WELL-POSEDNESS AND SHALLOW-WATER STABILITY FOR A NEW HAMILTONIAN FORMULATION OF THE WATER WAVES EQUATIONS WITH VORTICITY

In this paper we derive a new formulation of the water waves equa-tions with vorticity that generalizes the well-known Zakharov-Craig-Sulem for-mulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new formu-lation is cast in Eulerian variables, and in finite depth; we show that it can be used to provide uniform bounds on the lifespan and on the norms of the solutions in the singular shallow water regime. As an application to these re-sults, we derive and provide the first rigorous justification of a shallow water model for water waves in presence of vorticity; we show in particular that a third equation must be added to the standard model to recover the velocity at the surface from the averaged velocity. The estimates of the present paper also justify the formal computations of [15] where higher order shallow water models with vorticity (of Green-Naghdi type) are derived