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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:
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