Local solvability of free boundary problems for the two-phase navier-stokes equations with surface tension in the whole space

Abstract

We consider the free boundary problem of the two-phase Navier-Stokes equation with surface tension and gravity in the whole space. We prove a local-in-time unique existence theorem in the space W 2,1q,p with 2 < p < ∞ and n < q < ∞ for any initial data satisfying certain compatibility conditions. Our theorem is proved by the standard fixed point argument based on the maximal Lp-Lq regularity theorem for the corresponding linearized equations.