Stability of hydraulic shock profiles

TL;DR

This paper proves nonlinear stability with precise decay rates for hydraulic shock profiles in shallow water equations, including those with subshocks, under Evans-Lopatinsky stability, verified numerically.

Contribution

It provides the first nonlinear stability results for shock profiles with subshocks in a hyperbolic relaxation system, with verified Evans-Lopatinsky stability.

Findings

Established nonlinear $H^2\cap L^1 \to H^2$ stability with sharp decay rates.

Numerically verified Evans-Lopatinsky stability for all profiles.

First nonlinear stability results for shock profiles with subshocks.

Abstract

We establish nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans-Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.