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

Mucha, Piotr; Zajączkowski, Wojciech

Mucha, Piotr ; Zajączkowski, Wojciech

Applicationes Mathematicae, Tome 27 (2000), p. 319-333 / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

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 $u \in W_{r}^{2, 1} ({\tilde{Ω}}^{T})$ 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.

Publié le : 2000-01-01

Zbl 0996.35050

EUDML-ID : urn:eudml:doc:219276

@article{bwmeta1.element.bwnjournal-article-zmv27i3p319bwm,
     author = {Piotr Mucha and Wojciech Zaj\k aczkowski},
     title = {On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion},
     journal = {Applicationes Mathematicae},
     volume = {27},
     year = {2000},
     pages = {319-333},
     zbl = {0996.35050},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv27i3p319bwm}
}

Mucha, Piotr; Zajączkowski, Wojciech. On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion. Applicationes Mathematicae, Tome 27 (2000) pp. 319-333. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv27i3p319bwm/

Bibliographie

[000] [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).

[001] [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.

[002] [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. | Zbl 0970.35107

[003]

[004] [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).