Existence and uniqueness for a viscoelastic Kelvin–Voigt model with nonconvex stored energy

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2020-12-18 DOI:10.1142/s0219891623500133

K. Koumatos, Corrado Lattanzio, S. Spirito, A. Tzavaras

引用次数: 2

Abstract

We consider nonlinear viscoelastic materials of Kelvin–Voigt-type with stored energies satisfying an Andrews–Ball condition, allowing for nonconvexity in a compact set. Existence of weak solutions with deformation gradients in [Formula: see text] is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.

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具有非凸存储能量的粘弹性Kelvin-Voigt模型的存在唯一性

考虑具有满足Andrews-Ball条件的kelvin - voigt型非线性粘弹性材料，该材料在紧集中具有非凸性。对于任意超二次增长的能量，建立了[公式:见文]中带变形梯度的弱解的存在性。在二维空间中，弱解在这类中被证明是唯一的。在对存储能量增长的附加温和限制下，建立了二维和三维弱解的能量守恒以及二维光滑初始数据的全局正则性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Hyperbolic Differential Equations 数学-物理：数学物理

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.