A priori estimates for the free-boundary Euler equations with surface tension in three dimensions

TL;DR

This paper establishes a priori estimates for the 3D free-boundary Euler equations with surface tension, improving initial data regularity requirements using direct methods involving pressure, curvature, and invariance.

Contribution

It provides new a priori estimates for the free-boundary Euler equations in 3D, lowering initial regularity requirements in Lagrangian coordinates.

Findings

A priori estimates for local existence in 3D free-boundary Euler equations.

Reduced regularity requirements for initial velocity and boundary data.

Methodology involving pressure estimates, boundary curvature, and Cauchy invariance.

Abstract

We derive a priori estimates for the incompressible free-boundary Euler equations with surface tension in three spatial dimensions. Working in Lagrangian coordinates, we provide a priori estimates for the local existence when the initial velocity, which is rotational, belongs to $H^3$ and the trace of initial velocity on the free boundary to $H^{3.5}$, thus lowering the requirement on the regularity of initial data in the Lagrangian setting. Our methods are direct and involve three key elements: estimates for the pressure, the boundary regularity provided by the mean curvature, and the Cauchy invariance.