Global gradient catastrophe and its unfolding in solutions of the Airy's model of shallow water waves (Workshop on Nonlinear Water Waves)

Abstract

It is well known that under certain circumstances (see, e.g., [1, 2]), model equations such as the Airy's shallow water system, eta_{t}+(ueta)_{x}=0, u_{t}+uu_{x}+eta_{x}=0, xinGamma t, tin mathbb{R}^{ +}, (1.1) here written in suitable nondimensional space-time (x, t) coordinates with eta representing the water layer thickness and u the layer-averaged horizontal velocity, can capture some of the fundamental dynamics of the parent Euler equations for a free-surface, inviscid fluid under gravity, extending laterally to infinity and confined below by a flat bottom. Much of the model effectiveness at the qualitative and even quantitive level ultimately resides in the fact that the conservation laws of mass and horizontal momentum are captured either exactly (as in the first equation) or asymptotically in the limit of long waves, as in the second equation. With this in mind, we have recently found and studied exact solutions that can be used to shed some light on the peculiar features of the dynamics when the layer thickness vanishes (as set by initial data). In fact, it can be shown, even for the parent Euler equations, that "dry"' points where eta(x, t)=0, so that the free surface touches the bottom of the fluid layer, tend to persist as long as the function eta(x, cdot) remains sufficiently regular at these contact points. As a consequence, detachment of the free surface from the bottom can only happen trough a loss of regularity of the solution.