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 $\mathbb{H}^+\equiv\{\kappa\in\mathbb{C}:\text{Re}\,\kappa \geq 0\}$, 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] .