Existence of local solutions for free boundary problems for viscous compressible barotropic fluids

• Autorzy: W. M. Zajączkowski
• Języki publikacji: EN

Abstrakty

EN

We prove the local existence of solutions for equations of motion of a viscous compressible barotropic fluid in a domain bounded by a free surface. The solutions are shown to exist in exactly those function spaces where global solutions were found in our previous papers [14, 15].

Słowa kluczowe

EN

local existence; free boundary; anisotropic Sobolev spaces; viscous compressible barotropic fluids

Kategorie tematyczne

• 35R35: Free boundary problems
• 35G30: Boundary value problems for nonlinear higher-order equations
• 76N10: Existence, uniqueness, and regularity theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1994-1995
• Tom: 60
• Numer: 3
• Strony: 255-287

Daty

wydano

1995

otrzymano

1993-11-03

poprawiono

1994-03-15

Twórcy

autor

W. M. Zajączkowski

• Institute of Mathematics, Polish Academy of Sciences, P.O. Box 137, 00-950 Warszawa, Poland

Bibliografia

• [1] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representation of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian).
• [2] D. K. Faddeev, V. A. Solonnikov and N. N. Ural'tseva, Selected Topics of Analysis and Algebra, Leningrad, 1981 (in Russian).
• [3] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian).
• [4] L. Landau and E. Lifschitz, Mechanics of Continuum Media, Nauka, Moscow, 1954 (in Russian); English transl.: Pergamon Press, Oxford, 1959; new edition: Hydrodynamics, Nauka, Moscow, 1986 (in Russian).
• [5] V. A. Solonnikov, On an initial-boundary value problem for the Stokes system which appears in free boundary problems, Trudy Mat. Inst. Steklov. 188 (1990), 150-188 (in Russian).
• [6] V. A. Solonnikov, On boundary problems for linear parabolic systems of differential equations of general form, ibid. 83 (1965) (in Russian).
• [7] V. A. Solonnikov, A priori estimates for second order parabolic equations, ibid. 70 (1964), 133-212 (in Russian).
• [8] V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with a surface tension, in: Constantine Carathéeodory: An International Tribute, T. M. Rassias (ed.), World Scientific, 1991, 1270-1303.
• [9] E. Zadrzyńska and W. M. Zajączkowski, On local motion of a general compressible viscous heat conducting fluid bounded by a free surface, Ann. Polon. Math. 59 (1994), 133-170.
• [10] E. Zadrzyńska and W. M. Zajączkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid, to appear.
• [11] E. Zadrzyńska and W. M. Zajączkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting capillary fluid, to appear.
• [12] W. M. Zajączkowski, On an initial-boundary value problem for the parabolic system which appears in free boundary problems for compressible Navier-Stokes equations, Dissertationes Math. 304 (1990).
• [13] W. M. Zajączkowski, On local motion of a compressible barotropic viscous fluid bounded by a free surface, in: Banach Center Publ. 27, Inst. Math., Polish Acad. Sci., 1992, 511-553.
• [14] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993).
• [15] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface, SIAM J. Math. Anal. 25 (1994), 1-84.