Stokes equations and elliptic systems with nonstandard boundary conditions

Chérif Amrouche; Nour El Houda Seloula

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

Chérif Amrouche ¹ ; Nour El Houda Seloula ^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, Volume 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 :

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

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     author = {Ch\'erif Amrouche and Nour El Houda Seloula},
     title = {Stokes equations and elliptic systems with nonstandard boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {703--708},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.04.007},
     language = {en},
}

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Chérif Amrouche; Nour El Houda Seloula. Stokes equations and elliptic systems with nonstandard boundary conditions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 703-708. doi : 10.1016/j.crma.2011.04.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.007/

Bibliographie
Cité par

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[4] C. Amrouche, N. Seloula, $L^{p}$ -theory for vector potentials and Sobolevʼs inequalities for vector fields. Application to the Stokes problemʼs with pressure boundary conditions, submitted for publication.

[5] C. Conca; F. Murat; O. Pironneau The Stokes and Navier–Stokes equations with boundary conditions involving the pressure, Jpn. J. Math., Volume 20 (1994), pp. 263-318

[6] H. Kozono; T. Yanagisawa $L^{r}$ -variational inequality for vector fields and the Helmholtz–Weyl decomposition in bounded domains, Indiana Univ. Math. J., Volume 58 (2009) no. 4

[7] R. Temam Theory and Numerical Analysis of the Navier–Stokes Equations, North-Holland, Amsterdam, 1977

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