Conditional stability of unstable viscous shocks

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 ∩ H 2 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 Howard, Mascia, and Zumbrun.