Existence of equilibrium figures of a rotating capillary two-phase fluid

The paper deals with a solution of a stationary problem with unknown boundaries for the Navier–Stokes equations corresponding to the slow rigid rotation of a viscous two-phase drop consisting of compressible and incompressible embedded fluids. In this case, the internal fluid is incompressible. It is bounded by a closed interface that does not intersect the outer free surface. It is assumed that the compressible fluid is barotropic. Surface tension forces act at the boundaries. The existence of a family of equilibrium figures close to embedded balls is proved. The proof is carried out in Hölder spaces using implicit function theorem.