On the local wellposedness of 3-D water wave problem with vorticity

Abstract

In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible Euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that $$\frac{{\partial p}}{{\partial n}}(\xi )\left| {_{t = 0} } \right. \leqslant - 2c_0 < 0$$ being restricted to the initial surface.