Well-posedness in Sobolev spaces of the full water wave problem in 3-D

Abstract

We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion of the interface of the water wave is not subject to Taylor instability, as long as the interface separates the whole 3-D space into two simply connected C 2 C^{2} regions. We prove further the existence and uniqueness of solutions of the full 3-D water wave problem, locally in time, for any initial interface that separates the whole 3-D space into two simply connected regions.