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## Applicationes Mathematicae

2000 | 27 | 3 | 319-333
Tytuł artykułu

### On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that $u\in W^{2,1}_r(\widetilde{{\mitΩ}}^T)$ with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the $L_p$-approach the Lagrangian coordinates must be used. We are looking for solutions with lowest possible regularity because this simplifies the proof and decreases the number of compatibility conditions.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
319-333
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-28
poprawiono
2000-01-17
Twórcy
autor
• Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2 02-097 Warszawa, Poland
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
• [1] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian).
• [2] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1975.
• [3] P. B. Mucha and W. M. Zajączkowski, On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework, Studia Math., to appear.
• [4] V. A. Solonnikov, On nonstationary motion of an isolated volume of a viscous incompressible fluid, Izv. Akad. Nauk SSSR 51 (1987), 1065-1087 (in Russian).\vadjust
• [5] V. A. Solonnikov,Solvability on a finite time interval of the problem of evolution of a viscous incompressible fluid bounded by a free surface, Algebra Anal. 3 (1991), 222-257 (in Russian).
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Bibliografia
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