A sharp regularity result for the Euler equation with a free interface

Abstract

We establish a priori estimates for local-in-time existence of solutions to the water wave model consisting of the 3D incompressible Euler equations on a domain with a free surface, without surface tension, under minimal regularity assumptions on the initial data and the Rayleigh–Taylor sign condition. The initial data are allowed to be rotational and they are assumed to belong to [Formula: see text], where [Formula: see text] is arbitrary.

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

“…Moreover, the proof in [KTVW] uses in an essential way the preservation of Lagrangian vorticity, which is a property that does not hold in the 3D case. Another paper [KT4] considers the problem in ALE coordinates (see [MC] for the ALE coordinates in the fluid-structure interaction problem), but the proof requires an additional assumption on the initial data, due to additional regularity required by the Lagrangian variable on the boundary; for instance, the present paper shows that the conditions of theorem 1 (see below) suffice.…”

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

“…Moreover, the proof in [KTVW] uses in an essential way the preservation of Lagrangian vorticity, which is a property that does not hold in the 3D case. Another paper [KT4] considers the problem in ALE coordinates (cf. [MC] for the ALE coordinates in the fluid-structure interaction problem), but the proof requires an additional assumption on the initial data, due to additional regularity required by the Lagrangian variable on the boundary; for instance, the present paper shows that the conditions of Theorem 1 (see below) suffice.…”