Three-dimensional structural stability of shock waves in elastodynamics

TL;DR

This paper investigates the structural stability of shock waves in three-dimensional elastodynamics, establishing conditions for their stability and extending previous two-dimensional results to more complex elastic materials.

Contribution

It provides a comprehensive analysis of the stability conditions for 3D shock waves in elastodynamics, including necessary and sufficient conditions for uniform stability and the role of elastic forces.

Findings

Planar shock waves are at least weakly stable in 3D elastodynamics.

A necessary and sufficient condition for uniform stability of shock waves is identified.

Elastic forces have a stabilizing effect on shock wave stability.

Abstract

We study the three-dimensional structural stability of shock waves for the equations of elastodynamics governing isentropic flows of compressible inviscid elastic materials. By nonlinear structural stability of a shock wave we mean the local-in-time existence and uniqueness of the discontinuous shock front solution to the hyperbolic system of elastodynamics. By using equivalent formulations of the uniform and weak Kreiss-Lopatinski conditions for 1-shocks, we show that planar shock waves in three-dimensional elastodynamics are always at least weakly stable, and we find a condition necessary and sufficient for their uniform stability. Since the system of elastodynamics satisfies the Agranovich-Majda-Osher block structure condition, uniform stability implies structural stability of corresponding nonplanar shock waves. We also show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. This paper is a natural continuation of the previous two-dimensional analysis in [Morando A., Trakhinin Y., Trebeschi P., Math. Ann. 378 (2020), 1471-1504; Trakhinin Y., J. Hyperbolic Differ. Equ. 19 (2022), 157-173]. As in the two-dimensional case, we make the conclusion that the elastic force plays stabilizing role for uniform stability.