Low regularity ill-posedness for elastic waves driven by shock formation

IF 1.7 1区 数学 Q1 MATHEMATICS
Xinliang An, Hao-Lei Chen, Silu Yin
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引用次数: 4

Abstract

abstract:In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(\Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.
冲击波驱动弹性波的低正则性不适定性
文摘:本文构造了三维弹性波动方程低正则解局部存在性的反例。受Christodoulou最近工作的启发,我们推广了Lindblad关于标量波动方程的经典结果,证明了具有多个波速的三维弹性波的Cauchy问题在$H^3(\bb{R}^3)$中是不适定的。我们进一步证明了这种不适定性是由瞬时冲击形成引起的,其特征是反向叶理密度的消失。三维情况的主要困难来自于多个波速及其相关的非严格双曲性。我们通过设计和组合几何方法和代数方法,并对相应的非严格双曲系统的结构和系数进行详细的研究和计算,获得了预期的结果。此外,我们描述的不适定性也适用于二维弹性波,这对应于严格双曲的情况。
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
CiteScore
3.20
自引率
0.00%
发文量
35
审稿时长
24 months
期刊介绍: The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.
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