$H^{\frac{11}{4}}(\mathbb{R}^2)$ ill-posedness for 2D Elastic Wave system

TL;DR

This paper demonstrates that the 2D elastic wave system with multiple wave-speeds is ill-posed in the Sobolev space $H^{11/4}$ due to shock formation, extending known well-posedness results for single wave equations.

Contribution

It establishes the ill-posedness of the 2D elastic wave system in $H^{11/4}$, highlighting the impact of multiple wave-speeds on solution regularity and stability.

Findings

Ill-posedness in $H^{11/4}$ for 2D elastic waves

Shock formation causes ill-posedness

Extension of well-posedness results for single wave equations

Abstract

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.