  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.
