On the free boundary problem of three‐dimensional incompressible Euler equations

Abstract

Abstract In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well‐posedness of 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 ${{\partial p}\over {\partial {\bf n}}}(\xi)|_{t=0}\leq -2c_{0}<0$ with ξ being restricted to the initial surface. © 2007 Wiley Periodicals, Inc.