Global $L_2$-solvability of a problem governing two-phase fluid motion without surface tension

Abstract

The paper deals with an interface problem for the Navier–Stokes system governing the motion of two incompressible fluids in a container, one liquid being inside another one. We prove unique solvability of the problem in an infinite time interval provided that the data are small enough, surface tension effect being neglected on the interface between the fluids. The norms of the solution are shown to decay exponentially at infinity with respect to time. The proof is based on exponential energy estimate and on local existence theorem of the problem in anisotropic Sobolev–Slobodetskiĭ spaces.

We give also the main steps of the proof of the local theorem for the problem with and without including surface tension.