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

• Autorzy: Piotr Bogusław Mucha , Wojciech Zajączkowski
• 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

anisotropic Sobolev space; Navier-Stokes equations; local existence; sharp regularity; incompressible viscous barotropic self-gravitating fluid

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 2000
• Tom: 27
• Numer: 3
• Strony: 319-333

Daty

wydano

2000

otrzymano

1999-05-28

poprawiono

2000-01-17

Twórcy

autor

Piotr Bogusław Mucha

• Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2 02-097 Warszawa, Poland

autor

Wojciech Zajączkowski

• 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).