Problem on the motion of two compressible fluids separated by a closed free interface

A problem is considered on the simultaneous evolution of two barotropic capillary viscous compressible fluids occuping the whole space $\mathbb R^3$ and separated by a closed free interface. Under some restrictions on the viscosities of the eiquids, the local (in time) unique solvability of this problem is obtained in the Sobolev–Slobodetskii spaces. After the passage to the Lagrangean coordinates it is possible to exclude the renknown function of the fluid density from the system of equations. The proof of the solution existence of an nonlinear, non-coercive initial-boundary value problem received is based on the method of succesive approximations and on an explicite solution of a model linear problem with the plane interface between the eiquids. The restrictions on the viscosities mentioned above appear in the intermediate estimation of this explicit solution in the Sobolev spaces with an exponential weight.