Multi-dimensional stability of Lax shocks in hyperbolic-elliptic coupled systems

TL;DR

This paper proves that under certain stability conditions, small-amplitude Lax shocks in a coupled hyperbolic-elliptic system are nonlinearly stable, extending previous results to multi-dimensional radiative gas models.

Contribution

It establishes the nonlinear stability of Lax shocks in multi-dimensional hyperbolic-elliptic systems under the Evans stability condition, generalizing prior one-dimensional and viscous shock results.

Findings

Uniform Evans stability implies nonlinear stability.

Extension of stability results to multi-dimensional radiative systems.

Connection between hyperbolic-elliptic coupling and shock stability.

Abstract

We study nonlinear time-asymptotic stability of small--amplitude planar Lax shocks in a model consisting of a system of multi--dimensional conservation laws coupled with an elliptic system. Such a model can be found in context of dynamics of a gas in presence of radiation. Our main result asserts that the standard uniform Evans stability condition implies nonlinear stability. The main analysis is based on the earlier developments by Zumbrun for multi-dimensional viscous shock waves and by Lattanzio-Mascia-Nguyen-Plaza-Zumbrun for one--dimensional radiative shock profiles.