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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      C 3 2 + ε  C^{\frac {3}{2}+\varepsilon }      . 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.

Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary