  In this paper, we address the question of the hyperbolicity and the local
well- posedness 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 H^s(R^2), 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.
