Viscous and inviscid stability of multidimensional planar shock fronts

We explore the relation between viscous and inviscid stability of multidime nsional shock fronts, by studying the Evans function associated with the vi scous 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 invisci d case, multiplied by a constant measuring transversality of the shock conn ection in the underlying (viscous) traveling wave ODE. Remarkably, this res ult 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 criteri on even in the absence of a sensible inviscid problem. An immediate consequence is that inviscid stability is necessary (but not s ufficient) for viscous stability; this yields a number of interesting resul ts 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 solvab ility condition," determining the spectral expansion about zero of the line arized operator about the wave, and thereby the transverse propagation of s ignals along the front. This expansion is in general not analytic, due to a ccumulation of essential spectrum, but rather has conical structure. A cons equence, as in the inviscid case, is that stability is typically stronger f or systems than for scalar equations. In the indeterminate, "tangent" case (Majda's "weak stability"), we provide an appropriate higher order correcti on.