On the solvability of some inhomogeneous incompressible flow with free interface (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

In this note, we review some recent results in [4] on the solvability of some free boundary problem of the two phase inhomogeneous incompressible Navier-Stokes equations. In [4], we addressed some general approach to construct short time solutions in L_{p}-L_{q} maximal regularity class where q>N and p belongs to ] 2, infty[cup{overline{p}in] 1, 2[:1/overline{p}+N/q>3/2}. In particular, to handle the less regular initial data for 1<p<2, some new estimates are derived. Moreover, for the case of piecewise constant density, some long time solutions in the moving bounded droplet are establish within L_{p}-L_{q} maximal regularity. Furthermore, we can find some global solutions in the fixed bounded pool by applying the idea in [4] here.