Solvability of a free boundary problem for the Navier-Stokes equations describing the motion of viscous incompressible nonhomogeneous fluid

Abstract

We consider a time-dependent problem for a viscous incompressible nonhomogeneous fluid bounded by a free surface on which surface tension forces act. We prove the local in time solvability theorem for this problem in Sobolev function spaces. In the nonhomogeneous model the density of the fluid is unknown. Going over to Lagrange coordinates connected with the velocity vector field, we pass from the free boundary problem to the problem in the fixed boundary domain. Due to the continuity equation, in Lagrange coordinates the density is the same as at the initial moment of time. It gives us the possibility to apply the methods developed by V.A. Solonnikov for the case of incompressible fluid with constant density.