Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws

Abstract

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function \(\mathcal {E}:\Lambda \rightarrow \mathbb {C}\) with \(\Lambda \subset \mathbb {C}\) an open superset of the closed right half plane \(\mathbb {H}^+\equiv \{\kappa \in \mathbb {C}:\text {Re}\,\kappa \geqq 0\}\) has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).