Construction of the free-boundary 3D incompressible Euler flow under limited regularity

TL;DR

This paper constructs unique local-in-time solutions for the 3D free-boundary incompressible Euler equations with limited regularity, establishing optimal regularity conditions near the free boundary.

Contribution

It demonstrates the existence and uniqueness of solutions with minimal regularity assumptions, linking vorticity regularity to deformation regularity near the free boundary.

Findings

Solutions exist with initial velocity in H^{2.5+δ}

Regularity of deformation is optimal at H^{3+δ}

Vorticity regularity in H^{2+δ} is necessary for deformation regularity

Abstract

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that the Rayleigh-Taylor condition holds and $\mathrm{curl}\,u_0 \in H^{2+\delta }$ in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that $H^{3+\delta}$ regularity of the Lagrangian deformation near the free boundary can be ensured if and only if initial vorticity has $H^{2+\delta}$ regularity of vorticity near the free boundary.