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

Piotr Mucha; Wojciech Zajączkowski

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 3, page 319-333
  • ISSN: 1233-7234

Abstract

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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 ∈ W r 2 , 1 ( Ω ˜ 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.

How to cite

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Mucha, Piotr, and Zajączkowski, Wojciech. "On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion." Applicationes Mathematicae 27.3 (2000): 319-333. <http://eudml.org/doc/219276>.

@article{Mucha2000,
abstract = {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 $u\in W^\{2,1\}_r(\widetilde\{\{Ω\}\}^T)$ 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.},
author = {Mucha, Piotr, Zajączkowski, Wojciech},
journal = {Applicationes Mathematicae},
keywords = {anisotropic Sobolev space; Navier-Stokes equations; local existence; sharp regularity; incompressible viscous barotropic self-gravitating fluid; local-in-time existence; free boundary problem; self-gravitating fluid motion; lowest possible regularity; Cauchy-Neumann problem; Stokes system},
language = {eng},
number = {3},
pages = {319-333},
title = {On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion},
url = {http://eudml.org/doc/219276},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Mucha, Piotr
AU - Zajączkowski, Wojciech
TI - On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 3
SP - 319
EP - 333
AB - 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 $u\in W^{2,1}_r(\widetilde{{Ω}}^T)$ 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.
LA - eng
KW - anisotropic Sobolev space; Navier-Stokes equations; local existence; sharp regularity; incompressible viscous barotropic self-gravitating fluid; local-in-time existence; free boundary problem; self-gravitating fluid motion; lowest possible regularity; Cauchy-Neumann problem; Stokes system
UR - http://eudml.org/doc/219276
ER -

References

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  1. [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. [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. [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. Zbl0970.35107
  4.  
  5. [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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