Conditional stability of unstable viscous shocks

TL;DR

This paper demonstrates that certain linearly unstable viscous shock solutions can be nonlinearly stable within a specific manifold, with convergence to a shifted wave, extending stability results to some unstable cases.

Contribution

It establishes the existence of a center stable manifold for unstable viscous shocks, showing nonlinear orbital stability within this manifold, generalizing previous unconditional stability results.

Findings

Unstable shocks have a codimension-p stable manifold.

Solutions converge to a translated wave within this manifold.

Sharp decay rates are established in all L^p spaces.

Abstract

Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^2$ perturbatoins, 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 Howard, Mascia, and Zumbrun.