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

TL;DR

This paper proves local-in-time existence and uniqueness of 3D free-surface Euler flows with initial vorticity in $H^{2+ ext{delta}}$, using a Lagrangian approach under Rayleigh-Taylor stability.

Contribution

It establishes local existence and uniqueness for free-surface Euler flows with low regularity initial vorticity near the boundary, extending previous results.

Findings

Proves local-in-time existence of solutions.

Uses Lagrangian approach for a priori estimates.

Requires Rayleigh-Taylor stability condition.

Abstract

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that $u_0 \in H^{2.5+\delta }$ is such that $\mathrm{curl}\,u_0 \in H^{2+\delta }$ in an arbitrarily small neighborhood of the free boundary, and we use 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.