On the model problem arising in the study of motion of viscous compressible and incompressible fluids with a free interface

The model problem under study concerns the evolution of two viscous capillary fluids of different types: compressible and incompressible, contained in a bounded vessel and separated by a free interface. The solution is estimated in the Sobolev–Slobodetskiǐ function spaces; these estimates can be useful for the proof of stability for the rest state.