Local-in-time existence of a free-surface 3D Euler flow with H ^2+δ initial vorticity in a neighborhood of the free boundary

Abstract

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that u 0 ∈ H 2.5 + δ is such that c u r l u 0 ∈ H 2 + δ in an arbitrarily small neighborhood of the free boundary, and we use the Lagrangian approach to derive an a priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh–Taylor stability condition.