论文标题
$ h^{\ frac {11} {4}}}（\ mathbb {r}^2）
$H^{\frac{11}{4}}(\mathbb{R}^2)$ ill-posedness for 2D Elastic Wave system
论文作者
论文摘要
在本文中，我们证明，对于具有多个波速的物理系统，其cauchy问题无法在$ H^{\ frac {\ frac {11} {4}}}（\ Mathbb {r}^2）$中进行局部良好。这里的不良性是由瞬时冲击形成驱动的。在2D Smith-Tataru中，表明单个Quasilinear Wave方程的Cauchy问题在$ H^s $中以$ s> \ frac {11} {4} $为本。因此，我们的$ h^{\ frac {11} {4}} $不适当在此获得的结果是所需的结果。我们的证明依赖于结合几何方法和代数波分解方法，并配备了对相应双曲系统的详细分析。
In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H^{\frac{11}{4}}(\mathbb{R}^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s>\frac{11}{4}$. Hence our $H^{\frac{11}{4}}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, equipped with detailed analysis of the corresponding hyperbolic system.
