Zumbrun, K.; Serre, D. Viscous and inviscid stability of multidimensional planar shock fronts. (English) Zbl 0944.76027 Indiana Univ. Math. J. 48, No. 3, 937-992 (1999). We explore the relation between viscous and inviscid stability of multi-dimensional 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 A. Majda [Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, 53. New York etc.: Springer-Verlag (1984; Zbl 0537.76001)] 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 still more general: 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 J. J. Erpenbeck [Phys. Fluids 5, 1181-1187 (1962; Zbl 0111.38403)], A. 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 prove an appropriate higher-order correction. Reviewer: K.Zumbrun, D.Serre Cited in 2 ReviewsCited in 55 Documents MSC: 76E99 Hydrodynamic stability 76L05 Shock waves and blast waves in fluid mechanics 35B35 Stability in context of PDEs Keywords:viscous conservation law; generalized Fredholm solvability condition; Majda’s weak stability; Evans function; viscous shock; long-wave limit; Kreiss-Sakamoto-Lopatinski determinant; viscous regularization; overcompressive case; long wave stability criterion; inviscid stability; viscous stability; spectral expansion Citations:Zbl 0537.76001; Zbl 0111.38403 × Cite Format Result Cite Review PDF Full Text: DOI Link