Evolution of compressible and incompressible fluids separated by a closed interface

This work solves the problem governing the simultaneous motion of two viscous liquids of different kinds: compressible and incompressible. The boundary between the fluids is considered as an unknown (free) interface where the surface tension is taken into account. Although the fluids occupy the whole space $\mathbb{R}^3$, one of them should have a finite volume. Local (in time) unique solvability of this problem is obtained in the Sobolev--Slobodetskii spaces of functions. Estimates of the solution of a model problem for the Stokes equations are considered in detail, the interface between the fluids being a plane. The Schauder method is used to study a linear problem with a compact boundary. The passage to the nonlinear problem is made by successive approximations.