Stability of large-amplitude shock waves of compressible Navier--Stokes equations

TL;DR

This paper reviews recent advances in understanding the stability of large-amplitude viscous shock waves in compressible Navier--Stokes equations, providing rigorous conditions that distinguish stable from unstable regimes, especially in multi-dimensional settings.

Contribution

It establishes new necessary and sufficient conditions for shock wave stability, extending classical criteria and introducing a novel multi-dimensional stability condition.

Findings

Necessary and sufficient stability conditions are rigorously formulated.

The multi-dimensional sufficient condition is newly derived.

Stability criteria are numerically evaluable and depend on shock amplitude.

Abstract

We summarize recent progress on one- and multi-dimensional stability of viscous shock wave solutions of compressible Navier--Stokes equations and related symmetrizable hyperbolic--parabolic systems, with an emphasis on the large-amplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multi-dimensions by a codimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multi-dimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently small-amplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently large-amplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory.