Stokes equations and elliptic systems with nonstandard boundary conditions

Amrouche, Chérif; Seloula, Nour El Houda

doi:10.1016/j.crma.2011.04.007

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]

Présenté par : Philippe G. Ciarlet

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

Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 703-708.

Résumé
Abstract

Dans un ouvert borné tridimensionnel, éventuellement multiplement connexe de classe $C^{1, 1}$ , nous considérons les équations stationnaires de Stokes avec des conditions aux limites de la forme $u \cdot n = g$ et $curl u \times n = h \times n$ ou $u \times n = g \times n$ et $π = π_{0}$ sur le bord Γ. Nous prouvons lʼexistence et lʼunicité des solutions faibles, fortes et très faibles en théorie $L^{p}$ . Nos preuves sont basées sur lʼobtention de conditions $Inf – Sup$ qui jouent un rôle fondamental. Finalement, on donne deux décompositions dʼHelmholtz qui tiennent compte des deux types de conditions aux limites $u \cdot n$ et $u \times n$ sur Γ.

In a three-dimensional bounded possibly multiply-connected domain of class $C^{1, 1}$ , we consider the stationary Stokes equations with nonstandard boundary conditions of the form $u \cdot n = g$ and $curl u \times n = h \times n$ or $u \times n = g \times n$ and $π = π_{0}$ on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions corresponding to each boundary condition in $L^{p}$ theory. Our proofs are based on obtaining $Inf – Sup$ conditions that play a fundamental role. And finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as $u \cdot n$ and $u \times n$ on Γ.

Reçu le : 2010-07-07
Accepté le : 2011-03-30
Publié le : 2011-05-05

DOI : 10.1016/j.crma.2011.04.007

Affiliations des auteurs :

Amrouche, Chérif ¹ ; Seloula, Nour El Houda ^{1, 2}

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     title = {Stokes equations and elliptic systems with nonstandard boundary conditions},
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     pages = {703--708},
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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/

Bibliographie
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