Stability of hydraulic shock profiles

Abstract

We establish nonlinear $H^2\cap L^1\to H^2$ stability with sharp rates of decay in $L^p$, $p\ge 2$, 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.