Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD

TL;DR

This paper proves the conditional nonlinear stability of certain unstable viscous shock waves in hyperbolic-parabolic systems, extending previous results to more general cases without parabolic smoothing.

Contribution

It establishes the existence of a center stable manifold for unstable shock solutions in hyperbolic-parabolic systems, showing nonlinear orbital stability within this manifold.

Findings

Unstable shock waves are conditionally stable on a codimension-p manifold.

Constructs an invariant manifold without parabolic smoothing.

Recovers unconditional stability results for stable shocks.

Abstract

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.