×

Problem on the motion of two compressible fluids separated by a closed free interface. (English. Russian original) Zbl 0911.76079

J. Math. Sci., New York 99, No. 1, 837-853 (2000); translation from Zap. Nauchn. Semin. POMI 243, 61-86 (1997).
We consider the evolution of two barotropic capillary viscous compressible fluids occuping the whole space \(\mathbb{R}^3\) and separated by a closed free interface. Under some restrictions on the fluid viscosities, we obtain the local (in time) unique solvability of this problem in Sobolev-Slobodetskij spaces. The proof is based on the method of successive approximations and on an explicit solution of a model linear problem with the plane interface between the fluids.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] V. A. Solonnikov, ”On an initial-boundary value problem for the Stokes system arising in the study of a problem with a free boundary,”Tr. Mat. Inst. Akad. Nauk SSSR,188, 150–188 (1990).
[2] V. A. Solonnikov, ”Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval,”Algebra Analiz,3, 222–257 (1991).
[3] V. A. Solonnikov and A. Tani, ”Free boundary problem for a viscous compressible flow with surface tension,” in:Constantin Carathéodory: An International Tribute, World Scientific (1991), pp. 1270–1303. · Zbl 0752.35096
[4] I. V. Denisova, ”A priori estimates of a solution of a linear time-dependent problem connected with the motion of a drop in a fluid medium,”Tr. Mat. Inst. Akad. Nauk SSSR,188, 3–21 (1990). · Zbl 0737.35063
[5] I. V. Denisova and V. A. Solonnikov, ”Solvability of the linearized problem on motion of a drop in a fluid flow,”Zap. Nauchn. Semin. LOMI,171, 53–65 (1989). · Zbl 0796.76025
[6] I. V. Denisova, ”Problem of the motion of two viscous incompressible fluids separated by a closed free interface,”Acta Appl. Math.,37, 31–40 (1994). · Zbl 0814.35093 · doi:10.1007/BF00995127
[7] A. Tani, ”Two-phase free boundary problem for compressible viscous fluid motion,”J. Math. Kyoto Univ.,24, 243–267 (1984). · Zbl 0567.76098 · doi:10.1215/kjm/1250521328
[8] M. S. Agranovich and M. I. Vishik, ”Elliptic problems with a parameter and parabolic problems of general type,”Usp. Mat. Nauk,19, 53–152 (1964). · Zbl 0137.29602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.