Topography influence on the Lake equations in bounded domains

TL;DR

This paper studies how topography affects the lake equations, demonstrating stability under domain approximations and establishing weak solutions for complex domains, thus extending previous models to more realistic scenarios.

Contribution

It introduces stability results for the lake equations under domain and depth perturbations and proves existence of weak solutions in singular and rough bottom domains.

Findings

Lake equations are stable under Hausdorff domain approximations.

Existence of weak solutions in singular and rough bottom domains.

Extension of previous models to more complex and realistic topographies.

Abstract

We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and M\'etivier treating the lake equations with a fixed topography and by G\'erard-Varet and Lacave treating the Euler equations in singular domains.