Mathematical Problems in Mechanics
Stokes equations and elliptic systems with nonstandard boundary conditions
[Équations de Stokes et systèmes elliptiques avec des conditions aux limites non standard]
Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 703-708.

Dans un ouvert borné tridimensionnel, éventuellement multiplement connexe de classe C1,1, nous considérons les équations stationnaires de Stokes avec des conditions aux limites de la forme un=g et curlu×n=h×n ou u×n=g×n et π=π0 sur le bord Γ. Nous prouvons lʼexistence et lʼunicité des solutions faibles, fortes et très faibles en théorie Lp. Nos preuves sont basées sur lʼobtention de conditions InfSup qui jouent un rôle fondamental. Finalement, on donne deux décompositions dʼHelmholtz qui tiennent compte des deux types de conditions aux limites un et u×n sur Γ.

In a three-dimensional bounded possibly multiply-connected domain of class C1,1, we consider the stationary Stokes equations with nonstandard boundary conditions of the form un=g and curlu×n=h×n or u×n=g×n and π=π0 on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions corresponding to each boundary condition in Lp theory. Our proofs are based on obtaining InfSup conditions that play a fundamental role. And finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as un and u×n on Γ.

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DOI : 10.1016/j.crma.2011.04.007
Amrouche, Chérif 1 ; Seloula, Nour El Houda 1, 2

1 Laboratoire de mathématiques appliquées, CNRS UMR 5142, université de Pau et des Pays de lʼAdour, IPRA, avenue de lʼuniversité, 64000 Pau, France
2 EPI Concha, LMA UMR CNRS 5142, INRIA Bordeaux-Sud-Ouest, 64000 Pau, France
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Amrouche, Chérif; Seloula, Nour El Houda. Stokes equations and elliptic systems with nonstandard boundary conditions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 703-708. doi : 10.1016/j.crma.2011.04.007. http://www.numdam.org/articles/10.1016/j.crma.2011.04.007/

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