The 3D elastic wave system for admissible harmonic materials is shown to be locally well-posed with the divergence part in H^{3+} and the curl part in H^{4+}, the first low-regularity result for multi-wave-speed systems.
$H^{\frac{11}{4}}(\mathbb{R}^2)$ ill-posedness for 2D Elastic Wave system
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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.
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Low-Regularity Local Well-Posedness for the Elastic Wave System
The 3D elastic wave system for admissible harmonic materials is shown to be locally well-posed with the divergence part in H^{3+} and the curl part in H^{4+}, the first low-regularity result for multi-wave-speed systems.