Vortex motion for the lake equations

Abstract

The lake equations $$\{\begin{array}{rlrl}& & & \\ \nabla \cdot \left(b\mathbf{u}\right)& =0& & \text{on}\mathbb{R}\times D,\\ {\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}& =-\nabla h& & \text{on}\mathbb{R}\times D,\\ \mathbf{u}\cdot \mathbf{\nu }& =0& & \text{on}\mathbb{R}\times \partial D.\end{array}$$ model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b:D\to [0,+\infty )$ is small in comparison with the size of its two-dimensional projection $D\subset {\mathbb{R}}^2$. When the depth $b$ is positive everywhere in $D$ and constant on the boundary, we prove that the vorticity of solutions of the lake equations whose initial vorticity concentrates at an interior point is asympotically a multiple of a Dirac mass whose motion is governed by the depth function $b$.

Minor revision, 43 pages