On the lifespan of three-dimensional gravity water waves with vorticity

Abstract

We prove a long-term regularity result for three-dimensional gravity water waves with small initial data but nonzero initial vorticity. We consider solutions whose vorticity vanishes on the free boundary and use this to derive a system for the evolution of the free boundary which reduces to the Zakharov/Craig-Sulem formulation in the irrotational case. We are able to continue the solution until a time determined by the size of the initial vorticity in such a way that if the vorticity is zero, one recovers a lifespan $T\sim {\epsilon }^{-N}$ where $N$ can be taken arbitrarily large if the initial data is taken to be arbitrarily smooth.