Transition to Instability of Planar Viscous Shock Fronts: the Refined Stability Condition

Abstract

Classical inviscid stability analysis determines stability of shock waves only up to a region of neutral stability occupying an open set of physical parameters. To locate a precise transition point within this region, it has been variously suggested that nonlinear and or viscous effects should be taken into account. Recently, Zumbrun and Serre showed that transition under localized ($L^1\cap H^s$) perturbations is in fact entirely decided by viscous effects, and gave an abstract criterion for transition in terms of an effective viscosity coefficient $\beta$ determined by second derivatives of the Evans function associated with the linearized operator about the wave. Here, generalizing earlier results of Kapitula, Bertozzi et al, and Benzoni-Gavage et al., we develop a simplified perturbation formula for $\beta$, applicable to general shock waves, that is convenient for numerical and analytical investigation.