Topography influence on the Lake equations in bounded domains

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.