Low-Regularity Local Well-Posedness for the Elastic Wave System

TL;DR

This paper establishes low-regularity local well-posedness for the elastic wave system in three dimensions, handling multiple wave speeds and splitting dynamics into divergence and curl parts with optimal Sobolev regularity.

Contribution

It proves the first low-regularity well-posedness result for a wave system with multiple wave speeds, using a novel splitting approach for the elastic wave equations.

Findings

Control of Sobolev norms for divergence and curl parts in short time

Optimal regularity $H^{3+}$ for divergence part

First result for wave systems with multiple speeds

Abstract

We study the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a desired low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the divergence-part and the curl-part, and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm $H^{3+}$ of the divergence-part (the faster-wave part) and the $H^{4+}$ of the curl-part (the slower-wave part) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{3+}$ is optimal for the divergence-part. This marks the first favorable low-regularity local well-posedness result for a wave system with multiple wave speeds.