Waiting time solutions of the shallow water equations

Abstract

In this paper we present new solutions of the shallow water equations with waiting time properties. These are shown to represent the local behaviour of the solution of an initial-value problem for certain types of initial data which models the evolution of an initially stationary inviscid fluid layer of nonuniform thickness possessing bounded support. In this context the motion of the interfaces, where the layer thickness is zero, is shown to be governed solely by the local properties of the initial data which may remain stationary for a finite non-zero time before spreading. For other types of initial data we show that the interfaces may move immediately, the local behaviour being governed by another type of similarity solution. This behaviour is remarkably similar to that exhibited by degenerate parabolic equations which have also been used to model viscous thin film flows in the lubrication approximation.