IUMJ

Title: Well-Posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity

Authors: Angel Castro and David Lannes

Issue: Volume 64 (2015), Issue 4, 1169-1270

Abstract:

In this paper, we derive a new formulation of the water waves equations with vorticity that generalizes the well-known Zakharov-Craig-Sulem formulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new formulation 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 results, we derive and provide the first rigorous justification of a shallow-water model for water waves in the 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 [A. Castron and D. Lannes, \emph{Fully nonlinear long-wave models in the presence of vorticity}, J. Fluid Mech. \textbf{759} (2014), 642--675.], where higher-order shallow-water models with vorticity (of Green-Naghdi type) are derived.