A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid

TL;DR

This paper establishes a priori estimates for the 3D compressible free-boundary Euler equations with surface tension, enabling local existence results with lower initial regularity for a liquid's motion.

Contribution

It introduces a novel approach combining boundary regularity and a new compressible Cauchy invariance to lower initial data regularity requirements.

Findings

Provides a priori estimates for local existence in Lagrangian coordinates.

Reduces initial regularity requirements for velocity and density to H^3.

Introduces a new method leveraging boundary mean curvature and compressible invariance.

Abstract

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to $H^3$, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature, and a new compressible Cauchy invariance.