Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems

Abstract

We prove one-dimensional spectral stability of small amplitude shock profiles for degenerate viscosity conservation laws that are dissipative symmetric hyperbolic-parabolic in the sense of Kawashima [Ka.1], generalizing well-known results of Matsumura and Nishihara [MN] and of Kawashima, Matsumura, and Nishihara [KM,KMN] for the case of compressible gas dynamics. The proof follows an approach different from that of [MN,KM,KMN], instead combining the weighted energy method of Goodman [Go.2] with the auxiliary derivative estimates of Kawashima [Ka.1].