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Equations de Navier-Stokes stationnaires avec données peu régulières. (French) Zbl 0559.35064
One proves the existence of stationary solutions for the non-homogeneous Navier-Stokes equations in a bounded domain $$\Omega$$ with $$\partial \Omega$$ of class $$C^ 2$$, when the exterior forces field is $$f\in W^{- 1,p}(\Omega)$$, the velocity field on the boundary is $$\phi \in W^{1- 1/p,p}(\partial \Omega)$$ and $$n/2<p\leq 2$$ $$(n=2,3$$ being the dimension of $$\Omega)$$. The solutions then enjoy the same regularity properties as those of the corresponding Stokes problem, i.e. $$(u,p)\in W^{1,p}(\Omega)\times L^ p(\Omega).$$
The method used works as well in some cases where the boundary $$\partial \Omega$$ is only Lipschitz. It is shown in this way that there are stationay flows in a cavity $$(n=2)$$, where $$\phi\in \cap_{p<2}W^{1- 1/p,p}(\partial \Omega)$$ but $$\phi \not\in W^{1/2,2}(\partial \Omega)$$. The author proves also, it seems first, the existence of the stationary solutions for the Taylor problem $$(n=3)$$, for which alike $$\phi\in \cap_{p<2}W^{1-1/p,p}(\partial \Omega)$$; thus one gets solutions $$(u,p)\in \cap_{p<2}W^{1,p}(\Omega)\times L^ p(\Omega).\quad -$$ The nonstationary solutions in the same function spaces are studied too.
Reviewer: G.Minea

##### MSC:
 35Q30 Navier-Stokes equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35R05 PDEs with low regular coefficients and/or low regular data
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##### References:
 [1] J. Leray , Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l’hydrodynamique , J. Math. Pures Appl. , 12 ( 1933 ), pp. 1 - 82 . Article | Zbl 0006.16702 | JFM 59.0402.01 · Zbl 0006.16702 · eudml:235182 [2] E. Hopf Über die Aufangswertaufgabe für die hydrodynamischen Grundleichungen , Math. Nachr. , 4 ( 1951 ), pp. 213 - 231 . MR 50423 | Zbl 0042.10604 · Zbl 0042.10604 · doi:10.1002/mana.3210040121 [3] L. Cattabriga , Su un problema al contorno relativo al sistema di equazioni di Stokes , Rend. Sem. Mat. Univ. Padova , 31 ( 1961 ), pp. 308 - 340 . Numdam | MR 138894 | Zbl 0116.18002 · Zbl 0116.18002 · numdam:RSMUP_1961__31__308_0 · eudml:107065 [4] R. Temam , Navier-Stokes equations , North-Holland ( 1979 ). MR 603444 | Zbl 0383.35057 · Zbl 0383.35057
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