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

2024/11/24 by Xinliang An, Haoyang Chen, An, Xinliang +3 · 1 citation
Earth and Planetary Sciences · Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Seismic Imaging and Inversion Techniques #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2411.15886

openalex publication_date 2024/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 H³⁺ of the divergence-part (the faster-wave part) and the H⁴⁺ 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 H³⁺ 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.

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