Scattering and rigidity for nonlinear elastic waves

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI:10.1007/s00526-024-02736-2

Dongbing Zha

引用次数: 0

Abstract

For the Cauchy problem of nonlinear elastic wave equations of three-dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in Agemi (Invent Math 142:225–250, 2000) and Sideris (Ann Math 151:849–874, 2000), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

非线性弹性波的散射和刚度

对于满足空条件的三维各向同性、均质和超弹性材料的非线性弹性波方程的 Cauchy 问题，Agemi (Invent Math 142:225-250, 2000) 和 Sideris (Ann Math 151:849-874, 2000) 分别证明了小初始数据下经典解的全局存在性。在本文中，我们将考虑全局解的渐近行为。我们首先证明了全局解将会散射，即随着时间趋向无穷大，全局解将在能量意义上收敛于线性弹性波方程的某个解。我们还证明了以下刚性结果：如果散射数据消失，那么全局解也将同理消失。系统的变分结构将在我们的论证中发挥关键作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.