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

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.