Evolution of compressible and incompressible fluids separated by a closed interface (Q1584702)

The author proves local (in time) unique solvability of the free boundary value problem which describes the motion of an interface between compressible and incompressible fluids, taking into account the surface tension. The two fluids occupy the whole three-dimensional space, but the incompressible fluid must have a finite volume. The proof is based on Schauder estimates in Sobolev-Slobodeskij functional spaces for an auxiliary linear problem, which provides after iterations the solvability of the original nonlinear problem. The results are valid under restriction that the viscous fluid has a sufficient small viscosity, and the compressible fluid is barotropic.