# 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

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topMucha, 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

top- [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. Zbl0970.35107
- [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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