$ L_2$-theory for two viscous fluids of different types: Compressible and incompressible

We prove the stability of the rest state in the problem of evolution of two viscous fluids, compressible and incompressible, contained in a bounded vessel and separated by a free interface. The fluids are subject to mass and capillary forces. The proof of stability is based on “maximal regularity” estimates for the solution in the anisotropic Sobolev–Slobodetskiĭ spaces $W_2^{r,r/2}$ with an exponential weight.