Geodesic
Solvability in weighted H\"older spaces for a~problem governing the evolution of two compressible fluids
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 57-89

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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. 
@article{ZNSL_2003_295_a2,
     author = {I. V. Denisova},
     title = {Solvability in weighted {H\"older} spaces for a~problem governing the evolution of two compressible fluids},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {57--89},
     publisher = {mathdoc},
     volume = {295},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a2/}
}
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I. V. Denisova. Solvability in weighted H\"older spaces for a~problem governing the evolution of two compressible fluids. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 57-89. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a2/