Local Well-posedness of the Multi-layer Shallow Water Model with Free Surface∗

In this paper, we address the question of the hyperbolicity and the local wellposedness of the multi-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of this model, which implies hyperbolicity and local well-posedness in Hs(R2), with s > 2. Then, we analyze rigorously the eigenstructure associated to this model and prove a more general criterion of hyperbolicity and local well-posedness, under a particular asymptotic regime and a weak stratification assumptions of the densities and the velocities. Finally, we consider a new conservative multi-layer shallow water model, we prove the symmetrizability, the hyperbolicity and the local well-posedness and rely it to the basic multi-layer shallow water model.