A Regularity Result for the Incompressible Euler Equation with a Free Interface

We address the local existence of solutions of the 2D and 3D water wave problems. For the space dimension three, we consider the irrotational datum u ₀ and prove that the local in time existence holds for initial velocities belonging to H ^2.5+δ , where δ>0 is arbitrary. For the space dimension two, the data does not need to be irrotational. We prove the local in time existence when u ₀ belongs to H ^2+δ and $\operatorname{curl}u_{0}$ to H ^1.5+δ , where δ>0 is arbitrary.