  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.
