On the limit as the density ratio tends to zero for two perfect incompressible fluids separated by a surface of discontinuity

摘要

We study the asymptotic limit as the density ratio ρ~-/ρ~+ → 0, where ρ~+ and ρ~- are the densities of two perfect incompressible 2-D/3-D fluids, separated by a surface of discontinuity along which the pressure jump is proportional to the mean curvature of the moving surface. Mathematically, the fluid motion is governed by the two-phase incompressible Euler equations with vortex sheet data. By rescaling, we assume the density ρ~+ of the inner fluid is fixed, while the density ρ~- of the outer fluid is set to ε. We prove that solutions of the free-boundary Euler equations in vacuum are obtained in the limit as ε → 0.