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

Local (in time) unique solvability of a problem on the motion of two compressible fluids, one of which has finite volume, is obtained in Hölder spaces of functions with a power-like decay at infinity. After passage to Lagrangian coordinates, we arrive at a nonlinear initial boundary value problem with a given closed interface between the liquids. We establish an existence theorem for this problem on the basis of the solvability of a linearized problem by means of the fixed-point theorem. To obtain estimates and to prove the 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. The results are obtained under some restrictions on the fluid density and viscosities, which mean that the fluids are not much different from each other. Bibliography: 8 titles. © 2005 Springer Science+Business Media, Inc.