A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

TL;DR

This paper proves a new regularity result for the 3D incompressible free-boundary Euler equations with surface tension, using Lagrangian coordinates to derive a priori estimates under minimal initial regularity.

Contribution

It introduces a novel Lagrangian approach to establish interior regularity for free-boundary Euler equations with surface tension, requiring minimal initial data assumptions.

Findings

Established a priori estimates for solutions with minimal regularity

Demonstrated interior regularity for free-boundary Euler equations with surface tension

Used Lagrangian coordinates to simplify the analysis

Abstract

We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.