Viscous And Inviscid Stability Of Multidimensional Planar Shock Fronts

Abstract

. We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock profile. Our main result, generalizing earlier one-dimensional calculations, is that the Evans function reduces in the long-wave limit to the Kreiss--Sakamoto-- Lopatinski determinant obtained by Majda in the inviscid case, multiplied by a constant measuring transversality of the shock connection in the underlying (viscous) traveling wave ODE. Remarkably, this result holds independently of the nature of the viscous regularization, or the type of the shock connection. Indeed, the analysis is more general still: in the overcompressive case, we obtain a simple long-wave stability criterion even in the absence of a sensible inviscid problem. An immediate consequence is that inviscid stability is necessary (but not sufficient) for viscous stability; this yields a number of interesting results on viscous instability through the in..