Transition to instability of planar viscous shock fronts: the refined stability condition

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 [67, 62, 63] showed that transition under localized (L1∩Hs) perturbations is in fact entirely decided by viscous effects, and gave an abstract criterion for transition in terms of an effective viscosity coefficient β 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 [32, 6, 5], we develop a simplified perturbation formula for β, applicable to general shock waves, that is convenient for numerical and analytical investigation.