Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

“…Next, using a result concerning normal vector potential [4], we establish a similar Inf-Sup condition to (3), where the spaces X p T (Ω) and V p T (Ω) are replaced by the spaces X p N (Ω) and V p N (Ω) respectively. This conclude the proof of weak solution.…”

Section: The Stokes Equations With the Normal Boundary Conditionsmentioning

confidence: 99%

“…In the first time, we prove the existence of a unique π ∈ W −1,p (Ω), next we use [3] in order to prove that π ∈ L p (Ω).2…”

Section: The Stokes Equations With the Normal Boundary Conditionsmentioning

confidence: 99%

Stokes equations and elliptic systems with nonstandard boundary conditions

Amrouche

Seloula

2011

Comptes Rendus. Mathématique

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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 · n = g and curl u × 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 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 · n and u × n on Γ.To cite this article: C. Amrouche, N. Seloula, C. R. Acad. Sci. Paris, Ser. I 340 (2010). RésuméEquations de Stokes et systèmes elliptiques avec des conditions aux limites non standard. Dans un ouvert borné tridimensionnel,éventuellement multiplement connexe de classe C 1,1 , nous considérons leséquations stationnaires de Stokes avec des conditions aux limites de la forme u ·n = g et curl u ×n = h ×n ou u ×n = g ×n et π = π0 sur le bord Γ. Nous prouvons l'existence et l'unicité des solutions faibles, fortes et très faibles en théorie L p . Nos preuves sont basées sur l'obtention de conditions Inf − Sup qui jouent un rôle fondamental. Finalement, on donne deux décompositions d'Helmholtz qui tiennent compte des deux types de conditions aux limites u · n et u × n sur Γ.

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“…Observe that the study of the Stokes problem is fundamental for the study of the Oseen and Navier-Stokes equations. The detailed proofs of the results announced in this Note are given in [2].…”

Section: Introductionmentioning

confidence: 96%

Very weak solutions for the stationary Stokes equations

Amrouche¹,

Rodríguez-Bellido²

2010

Comptes Rendus. Mathématique

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The concept of very weak solution introduced by Giga [9] for the stationary Stokes equations has been intensively studied in the last years for the stationary Navier-Stokes equations. We give here a new and simpler proof of the existence of very weak solution for the stationary Navier-Stokes equations, based on density arguments and an adequate functional framework in order to define more rigourously the traces of non regular vector fields. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class C 1,1 of R 3 and can be extended to the Laplace's equation and to other dimensions. To cite this article: C. Amrouche, M.A. Rodríguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005). RésuméSolutions très faibles pour leséquations stationnaires de Stokes. Le concept de solution très faible introduit par Giga [9] pour leséquations stationnaires de Stokes aété beaucoupétudié ces dernières années pour leséquations stationnaires de Navier-Stokes. Nous donnons ici une nouvelle preuve plus simple de l'existence de solution très faible pour leséquations stationnaires de Navier-Stokes, qui s'appuie sur des arguments de densité et un cadre fonctionnel approprié pour définir de manière plus rigoureuse les traces des champs de vecteurs peu réguliers. On obtient aussi résultats de régularité dans des espaces de Sobolev fractionnaires. Tous les résultats sont obtenus dans le cas d'un ouvert connexe de classe C 1,1 de R 3 et peuventêtreétendusà l'équation de Laplace ainsi qu'aux autres dimensions. Pour citer cet article : C. Amrouche, M.A. Rodríguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

Cited by 62 publications

References 28 publications

On the very weak solution for the Oseen and Navier-Stokes equations

On the very weak solution for the Oseen and Navier-Stokes equations

Stokes equations and elliptic systems with nonstandard boundary conditions

Very weak solutions for the stationary Stokes equations

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