Global well-posedness for models of shallow water in a basin with a varying bottom

We prove global well-posedness for the great lake equations. These equations arise to first order in a low aspect ratio, low Froude number (i.e. low wave speed) and very small wave amplitude expansion of the three dimensional incompressible Euler equations in a basin with a free upper surface and a spatially varying bottom topography. On an abstract level, we consider a system that generalizes the two dimensional Euler equations in the following sense: while in the Euler system the vorticity field is given as the curl of the velocity field, here the two fields are related by a general linear operator enjoying analogous regularity properties. Moreover, the problem is posed in Sobolev spaces with a nondegenerate weight. In this setting, we follow the approach of Yudovitch and Bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the Navier-Stokes with respect to the Euler equations. The continuous dependence of the solutions on the initial data and the weight function is shown by a modification of the uniqueness estimate.

Global well-posedness for models of shallow water in a basin with a varying bottom Academic Article