On weak stability of shock waves in 2D compressible elastodynamics

TL;DR

This paper establishes that the stability condition for 2D shock waves in compressible elastodynamics is both necessary and sufficient for uniform stability, ensuring nonlinear stability and ruling out violent instability.

Contribution

It proves the equivalence of the energy method stability condition with the uniform Lopatinski condition for 2D shock waves in elastic materials, clarifying stability criteria.

Findings

Stability condition is necessary and sufficient for uniform stability.

Rectilinear shock waves are never violently unstable.

Analysis of transition between uniform and weak stability.

Abstract

By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin, P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471-1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e., they are always either uniformly or weakly stable.