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 $\mathcal{E}:\mathrm{\Lambda }\to \mathbb{C}$ with $\mathrm{\Lambda }\subset \mathbb{C}$ 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 [FS02] and [PZ04] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in [PZ04], [MZ09], [Ued09] .