, J. Differ. Equations 261, No. 10, 5670--5689 (2016; Zbl 1356.35181)].Analytical solutions of fractional Schrödinger equation and thermal properties of Morse potential for some diatomic molecules.https://zbmath.org/1460.810192021-06-15T18:09:00+00:00"Okorie, U. S."https://zbmath.org/authors/?q=ai:okorie.u-s"Ikot, A. N."https://zbmath.org/authors/?q=ai:ikot.akpan-ndem"Rampho, G. J."https://zbmath.org/authors/?q=ai:rampho.gaotsiwe-j"Amadi, P. O."https://zbmath.org/authors/?q=ai:amadi.p-o"Abdullah, Hewa Y."https://zbmath.org/authors/?q=ai:abdullah.hewa-yLow Mach number limit of some staggered schemes for compressible barotropic flows.https://zbmath.org/1460.352582021-06-15T18:09:00+00:00"Herbin, R."https://zbmath.org/authors/?q=ai:herbin.raphaele"Latché, J.-C."https://zbmath.org/authors/?q=ai:latche.jean-claude"Saleh, K."https://zbmath.org/authors/?q=ai:saleh.khaledSummary: In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one, so that the resulting scheme is unconditionally stable, and explicit for the second one, so that the scheme is stable under a CFL condition involving the material velocity only. We prove rigorously that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tends to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof of convergence of the solutions of the (continuous) barotropic Navier-Stokes equations to that of the incompressible Navier-Stokes equation as the Mach number vanishes. Numerical results performed with a hand-built analytical solution show the behaviour that is expected from the analysis. Additional numerical results are obtained for the shock solutions of problems which are not in the scope of the present non dimensionalization but are nevertheless interesting to understand the behaviour of the scheme.Quantum motion, coherent states and geometric phase of a generalized damped pendulum.https://zbmath.org/1460.810202021-06-15T18:09:00+00:00"Pedrosa, I. A."https://zbmath.org/authors/?q=ai:pedrosa.i-aOptimization of the matrix Fourier-filter for a class of nonlinear optical models with an integral objective functional.https://zbmath.org/1460.780242021-06-15T18:09:00+00:00"Sazonova, S. V."https://zbmath.org/authors/?q=ai:sazonova.sofia-v"Razgulin, A. V."https://zbmath.org/authors/?q=ai:razgulin.a-vSummary: We consider a new formulation of the Fourier-filtering problem that uses matrix Fourier-filters as the controls in nonlinear optical models described by quasi-linear functional-differential diffusion equations. Solvability of the control problem is proved for various classes of matrix Fourier-filters with a time-integral objective functional. Differentiability of the functional with respect to the matrix Fourier-filter and convergence of a variant of the gradient projection method are proved. Examples of numerical simulation of controlled structure formation are presented, and the advantages of matrix Fourier-filters compared with traditional multiplier filters are demonstrated.Expansion of gas by turning a sharp corner into vacuum for 2-D pseudo-steady compressible magnetohydrodynamics system.https://zbmath.org/1460.769222021-06-15T18:09:00+00:00"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjun"Yin, Gan"https://zbmath.org/authors/?q=ai:yin.gan"You, Shouke"https://zbmath.org/authors/?q=ai:you.shoukeSummary: This paper is concerned with the process of a gas expanding into vacuum by turning around a sharp corner for 2-D pseudo-steady compressible magnetohydrodynamics system. This problem actually can be interpreted as interaction of a centered wave and a planar rarefaction wave. As the gas touches the corner and starts expanding into the vacuum around the sharp corner, a centered wave and a planar rarefaction wave appear. In the estimates of solution, we utilize the characteristic analysis and deduce proper characteristic decompositions. Combining the \(C^0\) estimate, gradient estimates, hyperbolicity, and the interaction of centered wave and planar rarefaction wave, we constructively obtain the existence of global classical solution to the present problem.Uniqueness to inverse acoustic and electromagnetic scattering from locally perturbed rough surfaces.https://zbmath.org/1460.353962021-06-15T18:09:00+00:00"Zhao, Yu"https://zbmath.org/authors/?q=ai:zhao.yu"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Yan, Baoqiang"https://zbmath.org/authors/?q=ai:yan.baoqiangThe authors of this article consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. Precisely, the scattering interface is given by a graph of a Lipschitz continuous function having compact support. The authors prove that both an acoustically sound-soft and sound-hard scatterer can be uniquely reconstructed from given far-field patterns of infinite number of incident plane waves with distinct directions. Next, electromagnetic scattering from a scattering surface of polyhedral type with conductive boundary condition is investigated. It is shown that such a scatterer can be uniquely reconstructed by one single point source or one plane wave. The result is achieved by using the mixed reciprocity relation in a half space and the reflection principle for the Helmholtz and the Maxwell equations.
Reviewer: Andreas Kleefeld (Jülich)Existence theory for the Boussinesq equation in modulation spaces.https://zbmath.org/1460.352762021-06-15T18:09:00+00:00"Banquet, Carlos"https://zbmath.org/authors/?q=ai:banquet-brango.carlos"Villamizar-Roa, Élder J."https://zbmath.org/authors/?q=ai:villamizar-roa.elder-jesusSummary: In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces \(M^s_{p',q}(\mathbb{R}^n)\), \(n\geq 1\). After a decomposition of the Boussinesq equation in a \(2\times 2\)-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in \(M^s_{p,q}\times D^{-1}JM^s_{p,q}\)-spaces for suitable \(p,q\) and \(s\), including the special case \(p=2\), \(q=1\) and \(s=0\). Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.A regularity criterion for a 2D tropical climate model with fractional dissipation.https://zbmath.org/1460.353462021-06-15T18:09:00+00:00"Bisconti, Luca"https://zbmath.org/authors/?q=ai:bisconti.lucaSummary: Tropical climate model derived by \textit{D. M. W. Frierson} et al. [Commun. Math. Sci. 2, No. 4, 591--626 (2004; Zbl 1160.86303)] and its modified versions have been investigated in a number of papers [\textit{E. Titi} and \textit{J. Li}, Discrete Contin. Dyn. Syst. 36, No. 8, 4495--4516 (2016; Zbl 1339.35325); \textit{R. Wan}, J. Math. Phys. 57, No. 2, 021507, 13 p. (2016; Zbl 1338.86011); \textit{M. Wang} et al., Discrete Contin. Dyn. Syst., Ser. B 21, No. 3, 919--941 (2016; Zbl 1353.35007)] and more recently \textit{Q. Shi} and \textit{C. Peng} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 178, 133--144 (2019; Zbl 1406.35370)]. Here, we deal with the \(2D\) tropical climate model with fractional dissipative terms in the equation of the barotropic mode \(u\) and in the equation of the first baroclinic mode \(v\) of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system.Existence and uniqueness results for modeling jet flow of the antarctic circumpolar current.https://zbmath.org/1460.769122021-06-15T18:09:00+00:00"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Wen, Qian"https://zbmath.org/authors/?q=ai:wen.qian"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper we are concerned with the analysis of a mathematical model, a two point boundary value problem for a second-order differential equation, that is used to deal with the jet flow of the antarctic circumpolar current. We present some new existence and uniqueness results when the vorticity function satisfies either a Lipschitz condition or is continuous.Stability of the boundary layer expansion for the 3D plane parallel MHD flow.https://zbmath.org/1460.769252021-06-15T18:09:00+00:00"Ding, Shijin"https://zbmath.org/authors/?q=ai:ding.shijin"Lin, Zhilin"https://zbmath.org/authors/?q=ai:lin.zhilin"Niu, Dongjuan"https://zbmath.org/authors/?q=ai:niu.dongjuanSummary: In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic flow with the no-slip boundary condition of velocity and perfectly conducting walls for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm \(L^\infty (H^1)\). In addition, similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered.
{\copyright 2021 American Institute of Physics}Universality of capillary rising in corners.https://zbmath.org/1460.762742021-06-15T18:09:00+00:00"Zhou, Jiajia"https://zbmath.org/authors/?q=ai:zhou.jiajia"Doi, Masao"https://zbmath.org/authors/?q=ai:doi.masaoSummary: We study the dynamics of viscous capillary rising in small corners between two curved walls described by a function \(y=cx^n\) with \(n \geq 1\). Using the Onsager principle, we derive a partial differential equation that describes the time evolution of the meniscus profile. By solving the equation both numerically and analytically, we show that the capillary rising dynamics is quite universal. Our theory explains the surprising finding by \textit{A. Ponomarenko} et al. [ibid. 666, 146--154 (2011; Zbl 1225.76023)] that the time dependence of the height not only obeys the universal power-law of \(t^{1/3}\), but also that the prefactor is almost independent of \(n\).Positive vector solutions for a Schrödinger system with external source terms.https://zbmath.org/1460.351232021-06-15T18:09:00+00:00"Long, Wei"https://zbmath.org/authors/?q=ai:long.wei"Peng, Shuangjie"https://zbmath.org/authors/?q=ai:peng.shuangjieAuthors' abstract: This paper is concerned with the existence of many synchronized vector solutions for the following Schrödinger system with external source terms \[\begin{cases} - \Delta u + u=a(x) u^3+\beta u v^2+f(x), & \quad x \; \in \; {\mathbb{R}}^3,\\ - \Delta v + v=b(x) v^3+\beta u^2 v+g(x), & \quad x \; \in \; {\mathbb{R}}^3,\\ u,v >0, & \quad x\in{\mathbb{R}}^3, \end{cases}\] where \(\beta \in{\mathbb{R}}\) is a coupling constant, \(a, b\in C({\mathbb{R}}^3)\) and \(f, g\in L^2({\mathbb{R}}^3)\cap L^\infty ({\mathbb{R}}^3)\). This type of system arises in Bose-Einstein condensates and Kerr-like photo refractive media. This paper tries to reveal the influence of the external source terms \(f\) and \(g\) on the number of the solutions. It is shown that the level set of the corresponding functional has a quite rich topology and the system admits \(k\) spikes synchronized vector solutions for any \(k\in \mathbb{Z}^+\) when \(f\) and \(g\) are small and \(a(x), b(x)\) satisfy some additional assumptions at infinity. The proof is based on the Lyapunov-Schmidt reduction scheme and the main ingredient is to improve the estimate on the remainder term obtained in the reduction process.
Reviewer: Jiří Rákosník (Praha)Stabilization of swelling porous elastic soils with fluid saturation and delay time terms.https://zbmath.org/1460.740632021-06-15T18:09:00+00:00"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-junior.dilberto-da-silva|almeida.dilberto-s-jun"Freitas, M. M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Noé, A. S."https://zbmath.org/authors/?q=ai:noe.a-s"Santos, M. J. Dos"https://zbmath.org/authors/?q=ai:santos.manoel-j-dosSummary: In this article, we consider the swelling problem in porous elastic soils with fluid saturation. We study the well-posedness of the problem based on the semigroup theory, show that the energy associated with the system is dissipative, and establish the stability of the system in the exponential way. To guarantee the stability of the systems, we consider both viscous damping and the time delay term acting on the first equation of the system.
{\copyright 2021 American Institute of Physics}Semi-algebraic sets method in PDE and mathematical physics.https://zbmath.org/1460.350112021-06-15T18:09:00+00:00"Wang, W.-M."https://zbmath.org/authors/?q=ai:wang.weimin|wang.wenming|wang.wumin|wang.whei-ming|wang.wei-min|wang.weiming|wang.wenminSummary: This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein-Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on the lattice. We also review the related linear time-dependent problems.
{\copyright 2021 American Institute of Physics}Global solutions to the \(n\)-dimensional incompressible Oldroyd-B model without damping mechanism.https://zbmath.org/1460.760322021-06-15T18:09:00+00:00"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaopingSummary: The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in \(\mathbb{R}^n(n = 2, 3)\). This result allows us to construct global solutions for a class of highly oscillating initial velocities. The proof uses the special structure of the system. Moreover, our theorem extends the previous result of \textit{Y. Zhu} [J. Funct. Anal. 274, No. 7, 2039--2060 (2018; Zbl 1387.76004)] and covers the recent result of \textit{Q. Chen} and \textit{X. Hao} [J. Math. Fluid Mech. 21, No. 3, Paper No. 42, 23 p. (2019; Zbl 1418.76008)].
{\copyright 2021 American Institute of Physics}Asymptotics for optimal design problems for the Schrödinger equation with a potential.https://zbmath.org/1460.930202021-06-15T18:09:00+00:00"Waters, Alden"https://zbmath.org/authors/?q=ai:waters.alden"Merkurjev, Ekaterina"https://zbmath.org/authors/?q=ai:merkurjev.ekaterinaSummary: We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in \(L^{\infty} (\Omega)\), with \(\Omega \subset \mathbb{R}^d\), using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.Electric-field-induced instabilities in nematic liquid crystals.https://zbmath.org/1460.780092021-06-15T18:09:00+00:00"Gartland, Eugene C. jun."https://zbmath.org/authors/?q=ai:gartland.eugene-c-junExponential attractor for Hindmarsh-Rose equations in neurodynamics.https://zbmath.org/1460.350492021-06-15T18:09:00+00:00"Phan, Chi"https://zbmath.org/authors/?q=ai:phan.chi"You, Yuncheng"https://zbmath.org/authors/?q=ai:you.yunchengSummary: The existence of exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in the study of neurodynamics is proved through uniform estimates and a new theorem on the squeezing property of the abstract reaction-diffusion equation established in this paper. This result on the exponential attractor infers that the global attractor whose existence has been proved in [the authors and \textit{J. Su},``Global attractors for Hindmarsh-Rose equationsin neurodynamics'', Preprint, \url{arXiv:1907.13225}] for the diffusive Hindmarsh-Rose semiflow has a finite fractal dimension.Stochastic models of chemotaxis processes.https://zbmath.org/1460.600602021-06-15T18:09:00+00:00"Belopolskaya, Ya. I."https://zbmath.org/authors/?q=ai:belopolskaya.yana-iSummary: Probabilistic representations of weak solutions to the Cauchy problem are constructed for systems of nonlinear parabolic equations arising in chemotaxis. These equations include as a special case the Keller-Segel model.Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity.https://zbmath.org/1460.353422021-06-15T18:09:00+00:00"Łazuka, Jarosław"https://zbmath.org/authors/?q=ai:lazuka.jaroslawThe paper is devoted to thermoelasticity of non-simple materials in a three-dimensional space. The mathematical model is described by a system of partial differential equations of fourth order. After defining a suitable evolution equation, the existence of the solution to the Cauchy problem is proven by applying semigroup methods. An asymptotic analysis of the solution is developed. The explicit Fourier representation of the solution is derived. By employing Sobolev, Bessel and Besov spaces and by applying the interpolation method, the author shows the \(L^p-L^q\) time decay estimates for the solution.
Reviewer: Adina Chirila (Braşov)Continuity properties of the solution map for the Euler-Poisson equation.https://zbmath.org/1460.766602021-06-15T18:09:00+00:00"Holmes, J."https://zbmath.org/authors/?q=ai:holmes.john-m|holmes.jeff|holmes.john-p|holmes.justin-d|holmes.jim|holmes.jessica|holmes.james-m|holmes.jeffrey-w|holmes.john-h|holmes.j-a"Tığlay, F."https://zbmath.org/authors/?q=ai:tiglay.ferideSummary: We study the continuity properties of the data-to-solution map for the modified Euler-Poisson equation. We show that for initial data in the Sobolev space \(H^s\), \(s>3/2\), the data-to-solution map is not better than continuous. Furthermore, we consider the solution map in the \(H^\gamma \) topology for \(s>\gamma \) and find that the data-to-solution map is Hölder continuous.Kinetic equations. Volume 1: Boltzmann equation, Maxwell models, and hydrodynamics beyond Navier-Stokes.https://zbmath.org/1460.350012021-06-15T18:09:00+00:00"Bobylev, Alexander V."https://zbmath.org/authors/?q=ai:bobylev.alexandre-vasiljevitchThe methods of kinetic theory of gases are used in many different fields of science and technology. They are closely connected with particle methods of numerical modeling of various processes on modern computers.
We give a brief summary of the contents of the present book The chapters 1 and 2 of the book contain the standard material with all necessary material for beginners. The notion of \(N\)-particle distribution function and the Liouville equation follows. The Hamiltonian form of the Navier equations attached to the \(N\)-particle system. Then the two-body problem and pair collisions are discussed in details. At the end of Chapter 2 the Boltzmann equation for mixtures is introduced and discussed. The second part of the book (Chapters 3--6) is devoted to the general theory of kinetic Maxwell models. The theory of spatially homogenous Boltzmann equation for Maxwell molecules based on Fourier transform in the velocity space-an approach due to the author of this book [Sov. Phys., Dokl. 20 (1975), 820--822 (1976; Zbl 0361.76077); translation from Dokl. Akad. Nauk SSSR 225, 1041--1044 (1975)] represents the core of this chapter. The last part of this book (Chapter 7) is about the status of ``higher'' equations of hydrodynamics obtained by the famous Chapman-Enskog expansion. The Burnett equations which appeared in 1930 as a next step after the Navier-Stokes level being ill-posed are replaced by the so called ``generalized Burnett equations'' which are derived and discussed in detail.
This book will be a very important work-tool for both physicists and mathematicians working in kinetic theory and applications.
Reviewer: Titus Petrila (Cluj-Napoca)Quasi-steady-state reduction of a model for cytoplasmic transport of secretory vesicles in stimulated chromaffin cells.https://zbmath.org/1460.920672021-06-15T18:09:00+00:00"Oelz, Dietmar B."https://zbmath.org/authors/?q=ai:oelz.dietmar-bSummary: Neurosecretory cells spatially redistribute their pool of secretory vesicles upon stimulation. Recent observations suggest that in chromaffin cells vesicles move either freely or in a directed fashion by what appears to be a conveyor belt mechanism. We suggest that this observation reflects the transient active transport through molecular motors along cytoskeleton fibres and quantify this effect using a 1D mathematical model that couples a diffusion equation to advection equations. In agreement with recent observations the model predicts that random motion dominates towards the cell centre whereas directed motion prevails in the region abutting the cortical membrane. Furthermore the model explains the observed bias of directed transport towards the periphery upon stimulation. Our model suggests that even if vesicle transport is indifferent with respect to direction, stimulation creates a gradient of free vesicles at first and this triggers the bias of transport in forward direction. Using matched asymptotic expansion we derive an approximate drift-diffusion type model that is capable of quantifying this effect. Based on this model we compute the characteristic time for the system to adapt to stimulation and we identify a Michaelis-Menten-type law describing the flux of vesicles entering the pathway to exocytosis.One-dimensional multicomponent hemodynamics.https://zbmath.org/1460.353552021-06-15T18:09:00+00:00"Mamontov, A. E."https://zbmath.org/authors/?q=ai:mamontov.alexander-e"Prokudin, D. A."https://zbmath.org/authors/?q=ai:prokudin.dmitry-alexeyevichA one-dimensional model of blood flow in an artery has been developed in many works (in particular in [\textit{J. R. Womersley}, Philos. Mag., VII. Ser. 46, 199--221 (1955; Zbl 0064.43903); \textit{J. W. Lambert}, ``On the nonlinearities of fluid flow in nonrigid tubes'', J. Franklin Inst. 266, No. 2, 83--102 (1958); \textit{T. J. R. Hughes} and \textit{J. Lubliner}, Math. Biosci. 18, 161--170 (1973; Zbl 0262.92004); \textit{D. Bessems} et al., J. Fluid Mech. 580, 145--168 (2007; Zbl 1175.76171); \textit{R. Raghu} et al., ``Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow'', J. Biomech. Eng. 133, No. 8, 5--32 (2011); \textit{G. Mulder} et al., ``Patient-specific modeling of cerebral blood flow: geometrical variations in a 1D model'', Cardiovasc. Eng. Tech. 2, 334--348 (2011)]).
Under certain assumptions, the generalization of this model to the multicomponent case is studied.
After the transition from the Euler coordinate to the mass Lagrangian coordinates, a system of differential equations is obtained, which coincides in formulation with the system of differential equations of one-dimensional polytropic flows of viscous compressible multicomponent fluid. [\textit{A. E. Mamontov} and \textit{D. A. Prokudin}, Sib. Zh. Chist. Prikl. Mat. 17, No. 2, 52--68 (2017; Zbl 1438.76030); translation in J. Math. Sci., New York 231, No. 2, 227--242 (2018)].
The problem of initial-boundary value for this system is investigated and the conditions of existence and uniqueness of the classical solution are established.
Reviewer: Yaroslav Baranetskij (Lviv)Lie symmetries methods in boundary crossing problems for diffusion processes.https://zbmath.org/1460.600912021-06-15T18:09:00+00:00"Muravey, Dmitry"https://zbmath.org/authors/?q=ai:muravey.dmitrySummary: This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker-Planck-Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries' existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.A tumor growth model of Hele-Shaw type as a gradient flow.https://zbmath.org/1460.920502021-06-15T18:09:00+00:00"Di Marino, Simone"https://zbmath.org/authors/?q=ai:di-marino.simone"Chizat, Lénaïc"https://zbmath.org/authors/?q=ai:chizat.lenaicSummary: In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by \textit{B. Perthame} et al. [Arch. Ration. Mech. Anal. 212, No. 1, 93--127 (2014; Zbl 1293.35347)] as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the evolutional variational inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.Hybrid models of chemotaxis with application to leukocyte migration.https://zbmath.org/1460.920312021-06-15T18:09:00+00:00"Lu, Hannah"https://zbmath.org/authors/?q=ai:lu.hannah"Um, Kimoon"https://zbmath.org/authors/?q=ai:um.kimoon"Tartakovsky, Daniel M."https://zbmath.org/authors/?q=ai:tartakovsky.daniel-mA model of the immune response to inflammation processes is proposed and its efficiency is studied numerically. This hybrid type model consists of PDE-chemotaxis part describing diffusion and reaction, discrete stochastic simulation of movement of leukocytes and bacteria, and computation of biochemical transformations.
Reviewer: Piotr Biler (Wrocław)Comparative analysis of continuum angiogenesis models.https://zbmath.org/1460.920322021-06-15T18:09:00+00:00"Martinson, W. Duncan"https://zbmath.org/authors/?q=ai:martinson.w-duncan"Ninomiya, Hirokazu"https://zbmath.org/authors/?q=ai:ninomiya.hirokazu"Byrne, Helen Mary"https://zbmath.org/authors/?q=ai:byrne.helen-m"Maini, Philip Kumar"https://zbmath.org/authors/?q=ai:maini.philip-kThis paper is devoted to a comparison of two models of angiogenesis, i.e. the emergence of new blood vessels. Both models are based on partial differential equations. The first is a phenomenological ``snail-trail'' model, the second ``coarse-grained'' mean field type system is derived from microscopic considerations. The comparison is performed using analytic methods (perturbation theory) and numerical simulations.
Reviewer: Piotr Biler (Wrocław)On competition models under Allee effect: asymptotic behavior and traveling waves.https://zbmath.org/1460.350292021-06-15T18:09:00+00:00"Feng, Wei"https://zbmath.org/authors/?q=ai:feng.wei"Freeze, Michael"https://zbmath.org/authors/?q=ai:freeze.michael"Lu, Xin"https://zbmath.org/authors/?q=ai:lu.xinSummary: In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (\(u\) and \(v\)). Under one-side Allee effect on \(u\)-species, the model demonstrates complexity on its coexistence and \(u\)-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant \(K\) is large relative to other biological parameters, the asymptotic stability of the \(v\)-dominance state \((0,1)\) indicates the competitive exclusion of the \(u\)-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the \(u\)-dominance states), there exist traveling wave solutions flowing from the \(u\)-dominance states to the \(v\)-dominance state. The asymptotic rates of the traveling waves at \(\xi\rightarrow\mp\infty\) are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.Backward self-similar solutions for compressible Navier-Stokes equations.https://zbmath.org/1460.352562021-06-15T18:09:00+00:00"Germain, Pierre"https://zbmath.org/authors/?q=ai:germain.pierre"Iwabuchi, Tsukasa"https://zbmath.org/authors/?q=ai:iwabuchi.tsukasa"Léger, Tristan"https://zbmath.org/authors/?q=ai:leger.tristanMathematical analysis of bump to bucket problem.https://zbmath.org/1460.352792021-06-15T18:09:00+00:00"Chen, Min"https://zbmath.org/authors/?q=ai:chen.min|chen.min.3|chen.min.1|chen.min.2"Goubet, Olivier"https://zbmath.org/authors/?q=ai:goubet.olivier"Li, Shenghao"https://zbmath.org/authors/?q=ai:li.shenghaoSummary: In this article, several systems of equations which model surface water waves generated by a sudden bottom deformation (bump) are studied. Because the effect of such deformation are often approximated by assuming the initial water surface has a deformation (bucket), this procedure is investigated and we prove rigorously that by using the correct bucket, the solutions of the regularized bump problems converge to the solution of the bucket problem.Blow-up criterion and examples of global solutions of forced Navier-Stokes equations.https://zbmath.org/1460.352672021-06-15T18:09:00+00:00"Wu, Di"https://zbmath.org/authors/?q=ai:wu.diSummary: In this paper we first show a blow-up criterion for solutions to the Navier-Stokes equations with a time-independent force by using the profile decomposition method. Based on the orthogonal properties related to the profiles, we give some examples of global solutions to the Navier-Stokes equations with a time-independent force, whose initial data are large.Numerical study of the thermodynamic uncertainty relation for the KPZ-equation.https://zbmath.org/1460.820202021-06-15T18:09:00+00:00"Niggemann, Oliver"https://zbmath.org/authors/?q=ai:niggemann.oliver"Seifert, Udo"https://zbmath.org/authors/?q=ai:seifert.udoSummary: A general framework for the field-theoretic thermodynamic uncertainty relation was recently proposed and illustrated with the \((1+1)\) dimensional Kardar-Parisi-Zhang equation. In the present paper, the analytical results obtained there in the weak coupling limit are tested via a direct numerical simulation of the KPZ equation with good agreement. The accuracy of the numerical results varies with the respective choice of discretization of the KPZ non-linearity. Whereas the numerical simulations strongly support the analytical predictions, an inherent limitation to the accuracy of the approximation to the total entropy production is found. In an analytical treatment of a generalized discretization of the KPZ non-linearity, the origin of this limitation is explained and shown to be an intrinsic property of the employed discretization scheme.The Boussinesq system revisited.https://zbmath.org/1460.352952021-06-15T18:09:00+00:00"Molinet, Luc"https://zbmath.org/authors/?q=ai:molinet.luc"Talhouk, Raafat"https://zbmath.org/authors/?q=ai:talhouk.raafat"Zaiter, Ibtissam"https://zbmath.org/authors/?q=ai:zaiter.ibtissamLow regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation.https://zbmath.org/1460.353342021-06-15T18:09:00+00:00"Seong, Kihoon"https://zbmath.org/authors/?q=ai:seong.kihoonSummary: We consider the low regularity behavior of the fourth order cubic nonlinear Schrödinger equation (4NLS)
\[
\begin{cases}
i\partial_tu+\partial_x^4u=\pm\vert u\vert^2u,\quad(t,x)\in\mathbb{R}\times\mathbb{R}\\
u(x,0)=u_0(x)\in H^s(\mathbb{R}).
\end{cases}
\]
In [``Well-posedness and ill-posedness for the fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces'', Preprint, \url{arXiv: 1911.03253}], the author showed that this equation is globally well-posed in \(H^s(\mathbb{R})\), \(s\geq-\frac{1}{2}\) and mildly ill-posed in the sense that the solution map fails to be locally uniformly continuous for \(-\frac{15}{14}<s<-\frac{1}{2}\). Therefore, \(s=-\frac{1}{2}\) is the lowest regularity that can be handled by the contraction argument. In spite of this mild ill-posedness result, we obtain an a priori bound below \(s<-1/2\). This an a priori estimate guarantees the existence of a weak solution for \(-3/4<s<-1/2\). Our method is inspired by \textit{H. Koch} and \textit{D. Tataru} [Int. Math. Res. Not. 2007, No. 16, Article ID rnm053, 36 p. (2007; Zbl 1169.35055)]. We use the \(U^p\) and \(V^p\) based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.Connecting actin polymer dynamics across multiple scales.https://zbmath.org/1460.920882021-06-15T18:09:00+00:00"Copos, Calina"https://zbmath.org/authors/?q=ai:copos.calina-a"Bannish, Brittany"https://zbmath.org/authors/?q=ai:bannish.brittany-e"Gasior, Kelsey"https://zbmath.org/authors/?q=ai:gasior.kelsey"Pinals, Rebecca L."https://zbmath.org/authors/?q=ai:pinals.rebecca-l"Rostami, Minghao W."https://zbmath.org/authors/?q=ai:rostami.minghao-w"Dawes, Adriana T."https://zbmath.org/authors/?q=ai:dawes.adriana-tSummary: Actin is an intracellular protein that constitutes a primary component of the cellular cytoskeleton and is accordingly crucial for various cell functions. Actin assembles into semi-flexible filaments that cross-link to form higher order structures within the cytoskeleton. In turn, the actin cytoskeleton regulates cell shape, and participates in cell migration and division. A variety of theoretical models have been proposed to investigate actin dynamics across distinct scales, from the stochastic nature of protein and molecular motor dynamics to the deterministic macroscopic behavior of the cytoskeleton. Yet, the relationship between molecular-level actin processes and cellular-level actin network behavior remains understudied, where prior models do not holistically bridge the two scales together.
In this work, we focus on the dynamics of the formation of a branched actin structure as observed at the leading edge of motile eukaryotic cells. We construct a minimal agent-based model for the microscale branching actin dynamics, and a deterministic partial differential equation (PDE) model for the macroscopic network growth and bulk diffusion. The microscale model is stochastic, as its dynamics are based on molecular level effects. The effective diffusion constant and reaction rates of the deterministic model are calculated from averaged simulations of the microscale model, using the mean displacement of the network front and characteristics of the actin network density. With this method, we design concrete metrics that connect phenomenological parameters in the reaction-diffusion system to the biochemical molecular rates typically measured experimentally. A parameter sensitivity analysis in the stochastic agent-based model shows that the effective diffusion and growth constants vary with branching parameters in a complementary way to ensure that the outward speed of the network remains fixed. These results suggest that perturbations to microscale rates can have significant consequences at the macroscopic level, and these should be taken into account when proposing continuum models of actin network dynamics.
For the entire collection see [Zbl 1459.92003].Pointwise asymptotic behavior of a chemotaxis model.https://zbmath.org/1460.353572021-06-15T18:09:00+00:00"Rugamba, Jean"https://zbmath.org/authors/?q=ai:rugamba.jean"Zeng, Yanni"https://zbmath.org/authors/?q=ai:zeng.yanniThe authors consider the Cauchy problem for a parabolic-hyperbolic system in one space dimension derived from the logarithmic Keller-Segel-Fisher-KPP model via the inverse Hopf-Cole transformation. The diffusive long time pointwise asymptotic behavior of solutions is proved, i.e. the solutions are shown to converge to solutions of a linear diffusion equation. A mixture of the Green function and a priori energy type estimates is used in the proof.
For the entire collection see [Zbl 1453.35003].
Reviewer: Piotr Biler (Wrocław)Approximations of stochastic 3D tamed Navier-Stokes equations.https://zbmath.org/1460.352642021-06-15T18:09:00+00:00"Peng, Xuhui"https://zbmath.org/authors/?q=ai:peng.xuhui"Zhang, Rangrang"https://zbmath.org/authors/?q=ai:zhang.rangrangSummary: In this paper, we are concerned with 3D tamed Navier-Stokes equations with periodic boundary conditions, which can be viewed as an approximation of the classical 3D Navier-Stokes equations. We show that the strong solution of 3D tamed Navier-Stokes equations driven by Poisson random measure converges weakly to the strong solution of 3D tamed Navier-Stokes equations driven by Gaussian noise on the state space \(\mathcal{D}([0,T];\mathbb{H}^1)\).The spreading speed of an SIR epidemic model with nonlocal dispersal.https://zbmath.org/1460.353522021-06-15T18:09:00+00:00"Guo, Jong-Shenq"https://zbmath.org/authors/?q=ai:guo.jong-shenq"Poh, Amy Ai Ling"https://zbmath.org/authors/?q=ai:poh.amy-ai-ling"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahikoSummary: In this paper, we study an SIR epidemic model with nonlocal dispersal. We study the case with vital dynamics so that a renewal of the susceptible individuals is taken into account. We characterize the asymptotic spreading speed to estimate how fast the disease under consideration spreads. Due to the lack of comparison principle for the SIR model, our proof is based on a delicate analysis of related problems with nonlocal scalar equations.An improved regularity criterion for the 3D magneto-micropolar equations in homogeneous Besov space.https://zbmath.org/1460.769432021-06-15T18:09:00+00:00"Zhang, Panpan"https://zbmath.org/authors/?q=ai:zhang.panpan"Yuan, Baoquan"https://zbmath.org/authors/?q=ai:yuan.baoquanSummary: In this paper we consider the regularity criterion for the weak solutions to the 3D incompressible magneto-micropolar equations. We establish an improved regularity criterion for weak solutions in terms of two pairs of \(( \partial_i u_i, \partial_i b_i)\) (\(i = 1, 2, 3\)). More precisely, we show that for some \(i, j \in \{1, 2, 3 \}\) with \(i \neq j\), if
\[
( \partial_i u_i, \partial_i b_i),( \partial_j u_j, \partial_j b_j) \in L^p(0, T; \dot{B}_{q , \infty}^0( \mathbb{R}^3)), \quad \frac{ 2}{ p} + \frac{ 3}{ q} = 2, \quad 3 \leq q \leq \infty,
\]
then the solution \((u, w, b)\) to the magneto-micropolar equations is smooth on \([0, T]\).Construction of a solitary wave solution of the nonlinear focusing Schrödinger equation outside a strictly convex obstacle in the \(L^2\)-supercritical case.https://zbmath.org/1460.353292021-06-15T18:09:00+00:00"Landoulsi, Oussama"https://zbmath.org/authors/?q=ai:landoulsi.oussamaSummary: We consider the focusing \(L^2\)-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle \(\Theta\subset\mathbb{R}^3\). We construct a solution behaving asymptotically as a solitary wave on \(\mathbb{R}^3,\) for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by \textit{F. Merle} [Commun. Math. Phys. 129, No. 2, 223--240 (1990; Zbl 0707.35021)] to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of \textit{R. Killip} et al. [AMRX, Appl. Math. Res. Express 2016, No. 1, 146--180 (2016; Zbl 1345.35102)], which is the same as the one on the whole Euclidean space given by \textit{T. Duyckaerts} et al. [Math. Res. Lett. 15, No. 5--6, 1233--1250 (2008; Zbl 1171.35472)] and \textit{J. Holmer} and \textit{S. Roudenko} [Commun. Math. Phys. 282, No. 2, 435--467 (2008; Zbl 1155.35094)] in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.Modeling rabies transmission in spatially heterogeneous environments via \(\theta \)-diffusion.https://zbmath.org/1460.922242021-06-15T18:09:00+00:00"Wang, Xiunan"https://zbmath.org/authors/?q=ai:wang.xiunan"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4"Li, Michael Y."https://zbmath.org/authors/?q=ai:li.michael-yiThis paper studies a dog rabies model where the diffusion of dog population can be captured by the \(\theta\)-diffusion equation. Here, \(\theta\) describes the way each individual dog makes movement decisions in the random walk. The movements of infectious and non-infectious dogs are not the same. Infectious dogs have destroyed central nervous systems and they move randomly. Susceptible dogs will make their movement based on the conditions at the locations. It is assumed that the recruitment of new susceptible dogs may result from both new borns and the importation of dogs from outside. Susceptible and exposed dogs are vaccinated at the same rake. Moreover, if a susceptible dog is bitten by an infectious god, it will enter the exposed compartment and the risk factor of it to develop clinical rabies is introduced. The paper derives the basic reproduction number and numerically studied the dynamics of the system with homogeneous, city-wild and Gaussian-type diffusions.
Reviewer: Yilun Shang (Newcastle)Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth.https://zbmath.org/1460.740612021-06-15T18:09:00+00:00"Zhou, Jun"https://zbmath.org/authors/?q=ai:zhou.jun.1Summary: In this paper, we consider a class of fourth-order parabolic equation modeling epitaxial growth. For the evolution problem, we study the exponential decay of the solutions and its corresponding energy functional. For the stationary problem, we find a ground-state solution by Lagrange multiplier method. Moreover, the asymptotical behavior of the general global solution is also considered. The results of this paper extend some results got by \textit{Y. Han} [Nonlinear Anal., Real World Appl. 43, 451--466 (2018; Zbl 06892735)].Global dynamics and complex patterns in Lotka-Volterra systems: the effects of both local and nonlocal intraspecific and interspecific competitions.https://zbmath.org/1460.921642021-06-15T18:09:00+00:00"Chen, Xianyong"https://zbmath.org/authors/?q=ai:chen.xianyong"Jiang, Weihua"https://zbmath.org/authors/?q=ai:jiang.weihua.1"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: We consider a Lotka-Volterra system with both local and nonlocal intraspecific and interspecific competitions, where nonlocal competitions depend on both spatial and temporal effects in a general form. Firstly, global stability of two constant semi-trivial equilibria and global convergence of the coexistence equilibrium are derived by using the functional and energy method, which implies that strengths of nonlocal intraspecific competitions have great effects on these global dynamics but the nonlocal interspecific competitions not and extends global results of \textit{S. A. Gourley} and \textit{S. Ruan} [SIAM J. Math. Anal. 35, No. 3, 806--822 (2003; Zbl 1040.92045)]. Secondly, global attracting region of each constant semi-trivial equilibrium is limited by its environment capacity regardless of the distinction of local and nonlocal intraspecific competitions. Thirdly, in the weak competition case, the coexistence equilibrium becomes Turing unstable when the kernels are chosen as generally distributed delay functions in temporal and the nonlocal intraspecific competitions are suitably strong. Additionally, spatially homogeneous and inhomogeneous periodic solutions are found numerically.A central limit theorem for Gibbsian invariant measures of 2D Euler equations.https://zbmath.org/1460.601142021-06-15T18:09:00+00:00"Grotto, Francesco"https://zbmath.org/authors/?q=ai:grotto.francesco"Romito, Marco"https://zbmath.org/authors/?q=ai:romito.marcoSummary: We consider canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of \(\mathbb{R}^2\). We prove that under the central limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called energy-enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles of \textit{E. Caglioti} et al. [Commun. Math. Phys. 143, No. 3, 501--525 (1992; Zbl 0745.76001)] and \textit{M. K. H. Kiessling} and \textit{Y. Wang} [J. Stat. Phys. 148, No. 5, 896--932 (2012; Zbl 1263.82055)], and it generalises the result on fluctuations of \textit{T. Bodineau} and \textit{A. Guionnet} [Ann. Inst. Henri Poincaré, Probab. Stat. 35, No. 2, 205--237 (1999; Zbl 0920.60095)]. The main argument consists in proving convergence of partition functions of vortices.Property of the large densities in a two-species and two-stimuli chemotaxis system with competitive kinetics.https://zbmath.org/1460.920342021-06-15T18:09:00+00:00"Yang, Hongying"https://zbmath.org/authors/?q=ai:yang.hongying"Tu, Xinyu"https://zbmath.org/authors/?q=ai:tu.xinyu"Mu, Chunlai"https://zbmath.org/authors/?q=ai:mu.chunlaiThe authors consider a system of four parabolic equations describing interaction of two species via chemotactic influence of two chemicals and reactive terms of competitive type. The initial-boundary value problem in balls of \({\mathbb R}^n\), \(n\ge 3\), is shown to possess solutions of unbounded densities for ``small'' initial data. Energy arguments are used. However, finite time blow-up phenomena have not been detected.
Reviewer: Piotr Biler (Wrocław)Interior jump and contact singularity for compressible flows with inflow jump datum.https://zbmath.org/1460.352572021-06-15T18:09:00+00:00"Han, Joo Hyeong"https://zbmath.org/authors/?q=ai:han.joo-hyeong"Kweon, Jae Ryong"https://zbmath.org/authors/?q=ai:kweon.jae-ryongThis paper is concerning some discontinuous solutions for compressible viscous Navier-Stokes equations in a rectangle domain in \(\mathbb R^2\). A jump curve exists which meets some points of the boundary and brings us contact singularities. A new interesting procedure is used to manage pressure gradient in the momentum equation and the singularities appearing at the contact points where the interface curve meets with the boundary. This is based on a lifting mapping of the jump curve into a subregion of the domain. The existence of a unique solution is obtained, when the viscosity is large enough. For this, the specific decomposition (1.12) of the velocity filed is used, obtained by an iterative scheme. The convergence procedure is carefully analysed in section 4. An important point is to get the Rankine-Hugoniot jump conditions with the solution of the initial nonlinear problem by using the decomposition (1.12). Some specific Sobolev spaces and some results given in [\textit{G. H. Hardy} et al., Inequalities. 2nd ed. Cambridge: At the University Press (1952; Zbl 0047.05302)] are used.
Reviewer: Gelu Paşa (Bucureşti)Evolution equations for a wide range of Einstein-matter systems.https://zbmath.org/1460.830092021-06-15T18:09:00+00:00"Normann, M."https://zbmath.org/authors/?q=ai:normann.m"Valiente Kroon, J. A."https://zbmath.org/authors/?q=ai:valiente-kroon.juan-antonioSummary: We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and perfect fluid. We also provide a discussion of self-gravitating elastic matter. The frame is Fermi-Walker propagated and coordinates are chosen such as to satisfy the Lagrange condition. We show the propagation of the constraints of the Einstein-matter system.Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint.https://zbmath.org/1460.352802021-06-15T18:09:00+00:00"Druet, Pierre-Etienne"https://zbmath.org/authors/?q=ai:druet.pierre-etienneSummary: We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of \(N > 1\) chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible \textit{mixtures} is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in \(L^1\). In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density.https://zbmath.org/1460.351242021-06-15T18:09:00+00:00"Mercuri, Carlo"https://zbmath.org/authors/?q=ai:mercuri.carlo"Tyler, Teresa Megan"https://zbmath.org/authors/?q=ai:tyler.teresa-meganSummary: In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system
\[\begin{cases}
- \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, & x\in \mathbb{R}^3, \\
-\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3,
\end{cases}\]
under different assumptions on \(\rho\colon \mathbb{R}^3\rightarrow \mathbb{R}_+\) at infinity. Our results cover the range \(p\in(2,3)\) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a `limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem
\[\begin{cases}
- \epsilon^2\Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, & x\in \mathbb{R}^3, \\
-\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3,
\end{cases}\]
in various functional settings which are suitable for both variational and perturbation methods.Hilbert type problem for a Cauchy-Riemann equation with singularities on a circle and at a point in the lower-order coefficients.https://zbmath.org/1460.352482021-06-15T18:09:00+00:00"Fedorov, Yu. S."https://zbmath.org/authors/?q=ai:fedorov.yury-sergeevich"Rasulov, A. B."https://zbmath.org/authors/?q=ai:rasulov.abdurauf-babadzhanovichSummary: A Hilbert type problem is solved for a generalized Cauchy-Riemann system whose lower-order coefficients admit a strong singularity on a circle and a weak singularity at a point.Optimal feedback control for a model of motion of a nonlinearly viscous fluid.https://zbmath.org/1460.353042021-06-15T18:09:00+00:00"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich|zvyagin.victor-g"Zvyagin, A. V."https://zbmath.org/authors/?q=ai:zvyagin.alexander-v|zvyagin.andrey-v"Nguyen Minh Hong"https://zbmath.org/authors/?q=ai:nguyen-minh-hong.Summary: We consider an optimal feedback control problem for an initial-boundary value problem describing the motion of a nonlinearly viscous fluid. We prove the existence of an optimal solution minimizing a given performance functional. To prove the existence of an optimal solution, we use a topological approximation method for studying hydrodynamic problems.An existence and uniqueness theorem for the Navier-Stokes equations in dimension four.https://zbmath.org/1460.761622021-06-15T18:09:00+00:00"Coscia, Vincenzo"https://zbmath.org/authors/?q=ai:coscia.vincenzoSummary: We prove that the steady state Navier-Stokes equations have a solution in an exterior Lipschitz domain of \(\mathbb{R}^4\), vanishing at infinity, provided the boundary datum belongs to \(L^3(\partial\Omega)\).An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity.https://zbmath.org/1460.352622021-06-15T18:09:00+00:00"Mohan, M. T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: In this work, we consider the three dimensional Navier-Stokes equations on the whole space with a hereditary viscous term which depends on the past history. We study a blow-up criterion of smooth solutions to such systems. The existence and uniqueness of smooth solution is proved via a frequency truncation method. We also give the example of Maxwell's fluid flow equations, which is a linear viscoelastic fluid flow model.Uniqueness of dissipative solutions to the complete Euler system.https://zbmath.org/1460.352712021-06-15T18:09:00+00:00"Ghoshal, Shyam Sundar"https://zbmath.org/authors/?q=ai:ghoshal.shyam-sundar"Jana, Animesh"https://zbmath.org/authors/?q=ai:jana.animeshSummary: Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from \textit{E. Feireisl} et al. [Commun. Partial Differ. Equations 44, No. 12, 1285--1298 (2019; Zbl 1428.35325)], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space \(B^{\alpha ,\infty}_p\) with \(\alpha >1/2\). We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, \(B^{\alpha,\infty}_3\) for \(\alpha >1/3\). Our proof relies on commutator estimates for Besov functions and the relative entropy method.A space-time spectral method for multi-dimensional Sobolev equations.https://zbmath.org/1460.352972021-06-15T18:09:00+00:00"Tang, Siqin"https://zbmath.org/authors/?q=ai:tang.siqin"Li, Hong"https://zbmath.org/authors/?q=ai:li.hong"Yin, Baoli"https://zbmath.org/authors/?q=ai:yin.baoliSummary: In this paper, a space-time spectral method is applied to approximate the linear multi-dimensional Sobolev equations. The Legendre-Galerkin method and a dual-Petrov-Galerkin discretization are employed in space and time, respectively. Being different from the general Legendre-Galerkin method, the technique adopted in this study lies in diagonalization of the mass matrix by using Fourier-like basis functions to save the computing time and memory. Moreover, the detailed stability analysis and error estimates are provided in the weighted space-time norms. We also formulate the matrix forms of the fully discrete scheme. Finally, extensive numerical tests are implemented to verify the theoretical results and demonstrate the efficiency of our method.Reduction of a damped, driven Klein-Gordon equation into a discrete nonlinear Schrödinger equation: justification and numerical comparison.https://zbmath.org/1460.353302021-06-15T18:09:00+00:00"Muda, Yuslenita"https://zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, Fiki T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, Rudy"https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, Bobby E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, Hadi"https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a discrete nonlinear Klein-Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces.https://zbmath.org/1460.353272021-06-15T18:09:00+00:00"Hirayama, Hiroyuki"https://zbmath.org/authors/?q=ai:hirayama.hiroyuki"Kinoshita, Shinya"https://zbmath.org/authors/?q=ai:kinoshita.shinya"Okamoto, Mamoru"https://zbmath.org/authors/?q=ai:okamoto.mamoruSummary: In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by \textit{M. Colin} and \textit{T. Colin} [Differ. Integral Equ. 17, No. 3--4, 297--330 (2004; Zbl 1174.35528)]. We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, except for the scaling critical case. This result covers a gap left open in [the first author, Commun. Pure Appl. Anal. 13, No. 4, 1563--1591 (2014; Zbl 1294.35139); with the second author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 178, 205--226 (2019; Zbl 1406.35357)].Control of an axially moving viscoelastic Kirchhoff string.https://zbmath.org/1460.740682021-06-15T18:09:00+00:00"Kelleche, Abdelkarim"https://zbmath.org/authors/?q=ai:kelleche.abdelkarim"Tatar, Nasser-eddine"https://zbmath.org/authors/?q=ai:tatar.nasser-eddineSummary: The control problem of axially moving strings occurs in a large class of mechanical systems. In addition to the longitudinal displacement, the strings are subject to undesirable transversal vibrations. In this work, in order to suppress these vibrations, we consider a control by a hydraulic touch-roll actuator at the right boundary. We prove uniform stability of the system using a viscoelastic material and an appropriate boundary control force applied to the touch rolls of the actuator. The features of the present work are: taking into account the mass flow entering in and out at the boundaries due to the axial movement of the string and overcoming the difficulty raised by the Kirchhoff coefficient which does not allow us to profit from the dissipativity of the system (as in the existing works so far). We shall make use of an inequality which is new in this theory.Reduced equations for the hydroelastic waves in the cochlea: the membrane model.https://zbmath.org/1460.353562021-06-15T18:09:00+00:00"Rubinstein, Jacob"https://zbmath.org/authors/?q=ai:rubinstein.jacob"Sternberg, Peter"https://zbmath.org/authors/?q=ai:sternberg.peter-jSummary: We consider the hydroelastic waves in the cochlea for the case where the elastic partition between the \textit{scala media} and the \textit{scala timpani} is modeled as an elastic membrane. Specifically the cochlea is modeled as an elongated box in the \(x\) direction, with the membrane dividing the cochlea into two fluid-filled chambers. Since the basilar membrane vibrates in the present model in the orthogonal \(y\) direction, it is not possible to reduce the problem to a one-dimension equation in the \(x\) variable. Instead, we provide a rigorous reduction of the fluid-membrane coupled system of three-dimensional partial differential equations to an ordinary differential equation in the lateral \(y\) direction for the membrane and a second ordinary differential equation in \(x\) for the fluid pressure, where the first equation does not depend on the second.Normalized solutions for 3-coupled nonlinear Schrödinger equations.https://zbmath.org/1460.350932021-06-15T18:09:00+00:00"Liu, Chuangye"https://zbmath.org/authors/?q=ai:liu.chuangye"Tian, Rushun"https://zbmath.org/authors/?q=ai:tian.rushunSummary: In this paper, we study the existence of \(L^2\)-normalized solutions for the following 3-coupled nonlinear Schrödinger equations in \([H_r^1(\mathbb{R}^N)]^3\),
\[
\begin{cases}
-\Delta u_i=\lambda_iu_i+\mu_i|u_i|^{p_i-2}u_i+\beta r_i|u_i|^{r_i-2}\big(\sum\limits_{j\neq i}|u_j|^{r_j}\big)u_i,\\
|u_i|_2^2=a_i, \quad i,j=1,2,3,
\end{cases}
\]
where \(\mu_i,\beta\) and \(a_i\) are given positive constants, \(\lambda_i\) appear as unknown parameters, and \(H_r^1(\mathbb{R}^N)\) denotes the radial subspace of Hilbert space \(H^1(\mathbb{R}^N)\). For \(p_i, r_i\) satisfying \(L^2\)-subcritical or \(L^2\)-supercritical conditions, we obtain positive solutions of this system using variational methods and perturbation methods.Super compact equation for water waves.https://zbmath.org/1460.760922021-06-15T18:09:00+00:00"Dyachenko, A. I."https://zbmath.org/authors/?q=ai:dyachenko.aleksandr-ivanovich"Kachulin, D. I."https://zbmath.org/authors/?q=ai:kachulin.dmitriy-i"Zakharov, V. E."https://zbmath.org/authors/?q=ai:zakharov.vladimir-eSummary: Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the `miraculous' cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the `super compact water wave equation'. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the \textit{K. B. Dysthe}'s equations [Proc. R. Soc. Lond., Ser. A 369, 105--114 (1979; Zbl 0429.76014)] can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.Existence of a stationary Navier-Stokes flow past a rigid body, with application to starting problem in higher dimensions.https://zbmath.org/1460.352652021-06-15T18:09:00+00:00"Takahashi, Tomoki"https://zbmath.org/authors/?q=ai:takahashi.tomokiSummary: We consider the large time behavior of the Navier-Stokes flow past a rigid body in \(\mathbb{R}^n\) with \(n\geq 3\). We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time \(t\rightarrow \infty\) in \(L^q\) with \(q\in [n,\infty]\). This is called Finn's starting problem and the three-dimensional case was affirmatively solved by \textit{G. P. Galdi} et al. [Arch. Ration. Mech. Anal. 138, No. 4, 307--318 (1997; Zbl 0898.35071)]. The present paper extends the latter cited paper to the case of higher dimensions. Even for the three-dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.The two-phase Navier-Stokes equations with surface tension in cylindrical domains.https://zbmath.org/1460.352662021-06-15T18:09:00+00:00"Wilke, Mathias"https://zbmath.org/authors/?q=ai:wilke.mathiasSummary: This article is concerned with the well-posedness of a model for the dynamics of two immiscible and incompressible fluids in cylindrical domains, which are separated by a sharp interface, forming a contact angle with the solid wall of the container. We prove that the nonlinear system has a unique strong global solution in the \(L_p\)-sense, provided that the initial data is small. To this end, we show maximal \(L_p\)-regularity of the linearized problem and apply the contraction mapping principle in order to solve the nonlinear problem.Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier-Stokes systems.https://zbmath.org/1460.766612021-06-15T18:09:00+00:00"Kang, Moon-Jin"https://zbmath.org/authors/?q=ai:kang.moon-jin"Vasseur, Alexis F."https://zbmath.org/authors/?q=ai:vasseur.alexis-fSummary: We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier-Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier-Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exist a vanishing viscosity limit to solutions of the Navier-Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations.Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity.https://zbmath.org/1460.352902021-06-15T18:09:00+00:00"Kukavica, Igor"https://zbmath.org/authors/?q=ai:kukavica.igor"Wang, Weinan"https://zbmath.org/authors/?q=ai:wang.weinanSummary: We address the persistence of regularity for the 2D \(\alpha\)-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for \((u_0,\rho_0)\in W^{s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)\), where \(s>1\) and \(q\in(2,\infty)\). We prove that the solution \((u(t),\rho(t))\) exists and belongs to \(W^{s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)\) for all positive time \(t\) for \(q>2\), where \(\alpha\in (1,2)\) is arbitrary.On linear theory of thermoelasticity for an anisotropic medium under a recent exact heat conduction model.https://zbmath.org/1460.740212021-06-15T18:09:00+00:00"Gupta, Manushi"https://zbmath.org/authors/?q=ai:gupta.manushi"Mukhopadhyay, Santwana"https://zbmath.org/authors/?q=ai:mukhopadhyay.santwanaSummary: The aim of this paper is to discuss about a new thermoelasticity theory for a homogeneous and anisotropic medium in the context of a recent heat conduction model proposed by \textit{R. Quintanilla} [Mech. Res. Commun. 38, No. 5, 355--360 (2011; Zbl 1272.80007)]. The coupled thermoelasticity being the branch of science that deals with the mutual interactions between temperature and strain in an elastic medium had become the interest of researchers since 1956. In [loc. cit.], the author introduced a new model of heat conduction in order to reformulate the heat conduction law with three phase-lags and established mathematical consistency in this new model as compared to the three phase-lag model. This model has also been extended to thermoelasticity theory. Various Taylor's expansion of this model has gained the interest of many researchers in recent times. Hence, we considered the model's backward time expansion of Taylor's series up to second-order and establish some important theorems. Firstly, uniqueness theorem of a mixed type boundary and initial value problem is proved using specific internal energy function. Later, we give the alternative formulation of the problem using convolution which incorporates the initial conditions into the field equations. Using this formulation, the convolution type variational theorem is proved. Further, we establish a reciprocal relation for the model.
For the entire collection see [Zbl 1411.65006].A uniqueness result for 3D incompressible fluid-rigid body interaction problem.https://zbmath.org/1460.352962021-06-15T18:09:00+00:00"Muha, Boris"https://zbmath.org/authors/?q=ai:muha.boris"Nečasová, Šárka"https://zbmath.org/authors/?q=ai:necasova.sarka"Radošević, Ana"https://zbmath.org/authors/?q=ai:radosevic.anaSummary: We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy the Prodi-Serrin \(\mathrm{L}^r-\mathrm{L}^s\) condition are unique in the class of Leray-Hopf weak solutions.Global generalized solutions for a class of chemotaxis-consumption systems with generalized logistic source.https://zbmath.org/1460.353542021-06-15T18:09:00+00:00"Lyu, Wenbin"https://zbmath.org/authors/?q=ai:lyu.wenbinThe author studies a model of chemotaxis with chemoattractant consumption, a general smooth chemotactic sensitivity tensor function and a general logistic type reaction term. Under suitable assumptions the initial boundary value problem in bounded domains of \(\mathbb{R}^n\) is shown to be well posed globally in time so that finite time blowup cannot occur for those systems.
Reviewer: Piotr Biler (Wrocław)Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations.https://zbmath.org/1460.353062021-06-15T18:09:00+00:00"Almushaira, Mustafa"https://zbmath.org/authors/?q=ai:almushaira.mustafa"Liu, Fei"https://zbmath.org/authors/?q=ai:liu.fei|liu.fei.2|liu.fei.1Summary: In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schrödinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time.Development of rigorous methods in fluid mechanics and theory of water waves. (Abstract of thesis).https://zbmath.org/1460.760932021-06-15T18:09:00+00:00"Ermakov, Andrei"https://zbmath.org/authors/?q=ai:ermakov.andrei-m(no abstract)Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow.https://zbmath.org/1460.350682021-06-15T18:09:00+00:00"Tsuge, Naoki"https://zbmath.org/authors/?q=ai:tsuge.naokiThe auhtor studies the one-dimensional non-steady isentropic compressible Euler flow in a nozzle. The nozzle is infinitely long and described by an \(x\)-dependent cross-section function in the equations. Equations are written in terms of density and momentum. The gas is barotropic, so that pressure is a power function of density with the adiabatic exponent is from [1,5/3]. The Cauchy problem for arbitrarily large initial data is studied. The aim is to establish the global existence of the entropy solution with sonic state.
For the entire collection see [Zbl 1453.35003].
Reviewer: Ilya A. Chernov (Petrozavodsk)Multiple solutions for a class of quasilinear Schrödinger equations.https://zbmath.org/1460.351022021-06-15T18:09:00+00:00"Huang, Chen"https://zbmath.org/authors/?q=ai:huang.chen"Jia, Gao"https://zbmath.org/authors/?q=ai:jia.gaoSummary: In this paper, we investigate the quasilinear Schrödinger elliptic equations with infinitely many solutions:
\[
-\Delta u+V(x)u+\tau\Delta(\sqrt{1+u^2})\frac{u}{2\sqrt{1+u^2}}=W(x)f(x,u),\quad x\in\mathbb{R}^N,
\]
where \(N\geq 3\), \(\tau\geq 2\). It is assumed that the nonlinearity \(f(x,t)\) is sublinear in a neighbourhood of \(t=0\). Some techniques are used here, such as variational methods, \(Z_2\) genus and the Moser iteration.Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system.https://zbmath.org/1460.921632021-06-15T18:09:00+00:00"Chen, Mengxin"https://zbmath.org/authors/?q=ai:chen.mengxin"Wu, Ranchao"https://zbmath.org/authors/?q=ai:wu.ranchao"Chen, Liping"https://zbmath.org/authors/?q=ai:chen.lipingSummary: The Turing and Turing-Hopf bifurcations of a Leslie-Gower type predator-prey system with ratio-dependent Holling III functional response are investigated in this paper. Complex and interesting patterns induced by the bifurcations are identified theoretically and numerically. First the existence conditions of the Turing instability and the Turing-Hopf bifurcation are established from the theoretical analysis, respectively. Then by employing the technique of weakly nonlinear analysis, amplitude equations generated near the Turing instability critical value are derived. Various spatiotemporal patterns, such as homogeneous stationary state patterns, hexagonal patterns, coexisting patterns, stripe patterns, and their stability are determined via analyzing the obtained amplitude equations. Numerical simulations are presented to illustrate the theoretical analysis. Especially, the analogous-spiral and symmetrical wave patterns can be found near the codimension-two Turing-Hopf bifurcation point. A security center of the prey species can be found as well. These spatiotemporal patterns are explained from the perspective of the predators and prey species.Nonlinear stability of periodic-wave solutions for systems of dispersive equations.https://zbmath.org/1460.350282021-06-15T18:09:00+00:00"Cristófani, Fabrício"https://zbmath.org/authors/?q=ai:cristofani.fabricio"Pastor, Ademir"https://zbmath.org/authors/?q=ai:pastor.ademirSummary: We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.Non-linear bi-harmonic Choquard equations.https://zbmath.org/1460.353332021-06-15T18:09:00+00:00"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: This note studies the fourth-order Choquard equation
\[
i\dot u+\Delta^2 u\pm(I_\alpha *|u|^p)|u|^{p-2}u=0.
\]
In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible magnetohydrodynamic equations.https://zbmath.org/1460.352982021-06-15T18:09:00+00:00"Wang, Yongfu"https://zbmath.org/authors/?q=ai:wang.yongfuSummary: This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding \textit{L. Du} and the author [Nonlinearity 28, No. 8, 2959--2976 (2015; Zbl 1326.35263)] to bounded domain in \(\mathbb{R}^2\) when the initial density and the initial magnetic field are decay not too show at infinity, and \textit{R. Ji} and the author [Discrete Contin. Dyn. Syst. 39, No. 2, 1117--1133 (2019; Zbl 1404.35357)] to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.Periodic solutions of an age-structured epidemic model with periodic infection rate.https://zbmath.org/1460.353532021-06-15T18:09:00+00:00"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Huang, Qimin"https://zbmath.org/authors/?q=ai:huang.qimin"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.Pressure-dependent viscosity model for granular media obtained from compressible Navier-Stokes equations.https://zbmath.org/1460.768762021-06-15T18:09:00+00:00"Perrin, Charlotte"https://zbmath.org/authors/?q=ai:perrin.charlotteSummary: The aim of this article is to justify mathematically, in the two-dimensional periodic setting, a generalization of a two-phase model with pressure-dependent viscosity and memory effects first proposed by \textit{A. Lefebvre-Lepot} and \textit{B. Maury} [Adv. Math. Sci. Appl. 21, No. 2, 535--557 (2011; Zbl 1329.35136)] to describe a one-dimensional system of aligned spheres interacting through lubrication forces. This model involves an adhesion potential apparent only on the congested domain, which keeps track of history of the flow. The solutions are constructed (through a singular limit) from a compressible Navier-Stokes system with viscosity and pressure both singular close to a maximal volume fraction. Interestingly, this study can be seen as the first mathematical connection between incompressible models of granular flows and compressible models of suspension flows. As a by-product of this result, we also obtain global existence of weak solutions for a system of incompressible Navier-Stokes equations with pressure-dependent viscosity, the adhesion potential playing a crucial role in this result.Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density.https://zbmath.org/1460.353022021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinA posteriori error estimates for self-similar solutions to the Euler equations.https://zbmath.org/1460.352682021-06-15T18:09:00+00:00"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Shen, Wen"https://zbmath.org/authors/?q=ai:shen.wenSummary: The main goal of this paper is to analyze a family of ``simplest possible'' initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.Pseudo-transient ghost fluid methods for the Poisson-Boltzmann equation with a two-component regularization.https://zbmath.org/1460.920872021-06-15T18:09:00+00:00"Ahmed Ullah, Sheik"https://zbmath.org/authors/?q=ai:ullah.sheik-ahmed"Zhao, Shan"https://zbmath.org/authors/?q=ai:zhao.shanSummary: The Poisson-Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its exponential nonlinear term, strong singularity by the source terms, and distinct dielectric regions. In this paper, a new pseudo-transient approach is proposed, which combines an analytical treatment of singular charges in a two-component regularization, with an analytical integration of nonlinear term in pseudo-time solution. To ensure efficiency, both fully implicit alternating direction implicit (ADI) and unconditionally stable locally one-dimensional (LOD) methods have been constructed to decompose three-dimensional linear systems into one-dimensional (1D) ones in each pseudo-time step. Moreover, to accommodate the nonzero function and flux jumps across the dielectric interface, a modified ghost fluid method (GFM) has been introduced as a first order accurate sharp interface method in 1D style, which minimizes the information needed for the molecular surface. The 1D finite-difference matrix generated by the GFM is symmetric and diagonally dominant, so that the stability of ADI and LOD methods is boosted. The proposed pseudo-transient GFM schemes have been numerically validated by calculating solvation free energy, binding energy, and salt effect of various proteins. It has been found that with the augmentation of regularization and GFM interface treatment, the ADI method not only enhances the accuracy dramatically, but also improves the stability significantly. By using a large time increment, an efficient protein simulation can be realized in steady-state solutions. Therefore, the proposed GFM-ADI and GFM-LOD methods provide accurate, stable, and efficient tools for biomolecular simulations.Long-time Anderson localization for the nonlinear Schrödinger equation revisited.https://zbmath.org/1460.820062021-06-15T18:09:00+00:00"Cong, Hongzi"https://zbmath.org/authors/?q=ai:cong.hongzi"Shi, Yunfeng"https://zbmath.org/authors/?q=ai:shi.yunfeng"Zhang, Zhifei"https://zbmath.org/authors/?q=ai:zhang.zhifei.1|zhang.zhifeiSummary: In this paper, we confirm the conjecture of \textit{W. M. Wang} and \textit{Z. Zhang} [ibid. 134, No. 5--6, 953--968 (2009; Zbl 1193.82022)] in a long time scale, i.e., the displacement of the wavefront for \(1D\) nonlinear random Schrödinger equation is of logarithmic order in time \(|t|\).Generalized solutions to models of compressible viscous fluids.https://zbmath.org/1460.352742021-06-15T18:09:00+00:00"Abbatiello, Anna"https://zbmath.org/authors/?q=ai:abbatiello.anna"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Novotný, Antoní"https://zbmath.org/authors/?q=ai:novotny.antoninSummary: We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak-strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non-linear viscosity tensor and non-homogeneous boundary conditions, for which the problem of existence of global-in-time weak/strong solutions is largely open.Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part II: Stochastic Hopf bifurcation.https://zbmath.org/1460.600562021-06-15T18:09:00+00:00"Tantet, Alexis"https://zbmath.org/authors/?q=ai:tantet.alexis"Chekroun, Mickaël D."https://zbmath.org/authors/?q=ai:chekroun.mickael-d"Dijkstra, Henk A."https://zbmath.org/authors/?q=ai:dijkstra.henk-a"Neelin, J. David"https://zbmath.org/authors/?q=ai:neelin.j-davidSummary: The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution [\textit{M. D. Chekroun} et al., ibid. 179, No. 5--6, 1366--1402 (2020; Zbl 1460.60050)]. Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I [loc. cit.]. This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in [loc. cit.] is applied to a stochastic model relevant to climate dynamics in the third part of this contribution [the authors, ibid. 179, No. 5--6, 1449--1474 (2020; Zbl 1460.60057)].Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part I: Theory.https://zbmath.org/1460.600502021-06-15T18:09:00+00:00"Chekroun, Mickaël D."https://zbmath.org/authors/?q=ai:chekroun.mickael-d"Tantet, Alexis"https://zbmath.org/authors/?q=ai:tantet.alexis"Dijkstra, Henk A."https://zbmath.org/authors/?q=ai:dijkstra.henk-a"Neelin, J. David"https://zbmath.org/authors/?q=ai:neelin.j-davidSummary: A theory of Ruelle-Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space \(V\). These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space \(V\), and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in \(V\), is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in \(V\), i.e. the optimal reduced system in \(V\) obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II [\textit{A. Tantet} et al., ibid. 179, No. 5--6, 1403--1448 (2020; Zbl 1460.60056)] and Part III [\textit{A. Tantet} et al., ibid. 179, No. 5--6, 1449--1474 (2020; Zbl 1460.60057)], deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous ``signature'' of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.Optimal decay to the non-isentropic compressible micropolar fluids.https://zbmath.org/1460.352922021-06-15T18:09:00+00:00"liu, Lvqiao"https://zbmath.org/authors/?q=ai:liu.lvqiao"Zhang, Lan"https://zbmath.org/authors/?q=ai:zhang.lanSummary: In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate \((1+t)^{-3/4}\) in \(L^2\) norm and the micro-rotational velocity tends to the equilibrium state with the faster rate \((1+t)^{-5/4}\) in \(L^2\) norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential.https://zbmath.org/1460.350442021-06-15T18:09:00+00:00"Alouini, Brahim"https://zbmath.org/authors/?q=ai:alouini.brahimSummary: We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system.https://zbmath.org/1460.352852021-06-15T18:09:00+00:00"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin|han.bin.1"Zhao, Na"https://zbmath.org/authors/?q=ai:zhao.naSummary: In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [the authors, Nonlinear Anal., Real World Appl. 51, Article ID 103000, 26 p. (2020; Zbl 1430.35035)], in which the magnetic vector field is bounded in critical Sobolev spaces.Contraction property for large perturbations of shocks of the barotropic Navier-Stokes system.https://zbmath.org/1460.766142021-06-15T18:09:00+00:00"Kang, Moon-Jin"https://zbmath.org/authors/?q=ai:kang.moon-jin"Vasseur, Alexis F."https://zbmath.org/authors/?q=ai:vasseur.alexis-fSummary: This paper is dedicated to the construction of a pseudo-norm for which small shock profiles of the barotropic Navier-Stokes equations have a contraction property. This contraction property holds in the class of any large solutions to the barotropic Navier-Stokes equations. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is related to the notion of \(a\)-contraction first introduced by the authors in the hyperbolic case.A binary Darboux transformation for multicomponent NLS equations and their reductions.https://zbmath.org/1460.370672021-06-15T18:09:00+00:00"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiu"Batwa, Sumayah"https://zbmath.org/authors/?q=ai:batwa.sumayahAuthors' abstract: We present a binary Darboux transformation for multicomponent NLS equations and their reduced integrable counterparts. The starting point is to apply two pairs of eigenfunctions and adjoint eigenfunctions, and the resulting binary Darboux transformation can be decomposed into an \(N\)-fold Darboux transformation. By taking the zero potential as a seed solution, soliton solutions are generated from the binary Darboux transformation for multicomponent NLS equations and their reductions.
Reviewer: Ivan C. Sterling (St. Mary's City)The first eigenvalue for a quasilinear Schrödinger operator and its application.https://zbmath.org/1460.350952021-06-15T18:09:00+00:00"Miyagaki, Olimpio Hiroshi"https://zbmath.org/authors/?q=ai:miyagaki.olimpio-hiroshi"Moreira, Sandra Imaculada"https://zbmath.org/authors/?q=ai:moreira.sandra-imaculada"Ruviaro, Ricardo"https://zbmath.org/authors/?q=ai:ruviaro.ricardoSummary: In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.On a class of degenerate abstract parabolic problems and applications to some eddy current models.https://zbmath.org/1460.353392021-06-15T18:09:00+00:00"Pauly, Dirk"https://zbmath.org/authors/?q=ai:pauly.dirk"Picard, Rainer"https://zbmath.org/authors/?q=ai:picard.rainer-h"Trostorff, Sascha"https://zbmath.org/authors/?q=ai:trostorff.sascha"Waurick, Marcus"https://zbmath.org/authors/?q=ai:waurick.marcusThe authors develop a framework for parabolic problems which can be degenerate in certain spatial regions. The approach used by the authors is related to evolution equations in Hilbert spaces, and involves only minimal assumptions on the boundary. This framework is used to analyze the structure of the degenerate eddy current problem. This eddy current problem is then justified as a limiting model of Maxwell's equations.
Reviewer: Eric Stachura (Marietta)On special regularity properties of solutions of the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation.https://zbmath.org/1460.353132021-06-15T18:09:00+00:00"Nascimento, A. C."https://zbmath.org/authors/?q=ai:nascimento.anderson-c-aSummary: In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual \(L^2(\mathbb{R}^2)\)-based Sobolev spaces \(H^s(\mathbb{R}^2)\) for \(s>\frac{5}{4}\) which coincides with the best available result in the literature proved employing more complicated tools.On singular solutions of time-periodic and steady Stokes problems in a power cusp domain.https://zbmath.org/1460.352542021-06-15T18:09:00+00:00"Eismontaite, Alicija"https://zbmath.org/authors/?q=ai:eismontaite.alicija"Pileckas, Konstantin"https://zbmath.org/authors/?q=ai:pileckas.konstantinSummary: The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.A general decay result of a viscoelastic equation with infinite history and nonlinear damping.https://zbmath.org/1460.740102021-06-15T18:09:00+00:00"Al-Gharabli, Mohammad M."https://zbmath.org/authors/?q=ai:algharabli.mohammad-mSummary: In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases.Extended mean-field games.https://zbmath.org/1460.350202021-06-15T18:09:00+00:00"Lions, Pierre-Louis"https://zbmath.org/authors/?q=ai:lions.pierre-louis"Souganidis, Panagiotis E."https://zbmath.org/authors/?q=ai:souganidis.panagiotis-eSummary: We introduce a new class of coupled forward-backward in time systems consisting of a forward Hamilton-Jacobi and a backward quasilinear transport equation, which we call extended mean-field games system. This new class of equations strictly contains the classical mean-field games system with no common noise and its homogenization limit, and optimal transportation-type control problems. We also identify a new and meaningful ``monotonicity''-type condition that yields well-posedeness. The same condition yields uniqueness in the Hilbertian setting for the master equation without common noise as well as the hyperbolic system describing finite-state mean-field games.Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem.https://zbmath.org/1460.651472021-06-15T18:09:00+00:00"Gudi, Thirupathi"https://zbmath.org/authors/?q=ai:gudi.thirupathi"Sau, Ramesh Ch."https://zbmath.org/authors/?q=ai:sau.ramesh-chIn this article, an energy-space based approach is investigated for the Dirichlet boundary control problem governed by the Laplace equation with control constraints. The optimal control problem is shown to have a unique solution and the optimality system consists of a coupled simplified Signorini type boundary problem for the control variable. A finite element based numerical method is proposed by considering the linear Lagrange finite element spaces and with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the \(L^2\) cost functional. Using auxiliary systems of PDE optimal order error estimates are derived for the state, the control, and the adjoint state variables in the energy norm. The corresponding a priori error estimates are optimal in the \(H^1\)-norm up to the regularity. Numerical experiments have been performed to validate the priori error estimates of state, adjoint state variable, and control variable for gradient cost functional and \(L^2\) cost functional.
Reviewer: Bülent Karasözen (Ankara)Global well-posedness and infinite propagation speed for the \(N - abc\) family of Camassa-Holm type equation with both dissipation and dispersion.https://zbmath.org/1460.761182021-06-15T18:09:00+00:00"Zhang, Zaiyun"https://zbmath.org/authors/?q=ai:zhang.zaiyun"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhai"Deng, Youjun"https://zbmath.org/authors/?q=ai:deng.youjun"Huang, Chuangxia"https://zbmath.org/authors/?q=ai:huang.chuangxia"Lin, Shiyou"https://zbmath.org/authors/?q=ai:lin.shi-you"Zhu, Wen"https://zbmath.org/authors/?q=ai:zhu.wenSummary: In this paper, we consider the Cauchy problem for the \(N - abc\) family of the Camassa-Holm type equation with both dissipation and dispersion. First, we establish the global well-posedness of the strong solutions under certain conditions on the initial datum. Then, we investigate the propagation speed with compactly supported initial data. This result improves earlier ones reported in the literature, such as those by
\textit{E. Novruzov} and \textit{A. Hagverdiyev} [J. Differ. Equations 257, No. 12, 4525--4541 (2014; Zbl 1304.35140)],
\textit{G. Hwang} and \textit{B. Moon} [Electron Res. Arch. 28, No. 1, 15--25 (2020; Zbl 1436.35073)], and
Himonas and Thompson [J. Math. Phys. 55, 091503 (2014)].
{\copyright 2020 American Institute of Physics}Quadratic optimal control for bilinear systems.https://zbmath.org/1460.352372021-06-15T18:09:00+00:00"Yahyaoui, Soufiane"https://zbmath.org/authors/?q=ai:yahyaoui.soufiane"Ouzahra, Mohamed"https://zbmath.org/authors/?q=ai:ouzahra.mohamedSummary: In this work, we will investigate the quadratic optimal control for bilinear systems. We will first study the existence of a solution for the considered optimal control. Then, we will focus on a special class of bilinear systems for which the quadratic optimal control can be expressed in a feedback law form. The approach relies on the conditions of optimality and linear semi-group theory.
For the entire collection see [Zbl 1459.35003].Positive periodic solutions of Friedmann's equation for the acceleration of the cosmological scale factor.https://zbmath.org/1460.830142021-06-15T18:09:00+00:00"Belley, Jean-Marc"https://zbmath.org/authors/?q=ai:belley.jean-marcSummary: Given \(T > 0\), \(\Lambda \in \mathbb{R}\), \(k \in \{- 1, 0, 1 \}\) and \(T\)-periodic function \(p\) of bounded variation on \([0, T]\), we obtain conditions that guarantee the existence of a strictly positive \(T\)-periodic almost everywhere solution of Friedmann's nonlinear equation \(a'' = (\Lambda - 8 \pi p) a / 2 -( a^{\prime^2} + k) / 2 a\) with singularity. Both \(a\) and \(a^\prime\) will be absolutely continuous and \(a''\) Lebesgue integrable on \([0, T]\). This makes \(a\) a scale factor for an FRW cosmology with pressure \(p\), cosmological constant \(\Lambda\) and energy density equal to \(3( a^{\prime^2} + k) / 8 \pi a^2 - \Lambda / 8 \pi \). By way of Picard iterates, \(a\) and \(a^{\prime}\) can be obtained numerically.The first 180 Lyapunov exponents for two-dimensional complex Ginzburg-Landau-type equation.https://zbmath.org/1460.763002021-06-15T18:09:00+00:00"Kozitskiy, S. B."https://zbmath.org/authors/?q=ai:kozitskii.s-bSummary: Dynamic patterns of three-dimensional double-diffusive convection in horizontally infinite liquid layer at large Rayleigh numbers have been simulated with the use of the previously derived system of complex Ginzburg-Landau-type amplitude equations valid in the neighborhoods of Hopf bifurcation points. For the special case of convection the first 180 Lyapunov exponents of the system have been calculated and 164 of them are positive. The spatial autocorrelation function is shown to be localized. Thus the system exhibits spatiotemporal chaos.A hyper-block self-consistent approach to nonlinear Schrödinger equations: breeding, metamorphosis, and killing of Hofstadter butterflies.https://zbmath.org/1460.350972021-06-15T18:09:00+00:00"Solaimani, Mehdi"https://zbmath.org/authors/?q=ai:solaimani.mehdi"Aleomraninejad, S. M. A."https://zbmath.org/authors/?q=ai:aleomraninejad.seyed-m-a|aleomraninejad.s-mohammad-aliSummary: Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.On the existence of solutions of nonlinear boundary value problems for inhomogeneous isotropic shallow shells of the Timoshenko type with free edges.https://zbmath.org/1460.353402021-06-15T18:09:00+00:00"Akhmadiev, M. G."https://zbmath.org/authors/?q=ai:akhmadiev.m-g.1"Timergaliev, S. N."https://zbmath.org/authors/?q=ai:timergaliev.samat-n"Uglov, A. N."https://zbmath.org/authors/?q=ai:uglov.a-n"Yakushev, R. S."https://zbmath.org/authors/?q=ai:yakushev.rinat-sAuthors' abstract: The paper deals with the study of solvability to geometrically nonlinear boundary value problem for elastic inhomogeneous isotropic shallow shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation relative to deflection of shell in Sobolev space. Solvability of equation is proved with the use of contracting mappings principle.
Reviewer: Kaïs Ammari (Monastir)Anisotropic micropolar fluids subject to a uniform microtorque: the unstable case.https://zbmath.org/1460.763342021-06-15T18:09:00+00:00"Remond-Tiedrez, Antoine"https://zbmath.org/authors/?q=ai:remond-tiedrez.antoine"Tice, Ian"https://zbmath.org/authors/?q=ai:tice.ianSummary: We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that this equilibrium is nonlinearly unstable. Our proof relies on a nonlinear bootstrap instability argument which uses control of higher-order norms to identify the instability at the \(L^2\) level.Optimal feedback control problem for inhomogeneous Voigt fluid motion model.https://zbmath.org/1460.762922021-06-15T18:09:00+00:00"Zvyagin, Victor"https://zbmath.org/authors/?q=ai:zvyagin.victor-g"Turbin, Mikhail"https://zbmath.org/authors/?q=ai:turbin.mikhail-vSummary: In the present paper, we study weak solvability of the optimal feedback control problem for the inhomogeneous Voigt fluid motion model. The proof is based on the approximation-topological approach. This approach involves the approximation of the original problem by regularized operator inclusion with the consequent application of topological degree theory. Then, we show the convergence of the sequence of solutions for the approximation problem to the solution for the original problem. For this, we use independent on approximation parameter a priori estimates. Finally, we prove that the cost functional achieves its minimum on the weak solution set.Perturbation method in the theory of propagation of two-frequency electromagnetic waves in a nonlinear waveguide. I: TE-TE waves.https://zbmath.org/1460.780102021-06-15T18:09:00+00:00"Valovik, D. V."https://zbmath.org/authors/?q=ai:valovik.dmitry-vSummary: The article deals with the problem of propagation of a two-frequency electromagnetic wave in a waveguide filled with a nonlinear medium. A two-frequency wave is the sum of two monochromatic TE waves with different frequencies. The permittivity of the waveguide is characterized by a very general nonlinearity function corresponding to self-action effects. It is shown that, under certain conditions, the two-frequency wave is an eigenmode of the waveguide. From a mathematical point of view, the problem reduces to a nonlinear two-parameter eigenvalue problem for the system of (nonlinear) Maxwell's equations. The main result of the article is the proof of the existence of nonlinearizable solutions of the problem.Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation.https://zbmath.org/1460.353582021-06-15T18:09:00+00:00"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThe goal of this paper is to study effects of the Stokes flow on the fully parabolic chemotaxis system with suitable flux limitation in the equation for the evolution of the density of population. Results on the absence of blowup of solutions obtained in the three-dimensional case are similar to those for the chemotaxis system without fluid. General estimates derived for fluid motion and taxis gradients have also an independent interest for study of global-in-time existence of bounded solutions in related problems.
Reviewer: Piotr Biler (Wrocław)On the existence of a scalar pressure field in the Brödinger problem.https://zbmath.org/1460.766522021-06-15T18:09:00+00:00"Baradat, Aymeric"https://zbmath.org/authors/?q=ai:baradat.aymericThe global supersonic flow with vacuum state in a 2D convex duct.https://zbmath.org/1460.352722021-06-15T18:09:00+00:00"Li, Jintao"https://zbmath.org/authors/?q=ai:li.jintao"Shen, Jindou"https://zbmath.org/authors/?q=ai:shen.jindou"Xu, Gang"https://zbmath.org/authors/?q=ai:xu.gangSummary: This paper concerns the motion of the supersonic potential flow in a two-dimensional expanding duct. In the case that two Riemann invariants are both monotonically increasing along the inlet, which means the gases are spread at the inlet, we obtain the global solution by solving the problem in those inner and border regions divided by two characteristics in \((x, y)\)-plane, and the vacuum will appear in some finite place adjacent to the boundary of the duct. In addition, we point out that the vacuum here is not the so-called physical vacuum. On the other hand, for the case that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet, which means the gases have some local squeezed properties at the inlet, we show that the \(C^1\) solution to the problem will blow up at some finite location in the non-convex duct.Existence and orbital stability of standing waves for the 1D Schrödinger-Kirchhoff equation.https://zbmath.org/1460.350302021-06-15T18:09:00+00:00"Natali, Fábio"https://zbmath.org/authors/?q=ai:natali.fabio-m-amorin"Cardoso, Eleomar"https://zbmath.org/authors/?q=ai:cardoso.eleomar-junSummary: In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.Stationary solutions to the Anderson-Witting model of the relativistic Boltzmann equation in a bounded interval.https://zbmath.org/1460.352502021-06-15T18:09:00+00:00"Hwang, Byung-Hoon"https://zbmath.org/authors/?q=ai:hwang.byung-hoon"Yun, Seok-Bae"https://zbmath.org/authors/?q=ai:yun.seok-baeMesoscopic description of the adiabatic piston: kinetic equations and \(\mathcal{H}\)-theorem.https://zbmath.org/1460.820042021-06-15T18:09:00+00:00"Khalil, Nagi"https://zbmath.org/authors/?q=ai:khalil.nagiSummary: The adiabatic piston problem is solved at the mesoscale using a kinetic theory approach. The problem is to determine the evolution towards equilibrium of two gases separated by a wall with only one degree of freedom (the adiabatic piston). A closed system of equations for the distribution functions of the gases conditioned to a position of the piston and the distribution function of the piston is derived, under the assumption of a generalized molecular chaos. It is shown that the resulting kinetic description has the canonical equilibrium as a steady-state solution. Moreover, the Boltzmann entropy, which includes the motion of the piston, verifies the \(\mathcal{H}\)-theorem. The kinetic description is not limited to the thermodynamic limit nor to a small ratio between the masses of the particle and the piston, and collisions among particles are explicitly considered.Global well-posedness and inviscid limit for the generalized Benjamin-Ono-Burgers equation.https://zbmath.org/1460.353122021-06-15T18:09:00+00:00"Chen, Mingjuan"https://zbmath.org/authors/?q=ai:chen.mingjuan"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Han, Lijia"https://zbmath.org/authors/?q=ai:han.lijiaSummary: This paper deals with the Cauchy problem for the generalized Benjamin-Ono-Burgers equation \(\partial_tu+\mathcal{H}\partial_x^2u-vu_{xx}+\partial_x(u^{k+1}/(k+1))=0,k\geq 4\), where \(\mathcal{H}\) denotes Hilbert transform. We obtain its global well-posedness results in Besov Spaces if \(k\geq 4\) and the initial data in \(\dot B^{s_k}_{2,1}\) are sufficiently small, where \(s_k:=1/2-1/k\) corresponds to the critical scaling regularity index. Furthermore, we prove its global well-posedness and inviscid limit behavior in Sobolev spaces.The sharp time decay rate of the isentropic Navier-Stokes system in \(\mathbb{R}\).https://zbmath.org/1460.352522021-06-15T18:09:00+00:00"Chen, Yuhui"https://zbmath.org/authors/?q=ai:chen.yuhui"Pan, Ronghua"https://zbmath.org/authors/?q=ai:pan.ronghua"Tong, Leilei"https://zbmath.org/authors/?q=ai:tong.leileiSummary: We investigate the sharp time decay rates of the solution \(U\) for the compressible Navier-Stokes system (1.1) in \(\mathbb{R}^3\) to the constant equilibrium \((\bar\rho>0,0)\) when the initial data is a small smooth perturbation of \((\bar\rho, 0)\). Let \(\widetilde{U}\) be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that \(\|U-\widetilde{U}\|_{L^2}\) decays at least at the rate of \((1+t)^{-\frac{5}{4}}\), which is faster than the rate \((1+t)^{-\frac{3}{4}}\) for the \(\widetilde{U}\) to its equilibrium \((\bar\rho,0)\). Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.Non-local to local transition for ground states of fractional Schrödinger equations on \(\mathbb{R}^N\).https://zbmath.org/1460.353172021-06-15T18:09:00+00:00"Bieganowski, Bartosz"https://zbmath.org/authors/?q=ai:bieganowski.bartosz"Secchi, Simone"https://zbmath.org/authors/?q=ai:secchi.simoneSummary: We consider the nonlinear fractional problem
\[
(-\Delta )^su+V(x)u=f(x,u)\text{ in }\mathbb{R}^N
\]
We show that ground state solutions converge (along a subsequence) in \(L^2_{\text{loc}}(\mathbb{R}^N)\), under suitable conditions on \(f\) and \(V\), to a ground state solution of the local problem as \(s\rightarrow 1^-\).Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates.https://zbmath.org/1460.353662021-06-15T18:09:00+00:00"Kawamoto, Atsushi"https://zbmath.org/authors/?q=ai:kawamoto.atsushi.1|kawamoto.atsushi"Machida, Manabu"https://zbmath.org/authors/?q=ai:machida.manabuSummary: We consider a fractional radiative transport equation, where the time derivative is of half order in the Caputo sense. By establishing Carleman estimates, we prove the global Lipschitz stability in determining the coefficients of the one-dimensional time-fractional radiative transport equation of half-order.Strongly localized semiclassical states for nonlinear Dirac equations.https://zbmath.org/1460.353082021-06-15T18:09:00+00:00"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.1|bartsch.thomas.2"Xu, Tian"https://zbmath.org/authors/?q=ai:xu.tianSummary: We study semiclassical states of the nonlinear Dirac equation
\[
-i\hbar\partial_t\psi=ic\hbar\sum\limits_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta\psi-M(x)\psi+f(|\psi|)\psi,\quad t\in\mathbb{R},\, x\in\mathbb{R}^3,
\]
where \(V\) is a bounded continuous potential function and the nonlinear term \(f(|\psi|)\psi\) is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity \(f(s)=s^p\). We develop a variational method for the strongly indefinite functional associated to the problem.Positive multipeak solutions to a zero mass problem in exterior domains.https://zbmath.org/1460.353622021-06-15T18:09:00+00:00"Clapp, Mónica"https://zbmath.org/authors/?q=ai:clapp.monica"Maia, Liliane A."https://zbmath.org/authors/?q=ai:maia.liliane-a"Pellacci, Benedetta"https://zbmath.org/authors/?q=ai:pellacci.benedettaThe authors establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass, \[- \triangle u = f(u), \quad u \in D_0^{1, 2} (\Omega_R),\] where \(\Omega _R := \{x \in \mathbb R^N : \; \vert u \vert > R \}\) with \(R > 0, \; N \geq 4\), and the nonlinearity \(f\) is subcritical at infinity and supercritical near the origin. They show that the number of positive multipeak solutions becomes arbitrarily large as \(R \to \infty\).
Reviewer: Anthony D. Osborne (Keele)Non-linear anti-symmetric shear motion: a comparative study of non-homogeneous and homogeneous plates.https://zbmath.org/1460.353412021-06-15T18:09:00+00:00"Demirkuş, Dilek"https://zbmath.org/authors/?q=ai:demirkus.dilekSummary: In this article, the non-linear anti-symmetric shear motion for some comparative studies between the non-homogeneous and homogeneous plates, having two free surfaces with stress-free, is considered. Assuming that one plate contains hyper-elastic, non-homogeneous, isotropic, and generalized neo-Hookean materials and the other one consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the method of multiple scales, the self-modulation of the non-linear anti-symmetric shear motion in these plates, as the non-linear Schrödinger (NLS) equations, can be given. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to NLS equations, these comparative studies in terms of the non-homogeneous and non-linear effects are made. All numerical results, based on the asymptotic analyses, are graphically presented for the lowest anti-symmetric branches of both dispersion relations, including the deformation fields of plates.Weak solutions to the Muskat problem with surface tension via optimal transport.https://zbmath.org/1460.651142021-06-15T18:09:00+00:00"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthew"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christina"Mészáros, Alpár R."https://zbmath.org/authors/?q=ai:meszaros.alpar-richardThe global existence of weak solutions for the Muskat problem with surface tension, based on its gradient flow structure is obtained. The paper is organized as follows. Section 1 is an introduction. In the same section, the statement of the problem and its variational formulation are given. The main theorem of the paper is also formulated in Section 1. In Section 2, the basic properties of the minimizing movements scheme are derived and discrete-time quantities are constructed. The existence of pressure as a Lagrange multiplier for the incompressibility constraint is derived and the Euler-Lagrange equation for the minimization problem is obtained. In Section 3, weak solutions to the Muskat problem are obtained, under the assumption that the internal energy of the discrete solutions converges to the internal energy of the limiting solutions. The main task in Section 3 amounts to showing that one can pass to the limit in the Euler-Lagrange equation obtained in Section 2. In Section 4, several numerical examples with illustrations are given and discussed. Finally, in Appendix A, the results that are used when passing to the limit the weak curvature equation are recalled from the following paper [\textit{T. Laux} and \textit{F. Otto}, Calc. Var. Partial Differ. Equ. 55, No. 5, Paper No. 129, 74 p. (2016; Zbl 1388.35121)].
Reviewer: Temur A. Jangveladze (Tbilisi)Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part III: Application to the Cane-Zebiak model of the El Niño-southern oscillation.https://zbmath.org/1460.600572021-06-15T18:09:00+00:00"Tantet, Alexis"https://zbmath.org/authors/?q=ai:tantet.alexis"Chekroun, Mickaël D."https://zbmath.org/authors/?q=ai:chekroun.mickael-d"Neelin, J. David"https://zbmath.org/authors/?q=ai:neelin.j-david"Dijkstra, Henk A."https://zbmath.org/authors/?q=ai:dijkstra.henk-aSummary: The response of a low-frequency mode of climate variability, El Niño-Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane-Zebiak model [\textit{S.E. Zebiak} and \textit{M.A. Cane}, ``A model of El Nino-Southern Oscillation'', Mon. Weather Rev. 115, 31, 2262--2278 (1987)], from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle-Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution [\textit{M. D. Chekroun} et al., J. Stat. Phys. 179, No. 5--6, 1366--1402 (2020; Zbl 1460.60050)] to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by
\textit{P. Gaspard} [J. Stat. Phys. 106, No. 1--2, 57--96 (2002; Zbl 1020.82007)] and made explicit in the second part of this contribution [the authors, J. Stat. Phys. 179, No. 5--6, 1403--1448 (2020; Zbl 1460.60056)]. Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in [Chekroun, loc. cit.], complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity.https://zbmath.org/1460.351422021-06-15T18:09:00+00:00"Liu, Chungen"https://zbmath.org/authors/?q=ai:liu.chungen"Zhang, Huabo"https://zbmath.org/authors/?q=ai:zhang.huaboSummary: In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem
\[
\begin{cases}
(a+b\int_{\mathbb{R}^3}(|(-\Delta)^{\alpha/2}u|^2)dx)(-\Delta)^{\alpha}u+V(x)u+K(x)\phi u=|u|^{2^{\ast}-2}u+\kappa f(x,u),\\
(-\Delta)^{\beta}\phi=K(x)u^2,\quad x\in\mathbb{R}^3,
\end{cases}
\]
where \(a,b,\kappa\) are positive parameters, \(\alpha\in(\frac{3}{4},1),\beta\in(0,1)\), and \(2^{\ast}_{\alpha}=\frac{6}{3-2\alpha}\), \((-\Delta)^{\alpha}\) stands for the fractional Laplacian. By the nodal Nehari manifold method, for each \(b>0\), we obtain a ground state nodal solution \(u_b\) and a ground-state solution \(v_b\) to this problem when \(\kappa\gg 1\), where the nonlinear function \(f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}\) is a Carathéodory function. We also give an analysis on the behavior of \(u_b\) as the parameter \(b\to 0\).Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation.https://zbmath.org/1460.352892021-06-15T18:09:00+00:00"Ipocoana, Erica"https://zbmath.org/authors/?q=ai:ipocoana.erica"Zafferi, Andrea"https://zbmath.org/authors/?q=ai:zafferi.andreaSummary: The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial \(L^{\infty}\) estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.Random data theory for the cubic fourth-order nonlinear Schrödinger equation.https://zbmath.org/1460.353232021-06-15T18:09:00+00:00"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongSummary: We consider the cubic nonlinear fourth-order Schrödinger equation
\[
i\partial_tu-\Delta^2u+\mu\Delta u=\pm |u|^2u, \quad \mu\geq 0
\]
on \(\mathbb{R}^N\), \(N\geq 5\) with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.Steady asymmetric vortex pairs for Euler equations.https://zbmath.org/1460.352862021-06-15T18:09:00+00:00"Hassainia, Zineb"https://zbmath.org/authors/?q=ai:hassainia.zineb"Hmidi, Taoufik"https://zbmath.org/authors/?q=ai:hmidi.taoufikSummary: In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter-rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behavior for the angular velocity and the translating speed close to the point vortex pairs.Modulation instability and optical solitons of Radhakrishnan-Kundu-Lakshmanan model.https://zbmath.org/1460.780232021-06-15T18:09:00+00:00"Raza, Nauman"https://zbmath.org/authors/?q=ai:raza.nauman"Javid, Ahmad"https://zbmath.org/authors/?q=ai:javid.ahmadSummary: This paper studies the solitons of Radhakrishnan-Kundu-Lakshmanan (RKL) model with power law nonlinearity. The modified simple equation method and \(\exp(-\varphi(q))\) method are presented as integration mechanisms. Dark, bright, singular and periodic soliton solutions are extracted as well as the constraint conditions for their existence. A prized discussion on the stability of these soliton profiles on the basis of index of the power law nonlinearity is also carried out with the help of physical description of solutions. The integration techniques have been proved to be extremely efficient and robust to find new optical solitary wave solutions for various nonlinear evolution equations describing optical pulse propagation. Moreover, using linear stability analysis, modulation instability of the RKL model is studied. Different effects contributing to the modulation instability spectrum gain are analyzed.Optimal resource allocation for a diffusive population model.https://zbmath.org/1460.921612021-06-15T18:09:00+00:00"Bintz, Jason"https://zbmath.org/authors/?q=ai:bintz.jason"Lenhart, Suzanne"https://zbmath.org/authors/?q=ai:lenhart.suzanne-mA priori estimates for the general dynamic Euler-Bernoulli beam equation: supported and cantilever beams.https://zbmath.org/1460.740512021-06-15T18:09:00+00:00"Hasanov, Alemdar"https://zbmath.org/authors/?q=ai:hasanoglu.alemdar"Itou, Hiromichi"https://zbmath.org/authors/?q=ai:itou.hiromichiSummary: This work is a further development of weak solution theory for the general Euler-Bernoulli beam equation \(\rho(x) u_{t t} + \mu(x) u_t + \left(r(x) u_{x x}\right)_{x x} -(T_r(x) u_x)_x = F(x, t)\) defined in the finite dimension domain \(\Omega_T := (0, l) \times(0, T) \subset \mathbb{R}^2\), based on the energy method. Here \(r(x) = E I(x)\), \(E > 0\) is the elasticity modulus and \(I(x) > 0\) is the moment of inertia of the cross-section, \(\rho(x) > 0\) is the mass density of the beam, \(\mu(x) > 0\) is the damping coefficient and \(T_r(x) \geq 0\) is the traction force along the beam. Two benchmark initial boundary value problems with mixed boundary conditions, corresponding to supported and cantilever beams, are analyzed. For the weak and regular weak solutions of these problems a priori estimates are derived under the minimal conditions. These estimates in particular imply the uniqueness of the solutions of both problems.Axisymmetric flows on the torus geometry.https://zbmath.org/1460.768832021-06-15T18:09:00+00:00"Busuioc, Sergiu"https://zbmath.org/authors/?q=ai:busuioc.sergiu"Kusumaatmaja, Halim"https://zbmath.org/authors/?q=ai:kusumaatmaja.halim"Ambruş, Victor E."https://zbmath.org/authors/?q=ai:ambrus.victor-eSummary: We present a series of analytically solvable axisymmetric flows on the torus geometry. For the single-component flows, we describe the propagation of sound waves for perfect fluids, as well as the viscous damping of shear and longitudinal waves for isothermal and thermal fluids. Unlike the case of planar geometry, the non-uniform curvature on a torus necessitates a distinct spectrum of eigenfrequencies and their corresponding basis functions. This has several interesting consequences, including breaking the degeneracy between even and odd modes, a lack of periodicity even in the flows of perfect fluids and the loss of Galilean invariance for flows with velocity components in the poloidal direction. For the multi-component flows, we study the equilibrium configurations and relaxation dynamics of axisymmetric fluid stripes, described using the Cahn-Hilliard equation. We find a second-order phase transition in the equilibrium location of the stripe as a function of its area \(\Delta A\). This phase transition leads to a complex dependence of the Laplace pressure on \(\Delta A\). We also derive the underdamped oscillatory dynamics as the stripes approach equilibrium. Furthermore, relaxing the assumption of axial symmetry, we derive the conditions under which the stripes become unstable. In all cases, the analytical results are confirmed numerically using a finite-difference Navier-Stokes solver.On the Cahn-Hilliard equation with mass source for biological applications.https://zbmath.org/1460.353512021-06-15T18:09:00+00:00"Fakih, Hussein"https://zbmath.org/authors/?q=ai:fakih.hussein"Mghames, Ragheb"https://zbmath.org/authors/?q=ai:mghames.ragheb"Nasreddine, Noura"https://zbmath.org/authors/?q=ai:nasreddine.nouraSummary: This article deals with some generalizations of the Cahn-Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn-Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn-Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.Numerical investigations of shallow water waves via generalized equal width (GEW) equation.https://zbmath.org/1460.651232021-06-15T18:09:00+00:00"Karakoc, Seydi Battal Gazi"https://zbmath.org/authors/?q=ai:karakoc.seydi-battal-gazi"Omrani, Khaled"https://zbmath.org/authors/?q=ai:omrani.khaled"Sucu, Derya"https://zbmath.org/authors/?q=ai:sucu.deryaSummary: In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing \(L_\infty\) and \(L_2\) norms and for the other problem computing three invariant quantities \(I_1\), \(I_2\) and \(I_3\). The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid.The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field.https://zbmath.org/1460.580092021-06-15T18:09:00+00:00"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Nemer, Rodrigo C. M."https://zbmath.org/authors/?q=ai:nemer.rodrigo-c-m"Soares, Sergio H. Monari"https://zbmath.org/authors/?q=ai:soares.sergio-h-monariSummary: Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent \(p\) are considered. It is established a lower bound to the number of nontrivial solutions to these equations in terms of the topology of the domains in which the problem is given if \(p\) is suitably close to the critical exponent \(2^*=2N/(N-2)\), \(N\geq 3\). To prove this lower bound, based on a proof of a result of Benci and Cerami, it is provided an abstract result that establishes Morse relations that are used to count solutions.Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food.https://zbmath.org/1460.350272021-06-15T18:09:00+00:00"Xing, Chao"https://zbmath.org/authors/?q=ai:xing.chao"Pan, Jiaojiao"https://zbmath.org/authors/?q=ai:pan.jiaojiao"Luo, Hong"https://zbmath.org/authors/?q=ai:luo.hongSummary: The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.On the stagnation point position of the flow impinging obliquely on a moving flat plate.https://zbmath.org/1460.761642021-06-15T18:09:00+00:00"Cheng, Sheng-Yin"https://zbmath.org/authors/?q=ai:cheng.sheng-yin"Chen, Falin"https://zbmath.org/authors/?q=ai:chen.falinSummary: To study the variation of the stagnation point position of the flow impinging obliquely on a moving flat plate, we follow the mathematical approach of \textit{J. M. Dorrepaal} [ibid. 163, 141--147 (1986; Zbl 0605.76033)] and obtain the analytical solution of the flow. Based on the solution, we derive an equation governing the variation of stagnation point position with both the plate velocity as well as the impinging angle. Results show that, when the plate is stationary, the stagnation point will stay in upstream if the flow is non-orthogonal, as concluded by previous studies. As soon as the plate starts to move, the stagnation point will move from upstream to downstream when the plate velocity increases beyond a small critical value, no matter whether the flow is orthogonal or non-orthogonal.A Roe-like reformulation of the HLLC Riemann solver and applications.https://zbmath.org/1460.651092021-06-15T18:09:00+00:00"Pelanti, Marica"https://zbmath.org/authors/?q=ai:pelanti.maricaThe author presents a reformulation of the HLLC (Harten-Lax-van Leer-Contact) Riemann solver, which shows that the HLLC wave structure can be seen as an averaged system eigenstructure. In particular, this new Roe-like form of the HLLC solver is used to develop a robust second-order well-balanced f-wave HLLC method for the solution of the Euler equations with gravitational source terms. The presented reformulation of the HLLC solver can be used for other applications and it could be derived also for more complex flow models.
For the entire collection see [Zbl 1453.35003].
Reviewer: Abdallah Bradji (Annaba)Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional.https://zbmath.org/1460.353372021-06-15T18:09:00+00:00"Chen, Bo"https://zbmath.org/authors/?q=ai:chen.bo.4|chen.bo.3|chen.bo.2|chen.bo.1"Wang, Youde"https://zbmath.org/authors/?q=ai:wang.youdeSummary: We follow the idea of \textit{Y.-D. Wang} [J. Math. Phys. 39, No. 1, 363--371 (1998; Zbl 0938.58017)] to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a \(n\)-dimensional Euclidean domain \(\Omega\) or a \(n\)-dimensional closed Riemannian manifold \(M\) into a 2-dimensional unit sphere \(\mathbb{S}^2\). Our conclusions extend a series of related results obtained in the previous literature.Equilibrium strategies in a multiregional transboundary pollution differential game with spatially distributed controls.https://zbmath.org/1460.911762021-06-15T18:09:00+00:00"de Frutos, Javier"https://zbmath.org/authors/?q=ai:de-frutos.javier"López-Pérez, Paula M."https://zbmath.org/authors/?q=ai:lopez-perez.paula-m"Martín-Herrán, Guiomar"https://zbmath.org/authors/?q=ai:martin-herran.guiomarSummary: We analyse a differential game with spatially distributed controls to study a multiregional transboundary pollution problem. The dynamics of the state variable (pollution stock) is defined by a two dimensional parabolic partial differential equation. The control variables (emissions) are spatially distributed variables. The model allows for a, possibly large, number of agents with predetermined geographical relationships. For a special functional form previously used in the literature of transboundary pollution dynamic games we analytically characterize the feedback Nash equilibrium. We show that at the equilibrium both the level and the location of emissions of each region depend on the particular geographical relationship among agents. We prove that, even in a simplified model, the geographical considerations can modify the players' optimal strategies and therefore, the spatial aspects of the model should not be overlooked.Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity. II: Supercritical blow-up profiles.https://zbmath.org/1460.353282021-06-15T18:09:00+00:00"Holmer, Justin"https://zbmath.org/authors/?q=ai:holmer.justin"Liu, Chang"https://zbmath.org/authors/?q=ai:liu.chang.1Summary: We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
\[
i\partial_t\psi+\partial_x^2\psi+\delta|\psi|^{p-1}\psi=0,\tag{(0.1)}
\]
where \(\delta=\delta(x)\) is the delta function supported at the origin. In the \(L^2\) supercritical setting \(p>3\), we construct self-similar blow-up solutions belonging to the energy space \(L_x^\infty\cap\dot H_x^1\). This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \(x=0\) imposed by the \(\delta\) term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \(0<p-3\ll 1\) using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.
For part I, see [the authors, J. Math. Anal. Appl. 483, No. 1, Article ID 123522, 20 p. (2020; Zbl 1436.35290)].Multi-region finite element modelling of drug release from hydrogel based ophthalmic lenses.https://zbmath.org/1460.920892021-06-15T18:09:00+00:00"Gudnason, Kristinn"https://zbmath.org/authors/?q=ai:gudnason.kristinn"Sigurdsson, Sven"https://zbmath.org/authors/?q=ai:sigurdsson.s-t"Jonsdottir, Fjola"https://zbmath.org/authors/?q=ai:jonsdottir.fjolaSummary: Understanding the way in which drug is released from drug carrying hydrogel based ophthalmic lenses aids in the development of efficient ophthalmic drug delivery. Various solute-polymer interactions affect solute diffusion within hydrogels as well as hydrogel-bulk partitioning. Additionally, surface modifications or coatings may add to resistance of mass transfer across the hydrogel interface. It is necessary to consider both interfacial resistances as well as the appropriate driving force when characterizing interface flux. Such a driving force is induced by a difference in concentration which deviates from equilibrium conditions. We present a Galerkin finite element approach for solute transport in hydrogels which accounts for diffusion within the gel, storage effects due to polymer-solute interaction, as well as partitioning and mass transfer resistance effects at the interface. The approach is formulated using a rotational symmetric model to account for realistic geometry. We show that although the resulting global system is not symmetric in the case of partitioning, it is similar to a symmetric negative semidefinite system. Thus, it has non-positive real eigenvalues and is coercive, ensuring the validity of the finite element formulation as well as the numerical stability of the implicit backward Euler time integration method employed. Two models demonstrating this approach are presented and verified with release experimental data. The first is the release of moxifloxacin from intraocular lenses (IOLs) plasma grafted with different polyacrylates. The second accounts for both loading as well as the release of diclofenac from disc shaped IOL material loaded for varied time periods and temperature.Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation.https://zbmath.org/1460.353152021-06-15T18:09:00+00:00"Ardila, Alex H."https://zbmath.org/authors/?q=ai:ardila.alex-hernandez"Cardoso, Mykael"https://zbmath.org/authors/?q=ai:cardoso.mykaelSummary: Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)
\[
i\partial_tu+\Delta u+|x|^{-b}|u|^{p-1}u=0.
\]
We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in \(H^1(\mathbb{R}^N)\) in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is \(L^2\)-supercritical, then the ground states are strongly unstable by blow-up.Scattering of the focusing energy-critical NLS with inverse square potential in the radial case.https://zbmath.org/1460.353352021-06-15T18:09:00+00:00"Yang, Kai"https://zbmath.org/authors/?q=ai:yang.kaiSummary: We consider the Cauchy problem of the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential. We prove that if any radial solution obeys the supreme of the kinetic energy over the maximal lifespan is below the kinetic energy of the ground state solution, then the solution exists globally in time and scatters in both time directions.Existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation.https://zbmath.org/1460.351532021-06-15T18:09:00+00:00"Feng, Binhua"https://zbmath.org/authors/?q=ai:feng.binhua"Cao, Leijin"https://zbmath.org/authors/?q=ai:cao.leijin"Liu, Jiayin"https://zbmath.org/authors/?q=ai:liu.jiayinSummary: In this paper, we study existence of stable standing waves for the following Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation with a partial harmonic confine
\[
i \partial_t \psi = - \Delta \psi + (x_1^2 + x_2^2) \psi + \lambda_1 | \psi |^2 \psi + \lambda_2 (K \ast | \psi |^2) \psi + \lambda_3 | \psi |^p \psi, \quad (t, x) \in [0, T^\ast) \times \mathbb{R}^3.
\]
When \(0 < p < 4\) and \(\lambda_3 < 0\), we can prove the existence of stable standing waves for this equation. Our results are a complementary to the ones of \textit{Y. Luo} and \textit{A. Stylianou} [Discrete Contin. Dyn. Syst. Ser. B 26(6), 3455-3477 (2021, \url{http://dx.doi.org/10.3934/dcdsb.2020239})], where existence and nonexistence of standing waves have been studied for the complete harmonic potential.The families of explicit solutions for the Hirota equation.https://zbmath.org/1460.353112021-06-15T18:09:00+00:00"Su, Ting"https://zbmath.org/authors/?q=ai:su.ting"Wang, Jia"https://zbmath.org/authors/?q=ai:wang.jiaThe authors present some explicit solutions for the Hirota equation derived from the Ablowitz-Kaup-Newell-Segur (AKNS) shallow water wave equation: A dark one soliton solution is obtained by using homogeneous balance for the Hirota equation; multiple soliton solutions and multiple singular solutions are derived by using the so-called Hirota bilinear form of the equation. Finally, the authors obtain one- and two-periodic solutions for the AKNS equations in the form of Riemann theta functions by employing the Hirota bilinear form of the AKNS equation. The asymptotic behaviour of the two periodic solutions is then indicated.
Reviewer: Catalin Popa (Iaşi)The controllability of Fokker-Planck equations with reflecting boundary conditions and controllers in diffusion term.https://zbmath.org/1460.930132021-06-15T18:09:00+00:00"Barbu, Viorel"https://zbmath.org/authors/?q=ai:barbu.viorelQualitative analysis of stationary generalized viscoplastic fluid flows.https://zbmath.org/1460.760102021-06-15T18:09:00+00:00"Elborhamy, M."https://zbmath.org/authors/?q=ai:elborhamy.mSummary: In this article, we consider the qualitative analysis of the generalized stationary regularized viscoplastic fluid equations with highly nonlinear viscosity. The existence and uniqueness of weak solutions are proved using the theory of monotone operators. Under certain conditions, the contractive property of the viscoplastic operator is used to be a tool to show the stability of solutions for the Dirichlet's problem with sources. Some technical properties are proved which have good fitting with the experimental results for some types of flows. A novel viscoplastic model is proposed to avoid the singularity of the nonlinear viscoplastic viscosity.The massive Thirring system in the quarter plane.https://zbmath.org/1460.353092021-06-15T18:09:00+00:00"Xia, Baoqiang"https://zbmath.org/authors/?q=ai:xia.baoqiangSummary: The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit \((x, t)\)-dependence and depend only on the given initial and boundary values; they do \textit{not} involve additional unknown boundary values.Vector particle with electric quadrupole moment in external Coulomb field.https://zbmath.org/1460.353052021-06-15T18:09:00+00:00"Koralkov, Artem"https://zbmath.org/authors/?q=ai:koralkov.artem"Voynova, Yanina"https://zbmath.org/authors/?q=ai:voynova.yanina"Krylova, Nina"https://zbmath.org/authors/?q=ai:krylova.nina-g"Ovsiyuk, Elena"https://zbmath.org/authors/?q=ai:ovsiyuk.elena-m"Balan, Vladimir"https://zbmath.org/authors/?q=ai:balan.vladimirSummary: We study the problem of vector particles with electric quadrupole moment in presence of an external Coulomb field. Starting with the relativistic Duffin-Kemmer theory, we search for solutions. To this aim, we diagonalize the operators of energy, square of the total angular momentum and its third projection. After separating the variables, we derive a system of 10 radial equations. According to the requirement of diagonalizing the spatial reection operator, we split the system into two subsystems of 4 and 6 equations respectively, for states with parities \(P=(-1)^{j+1}\) and \(P= (-1)^j\). Additional interaction terms enter both subsystems. The relativistic radial subsystem of 4 equations reduces to a second-order equation which contains two singular points \(x=0\) and \(x=\infty\) of ranks 3 and 2, respectively, and four regular points. The local Frobenius solutions near the point \(x=0\) are constructed. It is shown that there exist 8-term recurrence formulas for the involved power series. The condition of transcendency of solutions gives a certain quantization rule for energy levels, which seems to be only partially physically appropriate. The relativistic radial system of 6 equations for states with parity \(P=(-1)^j\), turns out to be very complicated. In order to simplify the problem, we perform the transition to the non-relativistic approximation, and consequently derive two associated second-order differential equations for two radial functions. We obtain the fourth-order equations for radial functions, and construct four different Frobenius type solutions of these equations. As well, the convergence of the involved power series with 8- and 9-terms recurrence relations, is studied. The transcendency condition gives the formula for energies, which does not depend on the quantum number and on the parameter of quadrupole electric moment, and therefore cannot describe the physical spectrum correctly. The non-relativistic analysis is performed for states with the parity \(P= (-1)^j\) as well, but the radial equation for the main function turns out to have more simple structure of singular points. The transcendency condition leads to a formula for energies which only partially correlates with the relativistic one. All the constructed solutions are exact, but they are formal because there exists no reliable rule for quantization of energy levels, and the transcendency condition solves this difficulty only partially.
We additionally apply the geometrical method based on the theory of KCC-invariants. The first and the second invariants are calculated, and it is shown that the distinct branches of the solutions converge near the singular points \(r=0,\infty,-\Gamma/2\). This correlates with the expected behavior of solutions for bound states. Within this framework, the explicit Lagrangians related to the geometrical problem are determined. It is shown that the Lagrangians have an arbitrariness degree up to certain terms, which may be considered as a specific gauge freedom.
For the entire collection see [Zbl 1452.00029].Optimal feedback control for one motion model of a nonlinearly viscous fluid.https://zbmath.org/1460.762932021-06-15T18:09:00+00:00"Zvyagin, Viktor Grigor'evich"https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich"Zvyagin, Andreĭ Viktorovich"https://zbmath.org/authors/?q=ai:zvyagin.andrey-v"Hong, Nguyen Minh"https://zbmath.org/authors/?q=ai:hong.nguyen-minhSummary: An optimal control problem with a feedback is considered for an initial boundary problem describing a motion of non-linearly viscous liquid. An existence of an optimal solution minimising a given quality functional is proved. A topological approximation approach to study of mathematical problems of hydrodynamics is used in the proof of existence of an optimal solution.A boundary value problem of sand transport equations: an existence and homogenization results.https://zbmath.org/1460.350242021-06-15T18:09:00+00:00"Thiam, B. K."https://zbmath.org/authors/?q=ai:thiam.b-k"Baldé, M. A. M. T."https://zbmath.org/authors/?q=ai:balde.mouhamadou-aliou-m-t"Faye, I."https://zbmath.org/authors/?q=ai:faye.ibrahima"Seck, D."https://zbmath.org/authors/?q=ai:seck.diarafSummary: In this paper, we consider degenerate parabolic sand transport equations in a non periodic domain with Neumann boundary condition. We give existence and uniqueness results for the models which are also homogenized. Finally some corrector results are given.
For the entire collection see [Zbl 1458.00035].Regularity of solutions of a fractional porous medium equation.https://zbmath.org/1460.352882021-06-15T18:09:00+00:00"Imbert, Cyril"https://zbmath.org/authors/?q=ai:imbert.cyril"Tarhini, Rana"https://zbmath.org/authors/?q=ai:tarhini.rana"Vigneron, François"https://zbmath.org/authors/?q=ai:vigneron.francoisThe authors study nonnegative solutions of the porous medium equation involving pressure function depending on the density in a nonlinear and nonlocal way considered in [\textit{P. Biler} et al., Arch. Ration. Mech. Anal. 215, No. 2, 497--529 (2015; Zbl 1308.35197)]. The regularity of solutions in the Hölder sense is proved using De Giorgi method together with subtle estimates in the spirit of [\textit{L. Caffarelli} et al., J. Am. Math. Soc. 24, No. 3, 849--869 (2011; Zbl 1223.35098); J. Eur. Math. Soc. (JEMS) 15, No. 5, 1701--1746 (2013; Zbl 1292.35312)], separately in different ranges of the fractional order of Laplacian appearing in the equation.
Reviewer: Piotr Biler (Wrocław)On the cutoff approximation for the Boltzmann equation with long-range interaction.https://zbmath.org/1460.352492021-06-15T18:09:00+00:00"He, Ling-Bing"https://zbmath.org/authors/?q=ai:he.lingbing"Jiang, Jin-Cheng"https://zbmath.org/authors/?q=ai:jiang.jin-cheng"Zhou, Yu-Long"https://zbmath.org/authors/?q=ai:zhou.yulongSummary: The Boltzmann collision operator for long-range interactions is usually employed in its ``weak form'' in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the \textit{principle value} sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies \textit{compatible conditions}. (2). A quasi-linear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case.On nonlinear profile decompositions and scattering for an NLS-ODE model.https://zbmath.org/1460.353222021-06-15T18:09:00+00:00"Cuccagna, Scipio"https://zbmath.org/authors/?q=ai:cuccagna.scipio"Maeda, Masaya"https://zbmath.org/authors/?q=ai:maeda.masayaSummary: In this paper, we consider a Hamiltonian system combining a nonlinear Schrödinger equation (NLS) and an ordinary differential equation. This system is a simplified model of the NLS around soliton solutions. Following \textit{K. Nakanishi} [J. Math. Soc. Japan 69, No. 4, 1353--1401 (2017; Zbl 1383.35213)], we show scattering of \(L^2\) small \(H^1\) radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on \(L^4\) in time norms.Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: convergence for classical solutions.https://zbmath.org/1460.352942021-06-15T18:09:00+00:00"Luo, Yi-Long"https://zbmath.org/authors/?q=ai:luo.yi-long"Ma, Yangjun"https://zbmath.org/authors/?q=ai:ma.yangjunSummary: In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number \(\epsilon\) for both the compressible system and its differential system with respect to time under uniformly in \(\epsilon\) small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as \(\epsilon\rightarrow 0\), so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of \textit{F. De Anna} and \textit{A. Zarnescu} [J. Differ. Equations 264, No. 2, 1080--1118 (2018; Zbl 1393.35165)]. Moreover, we also obtain the convergence rates associated with \(L^2\)-norm with well-prepared initial data.Linear and nonlinear wave propagation.https://zbmath.org/1460.350022021-06-15T18:09:00+00:00"Kuo, Spencer"https://zbmath.org/authors/?q=ai:kuo.spencerPublisher's description: Waves are essential phenomena in most scientific and engineering disciplines, such as electromagnetism and optics, and different mechanics including fluid, solid, structural, quantum, etc. They appear in linear and nonlinear systems. Some can be observed directly and others are not. The features of the waves are usually described by solutions to either linear or nonlinear partial differential equations, which are fundamental to the students and researchers.
Generic equations, describing wave and pulse propagation in linear and nonlinear systems, are introduced and analyzed as initial/boundary value problems. These systems cover the general properties of non-dispersive and dispersive, uniform and non-uniform, with/without dissipations. Methods of analyses are introduced and illustrated with analytical solutions. Wave-wave and wave-particle interactions ascribed to the nonlinearity of media (such as plasma) are discussed in the final chapter.
This interdisciplinary textbook is essential reading for anyone in above mentioned disciplines. It was prepared to provide students with an understanding of waves and methods of solving wave propagation problems. The presentation is self-contained and should be read without difficulty by those who have adequate preparation in classic mechanics. The selection of topics and the focus given to each provide essential materials for a lecturer to cover the bases in a linear/nonlinear wave course.Global dynamics of a Filippov plant disease model with an economic threshold of infected-susceptible ratio.https://zbmath.org/1460.922022021-06-15T18:09:00+00:00"Huang, Lihong"https://zbmath.org/authors/?q=ai:huang.lihong"Ma, Huili"https://zbmath.org/authors/?q=ai:ma.huili"Wang, Jiafu"https://zbmath.org/authors/?q=ai:wang.jiafu"Huang, Chuangxia"https://zbmath.org/authors/?q=ai:huang.chuangxiaThe authors investigated a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio. By using the Filippov approach, they analyzed the sliding mode dynamics and the global dynamics of the planar Filippov system. These results implies that control measures are effective by choosing appropriate removal rate and replanting rate.
Reviewer: Hongying Shu (Xi'an)Homogenization for the cubic nonlinear Schrödinger equation on \(\mathbb R^2\).https://zbmath.org/1460.350212021-06-15T18:09:00+00:00"Ntekoume, Maria"https://zbmath.org/authors/?q=ai:ntekoume.mariaThe paper under review deals with the homogenization of the cubic nonlinear Schrödinger equation on \(\mathbb{R}^2\). More precisely, the author considers the Cauchy problem for the cubic nonlinear Schrödinger equation with inhomogeneous nonlinearity
\[
\left\{ \begin{array}{l} i\partial_t u_n+\Delta u_n =g(nx)|u_n|^2u_n \\
u_n(0)=u_0 \in L^2(\mathbb{R}^2)\, , \end{array} \right.
\]
where \(g \in L^\infty(\mathbb{R}^2)\). The above equation describes the propagation of laser beams in an inhomogeneous medium. The author addresses the question of well posedness and the behavior of the solutions \(u_n\) as \(n\to \infty\) and obtains sufficient condition on \(g\) to ensure existence and uniqueness of global solutision for \(n\) large. Moreover, the author proves that under suitable conditions the solutions \(u_n\) converge as \(n \to \infty\) to the solution of the problem
\[
\left\{ \begin{array}{l} i\partial_t u+\Delta u =\overline{g}|u|^2u \\
u(0)=u_0 \in L^2(\mathbb{R}^2)\, . \end{array} \right.
\]
Reviewer: Paolo Musolino (Padova)Local well-posedness for 2D incompressible magneto-micropolar boundary layer system.https://zbmath.org/1460.769332021-06-15T18:09:00+00:00"Lin, Xueyun"https://zbmath.org/authors/?q=ai:lin.xueyun"Zhang, Ting"https://zbmath.org/authors/?q=ai:zhang.tingThe authors consider a two-dimensional incompressible magneto-micropolar boundary layer system. Using suitable variables and analytic energy estimates, they obtain the local well-posedness for the two-dimensional incompressible magneto-micropolar boundary layer system when the initial data is analytic in the \(x\) variable.
Reviewer: Panagiotis Koumantos (Athína)Optimal control of a stochastic system related to the Kermack-McKendrick model.https://zbmath.org/1460.922062021-06-15T18:09:00+00:00"Lefebvre, Mario"https://zbmath.org/authors/?q=ai:lefebvre.marioSummary: A stochastic optimal control problem for a two-dimensional system of differential equations related to the Kermack-McKendrick model for the spread of epidemics is considered. The aim is to maximize the expected value of the time during which the epidemic is under control, taking the quadratic control costs into account. An exact and explicit solution is found in a particular case.Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space.https://zbmath.org/1460.352912021-06-15T18:09:00+00:00"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Wu, Xing"https://zbmath.org/authors/?q=ai:wu.xing"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhu, Weipeng"https://zbmath.org/authors/?q=ai:zhu.weipengSummary: Whether or not the data-to-solution map of the Cauchy problem for the Camassa-Holm equation and Novikov equation in the critical Besov space \(B_{2,1}^{3/2}(\mathbb{R})\) is uniformly continuous remains open. In the paper, we aim at solving the open question left in the previous works [the first author et al., J. Differ. Equations 269, No. 10, 8686--8700 (2020; Zbl 1442.35344); J. Math. Fluid Mech. 22, No. 4, Paper No. 50, 10 p. (2020; Zbl 1448.35402)] and giving a negative answer to this problem.Generalized compressible flows and solutions of the \(H(\text{div})\) geodesic problem.https://zbmath.org/1460.766592021-06-15T18:09:00+00:00"Gallouët, Thomas O."https://zbmath.org/authors/?q=ai:gallouet.thomas-o"Natale, Andrea"https://zbmath.org/authors/?q=ai:natale.andrea"Vialard, François-Xavier"https://zbmath.org/authors/?q=ai:vialard.francois-xavierSummary: We study the geodesic problem on the group of diffeomorphism of a domain \(M\subset{\mathbb{R}}^d\), equipped with the \(H(\text{div})\) metric. The geodesic equations coincide with the Camassa-Holm equation when \(d=1\), and represent one of its possible multi-dimensional generalizations when \(d>1\). We propose a relaxation à la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over \(M\). We use this relaxation to prove that smooth \(H(\text{div})\) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.Some explicit solutions to the three-dimensional nonlinear water wave problem.https://zbmath.org/1460.352732021-06-15T18:09:00+00:00"Martin, Calin I."https://zbmath.org/authors/?q=ai:martin.calin-iulianSummary: We present some explicit solutions (given in Eulerian coordinates) to the three-dimensional nonlinear water wave problem. The velocity field of some of the solutions exhibits a non-constant vorticity vector. An added bonus of the solutions we find is the possibility of incorporating a variable (in time and space) surface pressure which has a radial structure. A special type of radial structure of the surface pressure (of exponential type) is one of the features displayed by hurricanes [\textit{J. E. Overland}, ``Providing winds for wave models'', Marine Sci. 8, 3--37 (1979; \url{doi:10.1007/978-1-4684-3399-9_1})].On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum.https://zbmath.org/1460.352932021-06-15T18:09:00+00:00"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.6|liu.yang.17|liu.yang.21|liu.yang.3|liu.yang.1|liu.yang.8|liu.yang.9|liu.yang.16|liu.yang.13|liu.yang.20|liu.yang.23|liu.yang.11|liu.yang.15|liu.yang|liu.yang.10|liu.yang.12|liu.yang.2|liu.yang.18|liu.yang.19|liu.yang.5|liu.yang.4|liu.yang.22|liu.yang.14"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that \(\|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3}\) is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.On the non-diffusive magneto-geostrophic equation.https://zbmath.org/1460.769322021-06-15T18:09:00+00:00"Lear, Daniel"https://zbmath.org/authors/?q=ai:lear.danielSummary: Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In [Nonlinearity 24, No. 11, 3019--3042 (2011; Zbl 1228.76036)], \textit{S. Friedlander} and \textit{V. Vicol} prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the \(H^{5/2^+}(\mathbb{T}^3)\) norm of the perturbation.Pullback attractors for non-Newtonian fluids with shear dependent viscosity.https://zbmath.org/1460.350482021-06-15T18:09:00+00:00"López-Lázaro, Heraclio Ledgar"https://zbmath.org/authors/?q=ai:lopez-lazaro.heraclio-ledgar"Marín-Rubio, Pedro"https://zbmath.org/authors/?q=ai:marin-rubio.pedro"Planas, Gabriela"https://zbmath.org/authors/?q=ai:planas.gabrielaSummary: The so-called Ladyzhenskaya model is analyzed from a non-autonomous dynamical system point of view, for weak and strong solutions. Existence of attractors when forces are time-dependent is ensured in several universes with different tempered parameter conditions, and also for fixed bounded sets. Attraction is proved in \(L^2\) and \(W^{1,p}\) norms and relationships between these families are also established.Waves trapped by semi-infinite Kirchhoff plate at ultra-low frequencies.https://zbmath.org/1460.740492021-06-15T18:09:00+00:00"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: In this paper, a semi-infinite Kirchhoff plate with a traction-free edge, which rests partially on a heterogeneous Winkler foundation (the Neumann problem for the biharmonic operator perturbed by a small free term with a compact support), is considered. It is shown that, for arbitrary small \(\epsilon > 0\), a variable foundation compliance coefficient (defined nonuniquely) of order \(\epsilon\) can be constructed, such that the plate obtains the eigenvalue \(\epsilon^4\) that is embedded into a continuous spectrum, and the corresponding eigenfunction decays exponentially at infinity. It is verified that no more than one small eigenvalue can exist. It is noteworthy that a small perturbation cannot prompt an emergence of an eigenvalue near the cutoff point of the continuous spectrum in an acoustic waveguide (the Neumann problem for the Laplace operator).Viscous flow around a rigid body performing a time-periodic motion.https://zbmath.org/1460.352552021-06-15T18:09:00+00:00"Eiter, Thomas"https://zbmath.org/authors/?q=ai:eiter.thomas-walter"Kyed, Mads"https://zbmath.org/authors/?q=ai:kyed.madsSummary: The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.Singular limit for rotating compressible fluids with centrifugal force in a finite cylinder.https://zbmath.org/1460.768912021-06-15T18:09:00+00:00"Li, Yang"https://zbmath.org/authors/?q=ai:li.yang.5|li.yang|li.yang.3|li.yang.4|li.yang.2Summary: In this paper, we consider a rotating compressible Navier-Stokes system in a finite cylinder, by incorporating the effect of centrifugal force. The Mach number and Froude number admit the same scale. By using the relative entropy method, we prove that the scaled problem approaches the static profile as the Mach and Reynolds numbers tend to zero and infinity respectively, with well-prepared initial data. In particular, the property of radial symmetry resulting from the centrifugal force is crucial in the whole analysis.Growth of Sobolev norms for abstract linear Schrödinger equations.https://zbmath.org/1460.353072021-06-15T18:09:00+00:00"Bambusi, Dario"https://zbmath.org/authors/?q=ai:bambusi.dario"Grébert, Benoît"https://zbmath.org/authors/?q=ai:grebert.benoit"Maspero, Alberto"https://zbmath.org/authors/?q=ai:maspero.alberto"Robert, Didier"https://zbmath.org/authors/?q=ai:robert.didierSummary: We prove an abstract theorem giving a \(\langle t\rangle^\epsilon\) bound (for all \(\epsilon > 0)\) on the growth of the Sobolev norms in linear Schrödinger equations of the form \(\mathrm i \dot{\psi} = H_0 \psi + V(t) \psi\) as \(t \to \infty\). The abstract theorem is applied to several cases, including the cases where (i) \(H_0\) is the Laplace operator on a Zoll manifold and \(V(t)\) a pseudodifferential operator of order smaller than 2; (ii) \(H_0\) is the (resonant or nonresonant) Harmonic oscillator in \(\mathbb R^d\) and \(V(t)\) a pseudodifferential operator of order smaller than that of \(H_0\) depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of \textit{A. Maspero} and the last author [J. Funct. Anal. 273, No. 2, 721--781 (2017; Zbl 1366.35153)].Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model.https://zbmath.org/1460.800012021-06-15T18:09:00+00:00"Sellitto, A."https://zbmath.org/authors/?q=ai:sellitto.antonio"Carlomagno, I."https://zbmath.org/authors/?q=ai:carlomagno.isabella"Di Domenico, M."https://zbmath.org/authors/?q=ai:di-domenico.maria-carlaManufacturing of micro- and nanodevices makes it necessary to analyze heat transfer on nanoscale. It is impossible to use classical Fourier law in this case because characteristic length scale of the problem is comparable with mean free path of heat carreers. For example, these difficulties take place while modelling of electrons and phonons passing through a crystal lattice.
In order to describe the physical system mentioned above, the authors use a two-temperature model that takes into account nonlocal and nonlinear effects as well. In particular, the authors introduce relaxation times and mean free paths of electrons and phonons as important model parameters in addition to the ``classical'' parameters such as specific heats and heat conductivities.
Though this model is not obtained by rigorous microscopic derivation, the authors discuss its consistency with statistics of Bose-Einstein and Fermi-Dirac as well as with the 2nd law of thermodynamics.
The techniques of acceleration waves is then used to describe the heat propagation in the two-temperature medium. The dependence of heat wave speeds and amplitudes in electronic and phononic gases on the physical system characteristics is found. In particular, conditions are derived that show when the waves under discussion become shock waves.
Reviewer: Aleksey Syromyasov (Saransk)On the parabolic equation for portfolio problems.https://zbmath.org/1460.912572021-06-15T18:09:00+00:00"Zawisza, Dariusz"https://zbmath.org/authors/?q=ai:zawisza.dariuszSummary: We consider a semilinear equation linked to the finite horizon consumption-investment problem under stochastic factor framework, prove it admits a classical solution and provide all obligatory estimates to successfully apply a verification reasoning. The paper covers the standard time additive utility, as well as the recursive utility framework. We extend existing results by considering more general factor dynamics including a nontrivial diffusion part and a stochastic correlation between assets and factors. In addition, this is the first paper which compromise many other optimization problems in finance, for example those related to the indifference pricing or the quadratic hedging problem. The extension of the result to the stochastic differential utility and robust portfolio optimization is provided as well. The essence of our paper lays in using improved stochastic methods to prove gradient estimates for suitable HJB equations with restricted control space.
For the entire collection see [Zbl 1460.93006].Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents.https://zbmath.org/1460.761102021-06-15T18:09:00+00:00"Steer, James N."https://zbmath.org/authors/?q=ai:steer.james-n"Borthwick, Alistair G. L."https://zbmath.org/authors/?q=ai:borthwick.alistair-g-l"Stagonas, Dimitris"https://zbmath.org/authors/?q=ai:stagonas.dimitris"Buldakov, Eugeny"https://zbmath.org/authors/?q=ai:buldakov.eugeny"Bremer, Ton S. van den"https://zbmath.org/authors/?q=ai:van-den-bremer.ton-sSummary: This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current profiles (with constant vorticity). Measurements are compared against predictions from the unidirectional \((1\text{D}+1)\) constant vorticity nonlinear Schrödinger equation (the vor-NLSE) derived by \textit{R. Thomas} [``A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity'', Phys. Fluids 24, No. 12, Paper No. 127102 (2012; \url{doi:10.1063/1.4768530})]. The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the effect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on five opposing current profiles with shear rates from \(0\) to \(-0.87\text{s}^{-1}\). For all current profiles, the linear constant vorticity dispersion relation predicts the wavenumber to within the 95\% confidence bounds associated with estimates of shear rate and surface current velocity. The effect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current profiles with shear rates between \(0\) and \(-0.48\text{s}^{-1}\). Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the amplification of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and significantly over-predicted by the uniform-current NLSE.Locking-free and gradient-robust \(H(\operatorname{div})\)-conforming HDG methods for linear elasticity.https://zbmath.org/1460.651462021-06-15T18:09:00+00:00"Fu, Guosheng"https://zbmath.org/authors/?q=ai:fu.guosheng"Lehrenfeld, Christoph"https://zbmath.org/authors/?q=ai:lehrenfeld.christoph"Linke, Alexander"https://zbmath.org/authors/?q=ai:linke.alexander"Streckenbach, Timo"https://zbmath.org/authors/?q=ai:streckenbach.timoIn this paper, the concept of gradient-robustness for numerical methods for linear elasticity is introduced. A special class of discretizations for linear elasticity is considered: \(H(\operatorname{div})\)-conforming Hybrid Discontinuous Galerkin (HDG) discretizations by keeping track of the volume-locking and gradient-robustness property of the method. Two efficient variants of a divergence-conforming HDG scheme with reduced globally coupled degrees of freedom are discussed and analyzed. The importance of gradient-robustness and the computational efficiency of the proposed relaxed \(H(\operatorname{div})\)-conforming HDG method is demonstrated on a linear multi-physics thermo-elasticity problem in 2D and 3D. It was shown that the gradient-robustness against strong gradients fields in \(L^2\) leads to much more accurate schemes for nearly incompressible materials
Reviewer: Bülent Karasözen (Ankara)A numerical approach for fluid deformable surfaces.https://zbmath.org/1460.760662021-06-15T18:09:00+00:00"Reuther, S."https://zbmath.org/authors/?q=ai:reuther.sebastian"Nitschke, I."https://zbmath.org/authors/?q=ai:nitschke.ingo"Voigt, Axel"https://zbmath.org/authors/?q=ai:voigt.axelSummary: Fluid deformable surfaces show a solid-fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where, in the presence of curvature, any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity.Codimension one minimizers of highly amphiphilic mixtures.https://zbmath.org/1460.820152021-06-15T18:09:00+00:00"Dai, Shibin"https://zbmath.org/authors/?q=ai:dai.shibin"Promislow, Keith"https://zbmath.org/authors/?q=ai:promislow.keithSummary: We present a modified form of the Functionalized Cahn-Hilliard (FCH) functional which models highly amphiphilic systems in solvent. A molecule is highly amphiphilic if the energy of a molecule isolated within the bulk solvent molecule is prohibitively high. For such systems once the amphiphilic molecules assemble into a structure it is very rare for a molecule to exchange back into the bulk. The highly amphiphilic FCH functional has a well with limited smoothness and admits compactly supported critical points. In the limit of molecular length \(\varepsilon \to 0\) we consider sequences with bounded energy whose support resides within an \(\varepsilon\)-neighborhood of a fixed codimension one interface. We show that the FCH energy is uniformly bounded from below, independent of \(\varepsilon>0\), and identify assumptions on tangential variation of sequences that guarantee the existence of subsequences that converge to a weak solution of a rescaled bilayer profile equation, and show that sequences with limited tangential variation enjoy a \(\operatorname{lim inf}\) inequality. For fixed codimension one interfaces we construct bounded energy sequences which converge to the bilayer profile and others with larger tangential variation which do not converge to the bilayer profile but whose limiting energy can violate the \(\operatorname{lim inf}\) inequality, depending upon the energy parameters.Energy-preserving methods for nonlinear Schrödinger equations.https://zbmath.org/1460.650992021-06-15T18:09:00+00:00"Besse, Christophe"https://zbmath.org/authors/?q=ai:besse.christophe"Descombes, Stéphane"https://zbmath.org/authors/?q=ai:descombes.stephane"Dujardin, Guillaume"https://zbmath.org/authors/?q=ai:dujardin.guillaume-michel"Lacroix-Violet, Ingrid"https://zbmath.org/authors/?q=ai:lacroix-violet.ingridSummary: This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analogue of it. The Crank-Nicolson method is a well-known method of order \(2\) but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [\textit{C. Besse}, Analyse numérique des systèmes de Davey-Stewartson. Bordeaux: Université Bordeaux (PhD Thesis) (1998)] for the cubic nonlinear Schrödinger equation. This method is also an energy-preserving method and numerical simulations have shown that its order is \(2\). In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows one to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.Multisoliton solutions of the Degasperis-Procesi equation and its shortwave limit: Darboux transformation approach.https://zbmath.org/1460.761242021-06-15T18:09:00+00:00"Li, Nianhua"https://zbmath.org/authors/?q=ai:li.nianhua"Wang, Gaihua"https://zbmath.org/authors/?q=ai:wang.gaihua"Kuang, Yonghui"https://zbmath.org/authors/?q=ai:kuang.yonghuiSummary: We propose a new approach for calculating multisoliton solutions of the Degasperis-Procesi equation and its shortwave limit by combining a reciprocal transformation with the Darboux transformation of the negative flow of the Kaup-Kupershmidt hierarchy. In particular, different specifications of the soliton parameters lead to two different types of soliton solutions of the Degasperis-Procesi equation.Lie point symmetries and conservation laws for a class of BBM-KdV systems.https://zbmath.org/1460.766542021-06-15T18:09:00+00:00"Silva Junior, Valter Aparecido"https://zbmath.org/authors/?q=ai:silva-junior.valter-aparecidoSummary: We determine the Lie point symmetries of a class of BBM-KdV systems and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport.https://zbmath.org/1460.651382021-06-15T18:09:00+00:00"Alvarez, Mario"https://zbmath.org/authors/?q=ai:alvarez.mario-m"Gatica, Gabriel N."https://zbmath.org/authors/?q=ai:gatica.gabriel-n"Ruiz-Baier, Ricardo"https://zbmath.org/authors/?q=ai:ruiz-baier.ricardoSummary: This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy's law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.The diffusive logistic equation on periodically evolving domains.https://zbmath.org/1460.352042021-06-15T18:09:00+00:00"Jiang, Dan-Hua"https://zbmath.org/authors/?q=ai:jiang.danhua"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.1|wang.zhi-cheng.2Summary: A diffusive logistic equation on \(n\)-dimensional periodically and isotropically evolving domains is investigated. We first derive the model and present the eigenvalue problem on evolving domains. Then we prove that the species persists if the diffusion rate \(d\) is below the critical value \(\underline{D}_0\), while the species become extinct if it is above the critical value \(\overline{D}_0\). Finally, we analyze the effect of domain evolution rate on the persistence of a species. Precisely, it depends on the average value \(\overline{\rho^{-2}}\), where \(\rho(t)\) is the domain evolution rate, and \(\overline{\rho^{-2}}=\frac{1}{T}\int_0^T\frac{1}{\rho^2(t)}\,dt\). If \(\overline{\rho^{-2}}>1\), the periodical domain evolution has a negative effect on the persistence of a species. If \(\overline{\rho^{-2}}<1\), the periodical domain evolution has a positive effect on the persistence of a species. If \(\overline{\rho^{-2}}=1\), the periodical domain evolution has no effect on the persistence of a species. Numerical simulations are also presented to illustrate the analytical results.Sparsified discrete wave problem involving a radiation condition on a prolate spheroidal surface.https://zbmath.org/1460.651392021-06-15T18:09:00+00:00"Barucq, Hélène"https://zbmath.org/authors/?q=ai:barucq.helene"Fares, M'Barek"https://zbmath.org/authors/?q=ai:fares.mbarek"Kruse, Carola"https://zbmath.org/authors/?q=ai:kruse.carola"Tordeux, Sébastien"https://zbmath.org/authors/?q=ai:tordeux.sebastienSummary: We develop and analyse a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet-to-Neumann condition is constructed using the classical method of separation of variables that allows one to define the scattered field in a truncated domain. It reads as an infinite series that is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix that involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman-Morrison algorithm that reduces the memory usage drastically. Despite being of high order, this method has only a low memory cost.A numerical-analysis-focused comparison of several finite volume schemes for a unipolar degenerate drift-diffusion model.https://zbmath.org/1460.651042021-06-15T18:09:00+00:00"Cancès, Clément"https://zbmath.org/authors/?q=ai:cances.clement"Chainais-Hillairet, Claire"https://zbmath.org/authors/?q=ai:chainais-hillairet.claire"Fuhrmann, Jürgen"https://zbmath.org/authors/?q=ai:fuhrmann.jurgen"Gaudeul, Benoît"https://zbmath.org/authors/?q=ai:gaudeul.benoitSummary: In this paper we consider a unipolar degenerate drift-diffusion system where the relation between the concentration of the charged species \(c\) and the chemical potential \(h\) is \(h(c)=\log \frac{c}{1-c}\). We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.Besov regularity for the stationary Navier-Stokes equation on bounded Lipschitz domains.https://zbmath.org/1460.352532021-06-15T18:09:00+00:00"Eckhardt, Frank"https://zbmath.org/authors/?q=ai:eckhardt.frank"Cioica-Licht, Petru A."https://zbmath.org/authors/?q=ai:cioica-licht.petru-a"Dahlke, Stephan"https://zbmath.org/authors/?q=ai:dahlke.stephanSummary: We use the scale \(B^s_\tau(L_\tau(\Omega)), 1/\tau=s/d+1/2,s> 0\), to study the regularity of the stationary Stokes equation on bounded Lipschitz domains \(\Omega\subset\mathbb{R}^d,d\geq 3\), with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. Using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.Convergence of some mean field games systems to aggregation and flocking models.https://zbmath.org/1460.353492021-06-15T18:09:00+00:00"Bardi, Martino"https://zbmath.org/authors/?q=ai:bardi.martino"Cardaliaguet, Pierre"https://zbmath.org/authors/?q=ai:cardaliaguet.pierreThe authors investigate the convergence of solutions for two classes of Mean Field Game (MFG) systems. The first class of MFG systems with control on the velocity is given by a parabolic system with a large parameter \(\lambda\) associated to a stochastic MFG, for which the solution converges to a solution of a aggregation model as \(\lambda\to\infty\). The second class of MFG systems with control on acceleration is a first order PDEs system for which the solution converges to the solution of a kinetic equation. In the main results, they use PDEs methods for the first model, and variational methods in the space of probability measures on trajectories for the second model.
Reviewer: Rodica Luca (Iaşi)Strichartz estimates and Fourier restriction theorems on the Heisenberg group.https://zbmath.org/1460.353652021-06-15T18:09:00+00:00"Bahouri, Hajer"https://zbmath.org/authors/?q=ai:bahouri.hajer"Barilari, Davide"https://zbmath.org/authors/?q=ai:barilari.davide"Gallagher, Isabelle"https://zbmath.org/authors/?q=ai:gallagher.isabelleSummary: This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group \(\mathbb{H}^d\) for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on \(\mathbb{H}^d\) is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in [\textit{P. A. Tomas}, Bull. Am. Math. Soc. 81, 477--478 (1975; Zbl 0298.42011)], is based on Fourier restriction theorems on \(\mathbb{H}^d\), using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.On the inelastic Boltzmann equation for soft potentials with diffusion.https://zbmath.org/1460.766852021-06-15T18:09:00+00:00"Meng, Fei"https://zbmath.org/authors/?q=ai:meng.fei"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1Summary: We are concerned with the Cauchy problem of the inelastic Boltzmann equation for soft potentials, with a Laplace term representing the random background forcing. The inelastic interaction here is characterized by the non-constant restitution coefficient. We prove that under the assumption that the initial datum has bounded mass, energy and entropy, there exists a weak solution to this equation. The smoothing effect of weak solutions is also studied. In addition, it is shown the solution is unique and stable with respect to the initial datum provided that the initial datum belongs to \(L^2(R^3)\).Wave and Klein-Gordon equations on certain locally symmetric spaces.https://zbmath.org/1460.353362021-06-15T18:09:00+00:00"Zhang, Hong-Wei"https://zbmath.org/authors/?q=ai:zhang.hongweiSummary: This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.Remarks on the derivation of finite energy weak solutions to the QHD system.https://zbmath.org/1460.352752021-06-15T18:09:00+00:00"Antonelli, Paolo"https://zbmath.org/authors/?q=ai:antonelli.paoloSummary: In this note we give an alternative proof of existence of finite energy weak solutions to the quantum hydrodynamics (QHD) system. The main novelty in our approach is that no regularization procedure or approximation is needed, as it is only based on the integral formulation of NLS equation and the a priori bounds given by the Strichartz estimates. The main advantage of this proof is that it can be applied to a wider class of QHD systems.On a stationary Schrödinger equation with periodic magnetic potential.https://zbmath.org/1460.353162021-06-15T18:09:00+00:00"Bégout, Pascal"https://zbmath.org/authors/?q=ai:begout.pascal"Schindler, Ian"https://zbmath.org/authors/?q=ai:schindler.ianSummary: We prove existence results for a stationary Schrödinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.A posteriori error estimates and an adaptive finite element solution for the system of unsteady convection-diffusion-reaction equations in fluidized beds.https://zbmath.org/1460.651272021-06-15T18:09:00+00:00"Varma, V. Dhanya"https://zbmath.org/authors/?q=ai:varma.v-dhanya"Nadupuri, Suresh Kumar"https://zbmath.org/authors/?q=ai:nadupuri.suresh-kumar"Chamakuri, Nagaiah"https://zbmath.org/authors/?q=ai:chamakuri.nagaiahThe main aim of this paper is the construction of an a posteriori error estimator for a system of five unsteady and strongly coupled convection-diffusion-reaction equations in fluidized bed spray granulation (FBSG). In order to solve this nonlinear challenging system the authors make use of an adaptive finite element method in space variable and an implicit Euler scheme in time variable, both based on a weak formulation. A total residual, which is a sum of the residual due to space discretization, time discretization and linearization, is introduced. The nonlinear reaction term is manipulated using its Lipschitz property in the framework of Newton's method where the Jacobian matrix of the system is based on the exact derivatives. Some numerical examples are carried out in order to underline the effectiveness of the a posteriori error analysis.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Some comparisons between heterogeneous and homogeneous plates for nonlinear symmetric SH waves in terms of heterogeneous and nonlinear effects.https://zbmath.org/1460.350672021-06-15T18:09:00+00:00"Demirkuş, Dilek"https://zbmath.org/authors/?q=ai:demirkus.dilekSummary: In this article, the propagation of nonlinear shear horizontal waves for some comparisons between the heterogeneous and homogeneous plates is considered. It is assumed that one plate is made of up hyper-elastic, heterogeneous, isotropic, and generalized neo-Hookean materials, and the other consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to the nonlinear Schrödinger equation, these comparisons are made in terms of the heterogeneous and nonlinear effects. All numerical results, based on the asymptotic analyses in which the method of multiple scales is used, are graphically presented for the lowest dispersive symmetric branch of both dispersion relations.Periodic solutions to Navier-Stokes equations on non-compact Einstein manifolds with negative curvature.https://zbmath.org/1460.352632021-06-15T18:09:00+00:00"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Pham, Truong Xuan"https://zbmath.org/authors/?q=ai:pham.truong-xuan"Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha."Vu, Thi Mai"https://zbmath.org/authors/?q=ai:vu.thi-maiSummary: Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold \((\mathbf{M},g)\) with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on \((\mathbf{M},g)\). Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold \((\mathbf{M},g)\). We also prove the stability of the periodic solution.Vanishing viscosity in the Navier-Stokes equations of compressible heat-conducting flows with the spherical symmetry.https://zbmath.org/1460.766622021-06-15T18:09:00+00:00"Song, Wenjing"https://zbmath.org/authors/?q=ai:song.wenjing"Su, Wenhuo"https://zbmath.org/authors/?q=ai:su.wenhuoSummary: In this paper, we study the limit process as the bulk viscosity goes to zero for global weak solutions to the Navier-Stokes equations of compressible heat-conducting flows with density-dependent viscosity coefficient and general heat conductivity. We prove that the limit of the global solutions is a weak solution of the corresponding system with zero bulk viscosity.Surfaces of revolution associated with the kink-type solutions of the SIdV equation.https://zbmath.org/1460.353142021-06-15T18:09:00+00:00"Zhang, Guofei"https://zbmath.org/authors/?q=ai:zhang.guofei"He, Jingsong"https://zbmath.org/authors/?q=ai:he.jingsong"Wang, Lihong"https://zbmath.org/authors/?q=ai:wang.lihong"Mihalache, Dumitru"https://zbmath.org/authors/?q=ai:mihalache.dumitruSummary: In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.Normalized solutions for nonlinear Schrödinger systems with linear couples.https://zbmath.org/1460.353202021-06-15T18:09:00+00:00"Chen, Zhen"https://zbmath.org/authors/?q=ai:chen.zhen"Zou, Wenming"https://zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, we study the normalized solutions to the following system
\[
\begin{cases}
- \Delta u + ( V_1 ( x ) + \lambda_1 ) u = \mu_1 | u |^{p - 2} u + \beta v \quad & \text{in } \mathbb{R}^N, \\
- \Delta v + ( V_2 ( x ) + \lambda_2 ) v = \mu_2 | v |^{q - 2} v + \beta u & \text{in } \mathbb{R}^N, \\
\int_{\mathbb{R}^N} u^2 = a, \quad \int_{\mathbb{R}^N} v^2 = b,
\end{cases}
\]
with the mass-subcritical condition \(2 < p, q < 2 + \frac{ 4}{ N} \), where \(\mu_1, \mu_2, a, b > 0\), \(\beta \in \mathbb{R} \setminus \{0 \}\) are prescribed; \( \lambda_1, \lambda_2 \in \mathbb{R}\) are to be determined. We prove the existence of a solution with prescribed \(L^2\)-norm under some various conditions on the potential \(V_1, V_2 : \mathbb{R}^N \to \mathbb{R} \). The proof is based on the refined energy estimates.Stability of non-constant equilibrium solutions for the full compressible Navier-Stokes-Maxwell system.https://zbmath.org/1460.352822021-06-15T18:09:00+00:00"Feng, Yue-Hong"https://zbmath.org/authors/?q=ai:feng.yuehong"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.9|li.xin.3|li.xin.7|li.xin.10|li.xin.6|li.xin.4|li.xin.2|li.xin.1|li.xin.12|li.xin.13|li.xin.5|li.xin.15|li.xin|li.xin.11|li.xin.14"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shuSummary: In this article we consider a Cauchy problem for the full compressible Navier-Stokes-Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic-parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier-Stokes-Poisson system in a three-dimensional torus.On the forced surface quasi-geostrophic equation: existence of steady states and sharp relaxation rates.https://zbmath.org/1460.352842021-06-15T18:09:00+00:00"Hadadifard, Fazel"https://zbmath.org/authors/?q=ai:hadadifard.fazel"Stefanov, Atanas G."https://zbmath.org/authors/?q=ai:stefanov.atanas-gSummary: We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all \(L^1\cap L^\infty\) localized initial data produces global solutions of the forced SQG, which converge to the steady states in \(L^p(\mathbb{R}^2)\), \(1<p\leq 2\) as time goes to infinity. This establishes that the steady states serve as one point attracting set. Moreover, by employing the method of scaling variables, we compute the sharp relaxation rates, by requiring slightly more localized initial data.A new instability for Boussinesq-type equations.https://zbmath.org/1460.763292021-06-15T18:09:00+00:00"Kirby, James T."https://zbmath.org/authors/?q=ai:kirby.james-tSummary: A wide class of problems for free-surface gravity waves fall into a weakly dispersive regime, in which wavelength is large compared to water depth, and wave phase speed differs by a small amount from the speed \(c_0=\sqrt{gh}\) of shallow-water waves. The resulting problem is treated naturally using Taylor series expansions of dependent variables in the vertical coordinate, leading to a class of models that are collectively referred to here as Boussinesq-type models. \textit{P. A. Madsen} and \textit{D. R. Fuhrman} [ibid. 889, Article ID A38, 25 p. (2020; Zbl 1460.76102)] have recently shown that certain members of this broad class of models are subject to a high-wavenumber instability, which can grow rapidly when the elevation of the wave trough is sufficiently depressed below the mean water surface. This newly revealed instability may provide an explanation for the modelling community's frequent observations of noisy behaviour in Boussinesq-type model calculations.Dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics.https://zbmath.org/1460.170402021-06-15T18:09:00+00:00"Hentosh, Oksana Ye."https://zbmath.org/authors/?q=ai:hentosh.oksana-ye"Prykarpatsky, Yarema A."https://zbmath.org/authors/?q=ai:prykarpatsky.yarema-anatoliyovych"Blackmore, Denis"https://zbmath.org/authors/?q=ai:blackmore.denis-l"Prykarpatski, Anatolij K."https://zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevychSummary: Using diffeomorphism group vector fields on \(\mathbb{C}\)-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.Relative decay conditions on Liouville type theorem for the steady Navier-Stokes system.https://zbmath.org/1460.352512021-06-15T18:09:00+00:00"Chae, Dongho"https://zbmath.org/authors/?q=ai:chae.donghoSummary: In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in \(\mathbb{R}^3\) under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth solution \((u,p)\) of the stationary Navier-Stokes equations satisfying \(u(x) \rightarrow 0\) as \(|x|\rightarrow +\infty\) and the condition of finite Dirichlet integral \(\int_{\mathbb{R}^3} | \nabla u|^2 dx <+\infty\) is trivial, if either \(|u(x)|/|Q(x)|=O(1)\) or \(|p(x)|/|Q(x)| =O(1)\) as \(|x|\rightarrow \infty \), where \(|Q|=\frac{1}{2} |u|^2 +p\) is the head pressure.Joint inversion of high-frequency induction and lateral logging sounding data in Earth models with tilted principal axes of the electrical resistivity tensor.https://zbmath.org/1460.353382021-06-15T18:09:00+00:00"Nechaev, Oleg"https://zbmath.org/authors/?q=ai:nechaev.oleg-valentinovich"Glinskikh, Viacheslav"https://zbmath.org/authors/?q=ai:glinskikh.viacheslav-nikolaevich"Mikhaylov, Igor"https://zbmath.org/authors/?q=ai:mikhaylov.igor"Moskaev, Ilya"https://zbmath.org/authors/?q=ai:moskaev.ilyaSummary: In this article, we are the first to formulate the direct and inverse problems of resistivity logging on determining the components of the electrical resistivity tensor of rocks from a set of high-frequency induction and lateral logging sounding measurements. Using a finite element approximation, high-order hierarchical basis functions, computationally efficient multilevel methods and a multistart algorithm with the DFO-LS local optimization method, we investigate the capability of reconstructing the horizontal and vertical resistivity components, as well as the tilt of the resistivity tensor principal axes with regard to the study of complex geological objects. A separate consideration is given to a realistic generalized geoelectric model of the unique hydrocarbon source with hard-to-recover reserves, the Bazhenov Formation.A partitioned finite element method for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control.https://zbmath.org/1460.353612021-06-15T18:09:00+00:00"Serhani, Anass"https://zbmath.org/authors/?q=ai:serhani.anass"Matignon, Denis"https://zbmath.org/authors/?q=ai:matignon.denis"Haine, Ghislain"https://zbmath.org/authors/?q=ai:haine.ghislainSummary: Many boundary controlled and observed partial differential equations can be represented as port-Hamiltonian systems with dissipation, involving a Stokes-Dirac geometrical structure together with constitutive relations. The partitioned finite element method, introduced in [\textit{F. L. Cardoso-Ribeiro} et al., ``A structure-preserving partitioned finite element method for the 2D wave equation'', IFAC-Papers OnLine 51, No. 3, 119--124 (2018; \url{doi:10.1016/j.ifacol.2018.06.033})], is a structure preserving numerical method which defines an underlying Dirac structure, and constitutive relations in weak form, leading to finite-dimensional port-Hamiltonian differential algebraic systems (pHDAE). Different types of dissipation are examined: internal damping, boundary damping and also diffusion models.
For the entire collection see [Zbl 1428.94016].Machine learning based data retrieval for inverse scattering problems with incomplete data.https://zbmath.org/1460.352832021-06-15T18:09:00+00:00"Gao, Yu"https://zbmath.org/authors/?q=ai:gao.yu"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kaiSummary: We are concerned with the inverse scattering problems associated with incomplete measurement data. It is a challenging topic of increasing importance that arise in many practical applications. Based on a prototypical working model, we propose a machine learning based inverse scattering scheme, which integrates a CNN (convolution neural network) for the data retrieval. The proposed method can effectively cope with the reconstruction under limited-aperture and/or phaseless far-field data. Numerical experiments verify the promising features of our new scheme.Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction.https://zbmath.org/1460.351002021-06-15T18:09:00+00:00"Chen, Sitong"https://zbmath.org/authors/?q=ai:chen.sitong"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-hua"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlinThe authors consider a quasilinear Schrödinger equation of the form
\[
-\Delta u+V(x)u-\frac{1}{2}\Delta \left(u^2\right)u=g(u),\quad x\in\mathbb{R}^N,
\]
where \(N \geq 3\), \(V\in C(\mathbb{R}^N,[0,\infty))\) and \(g\in C(\mathbb{R},\mathbb{R})\) is superlinear at infinity. Under very general assumptions on \(V\) and \(g\) and by using variational tools and some new analytic techniques, it is shown that the problem above admits a ground state solution, where a solution is called a ground state solution if its energy is minimal among all nontrivial solutions.
Reviewer: Patrick Winkert (Berlin)A class of inverse problems for fractional order degenerate evolution equations.https://zbmath.org/1460.353722021-06-15T18:09:00+00:00"Fedorov, Vladimir E."https://zbmath.org/authors/?q=ai:fedorov.v-e"Nagumanova, Anna V."https://zbmath.org/authors/?q=ai:nagumanova.anna-viktorovich"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: The criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter-Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases.Darboux transformation of the coupled nonisospectral Gross-Pitaevskii system and its multi-component generalization.https://zbmath.org/1460.761252021-06-15T18:09:00+00:00"Xu, Tao"https://zbmath.org/authors/?q=ai:xu.tao"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.2|chen.yong.5|chen.yong.7|chen.yong.6|chen.yong.3|chen.yong.4|chen.yong.1|chen.yongSummary: In this paper, we extend the one-component Gross-Pitaevskii (GP) equation to the two-component coupled GP system including damping term, linear and parabolic density profiles. The Lax pair with nonisospectral parameter and infinitely-many conservation laws of this coupled GP system are presented. Actually, the Darboux transformation (DT) for this kind of nonautonomous system is essentially different from the autonomous case. Consequently, we construct the DT of the coupled GP equations, besides, nonautonomous multi-solitons, one-breather and the first-order rogue wave are also obtained. Various kinds of one-soliton solution are constructed, which include stationary one-soliton and nonautonomous one-soliton propagating along the negative (positive) direction of \(x\)-axis. The interaction of two solitons and two-soliton bound state are demonstrated respectively. We get the nonautonomous one-breather on a curved background and this background is completely controlled by the parameter \(\beta\). Using a limiting process, the nonautonomous first-order rogue wave can be obtained. Furthermore, some dynamic structures of these analytical solutions are discussed in detail. In addition, the multi-component generalization of GP equations are given, then the corresponding Lax pair and DT are also constructed.Dynamics of nonlinear thermoelastic double-beam systems.https://zbmath.org/1460.740202021-06-15T18:09:00+00:00"Campo, M."https://zbmath.org/authors/?q=ai:campo.marco-a"Fernández, J. R."https://zbmath.org/authors/?q=ai:fernandez.jose-ramon"Naso, M. G."https://zbmath.org/authors/?q=ai:naso.maria-grazia"Vuk, E."https://zbmath.org/authors/?q=ai:vuk.elenaIn this paper the authors consider the system describing the transverse vibrations of a symmetric elastically-coupled thermoelastic double-beam under even compresssive axial loading. It is assumed that each beam is elastic, extensible and supported by the ends. Hinged conditions are imposed for the displacement and Dirichlet homogeneous conditions are assumed for the temperatures. The problem determines a system composed of two nonlinear hyperbolic equations for the displacement and two linear parabolic equations for the temperature. An existence and uniqueness result is suggested by means of the Faedo-Galerkin approximation scheme. Energy arguments are used to obtain the exponential stability of the solutions. Then, fully discrete approximations are introduced by means of the finite element method and the implicit Euler scheme. Discrete stability and a priori error estimates are later obtained. The paper concludes by numerical simulations where the accuracy of the approximations is shown.
Reviewer: Ramón Quintanilla De Latorre (Barcelona)On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials.https://zbmath.org/1460.351032021-06-15T18:09:00+00:00"Yang, Jianfu"https://zbmath.org/authors/?q=ai:yang.jianfu"Yang, Jinge"https://zbmath.org/authors/?q=ai:yang.jingeThe existence and concentration of solutions to a supercritical elliptic equation with ellipse-shaped potential in the plane are studied through a variational approach.
Reviewer: Dumitru Motreanu (Perpignan)Convergence and numerical simulations of prey-predator interactions via a meshless method.https://zbmath.org/1460.650982021-06-15T18:09:00+00:00"Benito, J. J."https://zbmath.org/authors/?q=ai:benito.juan-jose"García, A."https://zbmath.org/authors/?q=ai:garcia.angelo|garcia.angel"Gavete, L."https://zbmath.org/authors/?q=ai:gavete.luis"Negreanu, M."https://zbmath.org/authors/?q=ai:negreanu.mihaela"Ureña, F."https://zbmath.org/authors/?q=ai:urena.francisco"Vargas, A. M."https://zbmath.org/authors/?q=ai:vargas.antonio-manuelSummary: We study two mathematical models consisting of nonlinear systems of partial differential equations, a predator prey and a competitive two-species chemotaxis systems with two chemicals satisfying their corresponding elliptic equations in a smooth bounded domain. By introducing global factors, for different ranges of parameters and by deriving a discretization of the system by means of the Generalized Finite Difference Method (GFDM) we prove that any positive and bounded discrete solution converges to the analytical one, i.e., a spatially homogeneous state. We apply the meshless method over regular and irregular domains where we simulate the behavior of the solution with the tools of several numerical examples.Book review of: L. R. Evangelista and E. Kaminski Lenzi, Fractional diffusion equations and anomalous diffusion.https://zbmath.org/1460.000172021-06-15T18:09:00+00:00"Magin, Richard L."https://zbmath.org/authors/?q=ai:magin.richard-lReview of [Zbl 1457.35001]A robust nonstandard finite difference scheme for pricing real estate index options.https://zbmath.org/1460.353502021-06-15T18:09:00+00:00"Dube, Mbakisi"https://zbmath.org/authors/?q=ai:dube.mbakisi"Patidar, Kailash C."https://zbmath.org/authors/?q=ai:patidar.kailash-cSummary: Real estate assets can be used to store capital, generate income through rentals and can act as collateral for debt instruments. Common risk management mechanisms for real estate investments use portfolio diversification techniques, but these techniques require large amounts of capital to work effectively. Real estate index derivatives offer an alternative mechanism for managing risks associated with real estate investments. They also increase market liquidity by providing a path for individuals who do not own real estate assets to participate in the real estate market. We consider the problem of pricing real estate index derivative contracts. The market is incomplete, so it is completed by futures derivatives on the same real estate index. A dynamic hedging strategy is employed leading to a parabolic partial differential equation with coefficients which are dependent on time and the real estate index. We then construct a nonstandard finite difference method to price European and American real estate index options. The scheme utilizes complete cubic spline interpolants of the option prices at time-dependent backtrack points on the spatial grid. Bounds for the global error are theoretically established. Numerical experiments are carried out to illustrate the accuracy of the scheme.Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas.https://zbmath.org/1460.352692021-06-15T18:09:00+00:00"Březina, Jan"https://zbmath.org/authors/?q=ai:brezina.jan"Kreml, Ondřej"https://zbmath.org/authors/?q=ai:kreml.ondrej"Mácha, Václav"https://zbmath.org/authors/?q=ai:macha.vaclavSummary: We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical \textit{BV} solution, instead a \(\delta\)-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exist infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one. In particular, not only the criterion based on comparing dissipation rates of total energy but also a stronger version based on comparing dissipation measures fails to pick the 1D solution.Liouville theorem of the 3D stationary MHD system: for D-solutions converging to non-zero constant vectors.https://zbmath.org/1460.352612021-06-15T18:09:00+00:00"Li, Zijin"https://zbmath.org/authors/?q=ai:li.zijin"Pan, Xinghong"https://zbmath.org/authors/?q=ai:pan.xinghongSummary: In this paper, we derive the Liouville theorem of D-solutions to the stationary MHD system under the asymptotic assumption: one of the velocity field and the magnetic field approaches zero and the other approaches a non zero constant vector at infinity. Our result extends the corresponding one of D-solutions to the Navier-Stokes equations when the velocity approaches a non zero constant vector at infinity.On the global well-posedness of the quadratic NLS on \(H^1(\mathbb{T}) + L^2(\mathbb{R})\).https://zbmath.org/1460.353192021-06-15T18:09:00+00:00"Chaichenets, L."https://zbmath.org/authors/?q=ai:chaichenets.leonid"Hundertmark, D."https://zbmath.org/authors/?q=ai:hundertmark.dirk"Kunstmann, P."https://zbmath.org/authors/?q=ai:kunstmann.peer-christian"Pattakos, N."https://zbmath.org/authors/?q=ai:pattakos.nikolaosSummary: We study the one dimensional nonlinear Schrödinger equation with power nonlinearity \(|u|^{\alpha-1} u\) for \(\alpha \in [1,5]\) and initial data \(u_0 \in H^1(\mathbb{T}) + L^2 (\mathbb{R})\). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity \((\alpha=2)\) we obtain \textit{global} well-posedness in the space \(C(\mathbb{R}, H^1(\mathbb{T}) + L^2 (\mathbb{R}))\) via Gronwall's inequality.Nonexistence for hyperbolic problems on Riemannian manifolds.https://zbmath.org/1460.353642021-06-15T18:09:00+00:00"Monticelli, Dario D."https://zbmath.org/authors/?q=ai:monticelli.dario-daniele"Punzo, Fabio"https://zbmath.org/authors/?q=ai:punzo.fabio"Squassina, Marco"https://zbmath.org/authors/?q=ai:squassina.marcoSummary: We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.Stability analysis of anti-periodic solutions of the time-varying delayed hematopoiesis model with discontinuous harvesting terms.https://zbmath.org/1460.352052021-06-15T18:09:00+00:00"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchao"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Fu, Xiangying"https://zbmath.org/authors/?q=ai:fu.xiangyingSummary: This paper is concerned with a time-varying delayed hematopoiesis model with discontinuous harvesting terms. The harvesting terms considered in our hematopoiesis model are discontinuous which are totally different from the previous continuous, Lipschitz continuous or even smooth ones. By means of functional differential inclusions theory, inequality technique and the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence and globally exponential stability of the anti-periodic solution for our proposed hematopoiesis model. Some previously known works are significantly extended and complemented. Moreover, simulation results of two topical numerical examples are also delineated to demonstrate the effectiveness of the theoretical results.An efficient numerical approach to solve Schrödinger equations with space fractional derivative.https://zbmath.org/1460.353102021-06-15T18:09:00+00:00"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.2|zhang.jun.1|zhang.jun.9|zhang.jun.3|zhang.jun.6|zhang.jun|zhang.jun.7|zhang.jun.5|zhang.jun.10"Lin, Shimin"https://zbmath.org/authors/?q=ai:lin.shimin"Wang, Jinrong"https://zbmath.org/authors/?q=ai:wang.jinrongIn the article the authors propose an effective linearized numerical scheme for two coupled Schrödinger-type equations involving the fractional Laplacian with periodic boundary condition. The scheme consists of using the Crank-Nicolson method in the time variable and the Fourier-Galerkin method in the spatial variable. The authors mainly show that the numerical scheme is conservative (on the one hand) and second-order convergent in the time direction and exponentially convergent in the space direction (on the other hand). Although only the case of one spatial variable is treated, the scheme is effective in higher dimensions as well. Numerical examples are given at the end.
Reviewer: Catalin Popa (Iaşi)On endpoint regularity criterion of the 3D Navier-Stokes equations.https://zbmath.org/1460.352602021-06-15T18:09:00+00:00"Li, Zhouyu"https://zbmath.org/authors/?q=ai:li.zhouyu"Zhou, Daoguo"https://zbmath.org/authors/?q=ai:zhou.daoguoSummary: Let \((u,\pi)\) with \(u = (u_1, u_2, u_3)\) be a suitable weak solution of the three-dimensional Navier-Stokes equations in \(\mathbb{R}^3 \times (0, T)\). Denote by \(\dot{\mathcal{B}}^{-1}_{\infty,\infty}\) the closure of \(C^\infty_0\) in \(\dot{B}^{-1}_{\infty,\infty} \). We prove that if \(u \in L^\infty (0, T; \dot{B}^{-1}_{\infty,\infty})\), \(u(x, T) \in \dot{\mathcal{B}}^{-1}_{\infty,\infty})\), and \(u_3 \in L^\infty (0, T; L^{3,\infty})\) or \(u_3 \in L^\infty (0, T; \dot{B}^{-1+3/p}_{p,q})\) with \(3 < p, q < \infty \), then \(u\) is smooth in \(\mathbb{R}^3 \times (0, T]\). Our result improves a previous result established by \textit{W. Wang} and \textit{Z. Zhang} [Sci. China, Math. 60, No. 4, 637--650 (2017; Zbl 1387.35471)].Small data global regularity for the 3-D Ericksen-Leslie hyperbolic liquid crystal model without kinematic transport.https://zbmath.org/1460.352872021-06-15T18:09:00+00:00"Huang, Jiaxi"https://zbmath.org/authors/?q=ai:huang.jiaxi"Jiang, Ning"https://zbmath.org/authors/?q=ai:jiang.ning"Luo, Yi-Long"https://zbmath.org/authors/?q=ai:luo.yi-long"Zhao, Lifeng"https://zbmath.org/authors/?q=ai:zhao.lifengThis work considers the hyperbolic Ericksen-Leslie system, which combined the hydrodynamical equation of motion with a constitutive equation for the orientation field that, in complex, models the motion of liquid crystals. The main result reported in prooving the existence of a unique global solution for such a system satisfying the energy bounds, which are provided.
Reviewer: Eugene Postnikov (Kursk)The local well-posedness to the density-dependent magnetic Bénard system with nonnegative density.https://zbmath.org/1460.353012021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We study the Cauchy problem of density-dependent magnetic Bénard system with zero density at infinity on the whole two-dimensional (2D) space. Despite the degenerate nature of the problem, we show the local existence of a unique strong solution in weighted Sobolev spaces by energy method.Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates.https://zbmath.org/1460.353892021-06-15T18:09:00+00:00"Huang, Xinchi"https://zbmath.org/authors/?q=ai:huang.xinchiSummary: In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we state the stability results for our inverse coefficient problem. Secondly, we establish and prove two Carleman type inequalities both for the solutions and for the unknown coefficients. Finally, we complete the proofs of the stability results in terms of the above Carleman estimates.On the local regularity theory for the magnetohydrodynamic equations.https://zbmath.org/1460.352782021-06-15T18:09:00+00:00"Chamorro, Diego"https://zbmath.org/authors/?q=ai:chamorro.diego"Cortez, Fernando"https://zbmath.org/authors/?q=ai:cortez.fernando"He, Jiao"https://zbmath.org/authors/?q=ai:he.jiao"Jarrín, Oscar"https://zbmath.org/authors/?q=ai:jarrin.oscarSummary: Local regularity results are obtained for the MHD equations using as global framework the setting of parabolic Morrey spaces. Indeed, by assuming some local boundedness assumptions (in the sense of parabolic Morrey spaces) for weak solutions of the MHD equations it is possible to obtain a gain of regularity for such solutions in the general setting of the Serrin regularity theory. This is the first step of a wider program that aims to study both local and partial regularity theories for the MHD equations.Wavenumber-explicit \textit{hp}-FEM analysis for Maxwell's equations with transparent boundary conditions.https://zbmath.org/1460.350852021-06-15T18:09:00+00:00"Melenk, Jens M."https://zbmath.org/authors/?q=ai:melenk.jens-marcus|melenk.jens-markus"Sauter, Stefan A."https://zbmath.org/authors/?q=ai:sauter.stefan-aSummary: The time-harmonic Maxwell equations at high wavenumber \(k\) are discretized by edge elements of degree \(p\) on a mesh of width \(h\). For the case of a ball as the computational domain and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the \(k\)-explicit scale resolution condition that (a) \(kh/p\) is sufficient small and (b) \(p/\text{ln }k\) is bounded from below.On some weighted Stokes problems: applications on Smagorinsky models.https://zbmath.org/1460.761852021-06-15T18:09:00+00:00"Rappaz, Jacques"https://zbmath.org/authors/?q=ai:rappaz.jacques"Rochat, Jonathan"https://zbmath.org/authors/?q=ai:rochat.jonathanSummary: In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characterization of these equations is the viscosity, which depends on the strain rate of the velocity field and in some cases is related with a weight being the distance to the boundary of the domain. Such non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman's on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show properties from the theory.
For the entire collection see [Zbl 1411.35011].Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations.https://zbmath.org/1460.352702021-06-15T18:09:00+00:00"Fan, Lili"https://zbmath.org/authors/?q=ai:fan.lili"Ruan, Lizhi"https://zbmath.org/authors/?q=ai:ruan.lizhi"Xiang, Wei"https://zbmath.org/authors/?q=ai:xiang.weiSummary: This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [SIAM J. Math. Anal. 51, No. 1, 595--625 (2019; Zbl 1412.35239)], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation.https://zbmath.org/1460.352102021-06-15T18:09:00+00:00"Ansini, Nadia"https://zbmath.org/authors/?q=ai:ansini.nadia"Fagioli, Simone"https://zbmath.org/authors/?q=ai:fagioli.simoneSummary: We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density \(u\). In case of \textit{fast-decay} mobilities, namely mobilities functions under an Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density \(\rho\) is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density \(\rho\) allow us to motivate the aforementioned change of variable and to state the results in terms of the original density \(u\) without prescribing any boundary conditions.On three-dimensional geophysical capillary-gravity water flows with constant vorticity.https://zbmath.org/1460.352812021-06-15T18:09:00+00:00"Fan, Lili"https://zbmath.org/authors/?q=ai:fan.lili"Gao, Hongjun"https://zbmath.org/authors/?q=ai:gao.hongjunSummary: Consideration in this paper is three-dimensional capillary-gravity water flows governed by the geophysical water wave equations with all the Coriolis terms being retained. It is proved that the merely possible flow exhibiting a constant vorticity vector captures vanishing vertical velocity, constant horizontal velocity and flat free surface.Cross ownership and divestment incentives.https://zbmath.org/1460.911292021-06-15T18:09:00+00:00"Stenbacka, Rune"https://zbmath.org/authors/?q=ai:stenbacka.rune"Van Moer, Geert"https://zbmath.org/authors/?q=ai:van-moer.geertSummary: Even though cross ownership raises industry profits, we demonstrate that it is prone to a commitment problem. Specifically, we show that producers in a Cournot duopoly have unilateral incentives to resell their minority share-holdings in the rival to outside investors, leading to an equilibrium with complete divestments. This feature challenges the stability of cross ownership configurations.On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations.https://zbmath.org/1460.353212021-06-15T18:09:00+00:00"Comech, Andrew"https://zbmath.org/authors/?q=ai:comech.andrew"Cuccagna, Scipio"https://zbmath.org/authors/?q=ai:cuccagna.scipioSummary: We extend to a specific class of systems of nonlinear Schrödinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.Local well-posedness for the Klein-Gordon-Zakharov system in 3D.https://zbmath.org/1460.353322021-06-15T18:09:00+00:00"Pecher, Hartmut"https://zbmath.org/authors/?q=ai:pecher.hartmutSummary: We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by \textit{I. Bejenaru} and \textit{S. Herr} [J. Funct. Anal. 261, No. 2, 478--506 (2011; Zbl 1228.42027)] for the Zakharov system and already applied by \textit{S. Kinoshita} [Discrete Contin. Dyn. Syst. 38, No. 3, 1479--1504 (2018; Zbl 1397.35281)] to the Klein-Gordon-Zakharov system in 2D.Time-periodic solution to a two-phase model with magnetic field in a periodic domain.https://zbmath.org/1460.769412021-06-15T18:09:00+00:00"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Zhou, Yu"https://zbmath.org/authors/?q=ai:zhou.yuThis paper is concerned with a two-phase model with magnetic field and a time-periodic external force in a periodic domain in \(\mathbb{R}^N\). The first goal is to establish the existence of periodic solutions for the regularized problem, where a topological degree theory for compact operators is used. As a main result, the existence of time-periodic solutions to the model is proved by a limiting process.
Reviewer: In-Sook Kim (Suwon)Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication.https://zbmath.org/1460.353452021-06-15T18:09:00+00:00"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Jung, Jinwook"https://zbmath.org/authors/?q=ai:jung.jinwook"Röckner, Michael"https://zbmath.org/authors/?q=ai:rockner.michaelSummary: We study the collective dynamics of the Cucker-Smale (C-S) ensemble under random communication. As the effective modeling of the C-S ensemble with infinite size, we introduce a stochastic kinetic C-S equation with a multiplicative white noise. For the proposed stochastic kinetic model with a multiplicative noise, we present a global existence of strong solutions and their asymptotic flocking dynamics, when initial datum is sufficiently regular, and communication weight function has a positive lower bound.Stability transition of persistence and extinction in an avian influenza model with Allee effect and stochasticity.https://zbmath.org/1460.922102021-06-15T18:09:00+00:00"Liu, Yu"https://zbmath.org/authors/?q=ai:liu.yu|liu.yu.2|liu.yu.1"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shigui"Yang, Ling"https://zbmath.org/authors/?q=ai:yang.lingThis paper concerns with the structural transition in an avian influenza model caused by external environmental noise. Firstly, bifurcation diagram of the base model (Model A) (from [\textit{S. Liu} et al., Math. Biosci. 283, 118--135 (2017; Zbl 1398.92242)]) with Allee effect for avian population is provided. Then noise is introduced into Model A to get the Model B. Under the assumption that the two key parameters, the transmission rate and the natural death rate, are in the parametric zone such that Model A is bistable with the trivial steady state and the unique endemic equilibrium, numerical simulations indicate that the noise can bring a change to stability of Model A. In order to reveal the transition characteristics, the related Fokker-Planck (FPK) equation is derived to analytically describe the probability density distributions with long time evolution. As it is difficult to find theoretical solutions of FPK, it is solved numerically. Results confirm the stability transition observed before. Moreover, it is found that noise not only reduces the parametric zone of sustaining bistability but also drive the system to exhibit different monostability. In fact, noise can induce higher probabilities for the system to sustain persistence instead of extinction in Model B.
Reviewer: Yuming Chen (Waterloo)Effect of complex landscape geometry on the invasive species spread: invasion with stepping stones.https://zbmath.org/1460.921602021-06-15T18:09:00+00:00"Alharbi, Weam"https://zbmath.org/authors/?q=ai:alharbi.weam"Petrovskii, Sergei"https://zbmath.org/authors/?q=ai:petrovskii.sergei-vSummary: Spatial proliferation of invasive species often causes serious damage to agriculture, ecology and environment. Evaluation of the extent of the area potentially invadable by an alien species is an important problem. Landscape features that reduces dispersal space to narrow corridors can make some areas inaccessible to the invading species. On the other hand, the existence of stepping stones -- small areas or `patches' with better environmental conditions -- is known to assist species spread. How an interplay between these factors can affect the invasion success remains unclear. In this paper, we address this question theoretically using a mechanistic model of population dynamics. Such models have been generally successful in predicting the rate and pattern of invasive spread; however, they usually consider the spread in an unbounded, uniform space hence ignoring the complex geometry of a real landscape. In contrast, here we consider a reaction-diffusion model in a domain of a complex shape combining corridors and stepping stones. We show that the invasion success depends on a subtle interplay between the stepping stone size, location and the strength of the Allee effect inside. In particular, for a stepping stone of a small size, there is only a narrow range of locations where it can unblock the otherwise impassable corridor.Optimal exponentials of thickness in Korn's inequalities for parabolic and elliptic shells.https://zbmath.org/1460.740592021-06-15T18:09:00+00:00"Yao, Peng-Fei"https://zbmath.org/authors/?q=ai:yao.pengfeiSummary: We consider the scaling of the optimal constant in Korn's first inequality for elliptic and parabolic shells which was first given by \textit{Y. Grabovsky} and \textit{D. Harutyunyan} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 1, 267--282 (2018; Zbl 1395.74056)] with hints coming from the test functions constructed by \textit{P. E. Tovstik} and \textit{A. L. Smirnov} [Asymptotic methods in the buckling theory of elastic shells. River Edge, NJ: World Scientific (2001; Zbl 1066.74500)] on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation.https://zbmath.org/1460.353482021-06-15T18:09:00+00:00"You, Bo"https://zbmath.org/authors/?q=ai:you.boSummary: The objective of this paper is to study the well-posedness of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with additive noise. Since strong coupling terms and the noise term create some difficulties in the process of showing the existence of weak solutions, we will first show the existence of weak solutions by the monotonicity methods when the initial data satisfies some ``regular'' condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of ``regular'' initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. Finally, we establish the existence of weak \(\mathcal{D}\)-pullback mean random attractors in the framework developed in [\textit{P. E. Kloeden} and \textit{T. Lorenz}, J. Differ. Equations 253, No. 5, 1422--1438 (2012; Zbl 1267.37018); \textit{B. Wang}, J. Dyn. Differ. Equations 31, No. 4, 2177--2204 (2019; Zbl 1428.35052)].Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations.https://zbmath.org/1460.353312021-06-15T18:09:00+00:00"Murphy, Jason"https://zbmath.org/authors/?q=ai:murphy.jason"Nakanishi, Kenji"https://zbmath.org/authors/?q=ai:nakanishi.kenji.1|nakanishi.kenji.2Summary: We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system.https://zbmath.org/1460.353252021-06-15T18:09:00+00:00"Hamano, Masaru"https://zbmath.org/authors/?q=ai:hamano.masaru"Masaki, Satoshi"https://zbmath.org/authors/?q=ai:masaki.satoshiSummary: In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.Global large solutions and optimal time-decay estimates to the Korteweg system.https://zbmath.org/1460.353002021-06-15T18:09:00+00:00"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaoping"Li, Yongsheng"https://zbmath.org/authors/?q=ai:li.yongshengSummary: We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.Cohesive fracture in 1D: quasi-static evolution and derivation from static phase-field models.https://zbmath.org/1460.740022021-06-15T18:09:00+00:00"Bonacini, Marco"https://zbmath.org/authors/?q=ai:bonacini.marco"Conti, Sergio"https://zbmath.org/authors/?q=ai:conti.sergio"Iurlano, Flaviana"https://zbmath.org/authors/?q=ai:iurlano.flavianaSummary: In this paper we propose a notion of irreversibility for the evolution of cracks in the presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models, and we investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model arises naturally via \(\Gamma\)-convergence from a phase-field model of the generalized Ambrosio-Tortorelli type, which may be used as regularization for numerical simulations.Darcy's law with a source term.https://zbmath.org/1460.350102021-06-15T18:09:00+00:00"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthew"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christina"Tong, Jiajun"https://zbmath.org/authors/?q=ai:tong.jiajunSummary: We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform \(L^1\)-equicontinuity. Many of these properties are new, even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.Asymptotic stability in a chemotaxis-competition system with indirect signal production.https://zbmath.org/1460.353602021-06-15T18:09:00+00:00"Zheng, Pan"https://zbmath.org/authors/?q=ai:zheng.panA system of chemotaxis and competition with three parabolic equations combining the models of Keller-Segel and Lotka-Volterra type is studied in bounded domains with the homogeneous Neumann boundary conditions. Exponential convergence to a coexistence steady state is shown under some assumptions on the parameters of the problem, as well as a result on an algebraic decay to an extinction steady state in another parameter range.
Reviewer: Piotr Biler (Wrocław)Curvature-driven wrinkling of thin elastic shells.https://zbmath.org/1460.740572021-06-15T18:09:00+00:00"Tobasco, Ian"https://zbmath.org/authors/?q=ai:tobasco.ianSummary: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via \(\Gamma\)-convergence for answering this question to leading order in the shell's thickness and other small parameters. The observed patterns involve ``ordered'' regions of well-defined wrinkles alongside ``disordered'' regions whose local features are less robust; as little to no tension is applied, the preference for order is not \textit{a priori} clear. Rescaling by the energy of a typical pattern, we derive a limiting variational problem for the effective displacement of the shell. It asks, in a linearized way, to cover up a maximum area with a length-shortening map to the plane. Convex analysis yields a boundary value problem characterizing the accompanying patterns via their defect measures. Partial uniqueness and regularity theorems follow from the method of characteristics on the ordered part of the shell. In this way, we can deduce from the principle of minimum energy the leading order features of stamped elastic shells.