Stability of Hydraulic Shock Profiles

Abstract

We establish nonlinear $$H^2\cap L^1 \rightarrow H^2$$ 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.