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

Fabio Pusateri

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

Analysis of PDEs 2011-03-08 v2

Authors: Fabio Pusateri

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.

Keywords

two-phase Euler equations free-boundary motion surface tension density ratio Rayleigh-Taylor sign condition energy estimates convergence of solutions

Cite

@article{arxiv.0912.3296,
  title  = {On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations},
  author = {Fabio Pusateri},
  journal= {arXiv preprint arXiv:0912.3296},
  year   = {2011}
}

Comments

Revised version, to appear. In the condition $\rho_- \leq \epsilon^{7/3}$, the exponent is now improved to 3/2. The vorticity of the outer velocity field has been added to the Energy, and a few arguments have been changed accordingly. Other minor changes. Typos corrected

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

Abstract

Keywords

Cite

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