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

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 \(\mathbb {H}^+\equiv \{\kappa \in \mathbb {C}:\text {Re}\,\kappa \ge 0\}\) , has only one zero, namely a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan [11] and Plaza and Zumbrun [27] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun [26], Plaza and Zumbrun [27], Ueda [32].