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 equ ations 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 th ree dimensional incompressible Euler equations in a basin with a free upper surface and a spatially varying bottom topography. On an abstrac t level, we consider a system that generalizes the two dimensional Eul er equations in the following sense: while in the Euler system the vor ticity field is given as the curl of the velocity field, here the two fields are related by a general linear operator enjoying analogous reg ularity properties. Moreover, the problem is posed in Sobolev spaces w ith a nondegenerate weight. In this setting, we follow the approach of Yudovitch and Bardos in constructing the solutions as the inviscid li mit of solutions to a system with artificial viscosity which is the an alog of the Navier-Stokes with respect to the Euler equations. The con tinuous dependence of the solutions on the initial data and the weight function is shown by a modification of the uniqueness estimate.