Spectral stability of hydraulic shock profiles

TL;DR

This paper proves spectral stability for hydraulic shock profiles in shallow-water equations, covering all parameters and types, using a Sturm-Liouville reduction, leading to comprehensive stability results with explicit decay rates.

Contribution

It introduces a Sturm-Liouville approach to establish the spectral stability of large-amplitude hydraulic shock profiles, including both smooth and discontinuous types, in the Saint-Venant equations.

Findings

Spectral stability established for all parameter ranges.

Valid for both smooth and discontinuous shock profiles.

Provides explicit decay rates in $L^p$ norms.

Abstract

By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear and nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.