$H^{\frac{11}{4}}(\mathbb{R}^2)$ Ill-Posedness for 2D Elastic Wave System

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 $s \gt \frac{11}{4}$. 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, together with detailed analysis of the corresponding hyperbolic system.