Geometry and a priori estimates for free boundary problems of the Euler's equation

TL;DR

This paper establishes estimates for free boundary Euler equations with and without surface tension, demonstrating convergence to zero surface tension solutions under the Rayleigh-Taylor condition.

Contribution

It provides new a priori estimates and convergence results for free boundary Euler equations with surface tension, under the Rayleigh-Taylor sign condition.

Findings

Solutions with surface tension converge to zero surface tension solutions as surface tension vanishes.

The Rayleigh-Taylor sign condition ensures well-posedness and convergence.

Derived estimates are valid for both cases with and without surface tension.

Abstract

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension.