Viscous and inviscid stability of multidimensional planar shock fronts

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 inviscid analyses of Erpenbeck, Majda, and others. Moreover, in the viscous case, the Kreiss-Sakamoto-Lopatinski determinant is seen to play the key role of a "generalized Fredholm solvability condition," determining the spectral expansion about zero of the linearized operator about the wave, and thereby the transverse propagation of signals along the front. This expansion is in general not analytic, due to accumulation of essential spectrum, but rather has conical structure. A consequence, as in the inviscid case, is that stability is typically stronger for systems than for scalar equations. In the indeterminate, "tangent" case (Majda's "weak stability"), we provide an appropriate higher order correction.