Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary

In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to C3/2 + epsilon. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.