On the local existence for the Euler equations with free boundary for compressible and incompressible fluids

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H 3 , with the H 3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.