On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations

TL;DR

This paper analyzes the behavior of two-phase Euler equations with free boundary conditions as surface tension and density ratio approach zero, proving convergence to vacuum solutions without surface tension.

Contribution

It establishes uniform energy estimates and convergence results for the two-phase Euler equations in the zero surface tension and density ratio limit.

Findings

Proves energy estimates uniform in density ratio and surface tension parameter.

Shows convergence of solutions to vacuum Euler equations without surface tension.

Provides rigorous mathematical foundation for zero surface tension limit in two-phase flows.

Abstract

We consider the free-boundary motion of two perfect incompressible fluids with different densities $\rho_+$ and $\rho_-$, separated by a surface of discontinuity along which the pressure experiences a jump proportional to the mean curvature by a factor $\epsilon^2$. Assuming the Raileigh-Taylor sign condition and $\rho_- \leq \epsilon^{3/2}$ we prove energy estimates uniform in $\rho_-$ and $\epsilon$. As a consequence we obtain convergence of solutions of the interface problem to solutions of the free-boundary Euler equations in vacuum without surface tension as $\epsilon$ and $\rho_-$ tend to zero.