Exponential asymptotic stability of Riemann shocks of hyperbolic systems of balance laws

TL;DR

This paper proves that spectral stability of Riemann shocks in hyperbolic systems of balance laws guarantees their exponential asymptotic stability, using advanced spectral and nonlinear analysis techniques.

Contribution

It establishes a link between spectral and nonlinear stability for Riemann shocks and introduces new methods for analyzing constant solutions in hyperbolic systems.

Findings

Spectral stability implies exponential asymptotic stability for Riemann shocks.

New analysis techniques for constant solutions in hyperbolic systems.

Development of a hypocoercive structure with dissipative boundary conditions.

Abstract

For strictly entropic Riemann shock solutions of strictly hyperbolic systems of balance laws, we prove that exponential spectral stability implies large-time asymptotic orbital stability. As a preparation, we also prove similar results for constant solutions of initial value and initial boundary value problems, that seem to be new in this generality. Main key technical ingredients include the design of a nonlinear change of variables providing a hypocoercive Kawashima-type structure with dissipative boundary conditions in the high-frequency regime and the explicit identification of most singular parts of the linearized evolution, both being deduced from the mere spectral assumption.