Recent zbMATH articles in MSC 35Ghttps://zbmath.org/atom/cc/35G2021-06-15T18:09:00+00:00WerkzeugNon-universal Casimir forces at approach to Bose-Einstein condensation of an ideal gas: effect of Dirichlet boundary conditions.https://zbmath.org/1460.810962021-06-15T18:09:00+00:00"Napiórkowski, M."https://zbmath.org/authors/?q=ai:napiorkowski.marek"Piasecki, J."https://zbmath.org/authors/?q=ai:piasecki.jaroslaw"Turner, J. W."https://zbmath.org/authors/?q=ai:turner.john-wmSummary: We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance \(D\). The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose-Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of thermal Casimir force between the walls is \(\sim 1/D^2\) with a non-universal amplitude. The next order correction is \(\sim \ln D/D^3\). These observations remain in contrast with the decay law for both the periodic and Neumann boundary conditions for which the leading term is \(\sim 1/D^3\) with a universal amplitude. We associate this discrepancy with the D-dependent positive value of the one-particle ground state energy in the case of Dirichlet boundary conditions.
Reviewer: Reviewer (Berlin)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.
Reviewer: Reviewer (Berlin)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.
Reviewer: Reviewer (Berlin)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)Asymptotics of eigenfunctions of the bouncing ball type of the operator \(\nabla D(x)\nabla\) in a domain bounded by semirigid walls.https://zbmath.org/1460.810322021-06-15T18:09:00+00:00"Klevin, A. I."https://zbmath.org/authors/?q=ai:klevin.a-iSummary: We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla\) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.
Reviewer: Reviewer (Berlin)The peak model for finite rank supersingular perturbations.https://zbmath.org/1460.810282021-06-15T18:09:00+00:00"Juršėnas, Rytis"https://zbmath.org/authors/?q=ai:jursenas.rytisSummary: We review the peak model for finite rank supersingular perturbations of a lower semibounded self-adjoint operator by comparing the main aspects with the A-model. The exposition utilies the techniques based on the notion of boundary triples.
For the entire collection see [Zbl 1445.00026].
Reviewer: Reviewer (Berlin)A note on blow-up criteria for a class of nonlinear dispersive wave equations with dissipation.https://zbmath.org/1460.350522021-06-15T18:09:00+00:00"Deng, Xijun"https://zbmath.org/authors/?q=ai:deng.xijunSummary: In this note, we study the Cauchy problem for a class of nonlinear dispersive wave equation with dissipative term on the real line. We establish a new local-in-space blow-up criterion. Our results improve the corresponding ones in the previous paper [\textit{L. Brandolese} and \textit{M. F. Cortez}, J. Differ. Equations 256, No. 12, 3981--3998 (2014; Zbl 1293.35053)].
Reviewer: Reviewer (Berlin)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.
Reviewer: Reviewer (Berlin)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.
Reviewer: Reviewer (Berlin)On the critical exponent ``instantaneous blow-up'' versus ``local solubility'' in the Cauchy problem for a model equation of Sobolev type.https://zbmath.org/1460.350532021-06-15T18:09:00+00:00"Korpusov, Maxim O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Panin, Alexander A."https://zbmath.org/authors/?q=ai:panin.aleksandr-anatolevich"Shishkov, Andrey E."https://zbmath.org/authors/?q=ai:shishkov.andrey-eSummary: We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form \(|\nabla u|^q\). We prove that when \(q\in(1,3/2]\) the Cauchy problem in \(\mathbb{R}^3\) has no local-in-time weak solution for a large class of initial functions, while when \(q>3/2\) there is a local weak solution.
Reviewer: Reviewer (Berlin)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}
Reviewer: Reviewer (Berlin)Mathematical 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)On the solvability of a boundary-value problem for second-order partial differential operator equations.https://zbmath.org/1460.350752021-06-15T18:09:00+00:00"Mirzoev, S. S."https://zbmath.org/authors/?q=ai:mirzoev.sabir-s"Dzhafarov, I. Dzh."https://zbmath.org/authors/?q=ai:dzhafarov.i-dzhFrom the text: In this paper, we indicate sufficient conditions that ensure a regular solvability of the title problem. These conditions are expressed only by the coefficients of the operator differential equation.
Reviewer: Reviewer (Berlin)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.
Reviewer: Reviewer (Berlin)An accelerated solution for some classes of nonlinear partial differential equations.https://zbmath.org/1460.350762021-06-15T18:09:00+00:00"El-Kalla, Ibrahim L."https://zbmath.org/authors/?q=ai:el-kalla.ibrahim-l"Mohamed, E. M."https://zbmath.org/authors/?q=ai:mohamed.e-m-h|mohamed.emad-m"El-Saka, Hala A. A."https://zbmath.org/authors/?q=ai:el-saka.hala-a-aSummary: In this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.
Reviewer: Reviewer (Berlin)