Well-posedness of the shallow water equations in the presence of a front

Abstract

We consider the initial value problem for the inviscid shallow-water equations in the case where a "front" is present, i.e., a boundary where the fluid depth tends to zero. Since the wave speed in shallow water behaves like the square root of the depth, this results in a degenerate hyperbolic system "on the edge" of change of type. It is shown that smooth solutions exist for smooth initial data.