Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids

Local (in time) unique solvability of the problem on the motion of two compressible fluids, one of which has a finite volume, is obtained in Hölder spaces of functions with power-like decay at infinity. After the passage to Lagrangian coordinates, we arrive at a nonlinear initial-boundary value problem with a given closed interface between the liquids. We establish the existence theorem for this problem on the basis of the solvability of a linearized one by means of the fixed-point theorem. To obtain the estimates and to prove solvability for the linearized problem, we use the Schauder method and an explicit solution of a model linear problem with a plane interface between the liquids. All results are obtained under some restrictions to the fluid density and viscosities, which mean that the fluids are not so different from each other.