论文标题
低规律性不适用于弹性波驱动的弹性波
Low regularity ill-posedness for elastic waves driven by shock formation
论文作者
论文摘要
在本文中，我们对三个空间维度（3D）的弹性波方程的局部存在构建反示例。受Christodoulou的最新作品的启发，我们通过表明3D弹性波的Cauchy问题（一种具有多个波速的物理系统）在$ H^3（\ Mathbb {r}^3）中概括了3D弹性波的凯奇问题，从而概括了Lindblad的经典结果。我们进一步证明了拟态性不足是由瞬时冲击形成引起的，其特征是反叶密度消失。 3D情况的主要困难来自多个波速及其相关的非图案双曲线。我们通过设计和结合几何方法和代数方法来获得所需的结果，该方法配备了对相应非刻板双曲线系统的结构和系数的详细研究和计算。此外，我们描绘的不良性也适用于2D弹性波，这与严格的双曲线相对应。
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, is ill-posed in $H^3(\mathbb{R}^3)$. 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.
