A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W?1/q,q

Galdi, Giovanni P.; Simader, Christian G.; Sohr, Hermann

doi:10.1007/s00208-004-0573-7

Cited by 73 publications

(88 citation statements)

References 16 publications

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“…Moreover, our solution is obtained in the space T p,r (Ω), which has been clearly characterized, contrary to the space W 1,p (Ω) appearing in [8], which was not characterized, completely abstract and obtained as the closure of W 1,p (Ω) for the norm…”

Section: Definition 32 (Very Weak Solution For the Stokes Problem)mentioning

confidence: 81%

“…Remark 1 i) Observe that in [8] Theorem 3, the domain was of class C 2,1 (here it is of class C 1,1 ), and the divergence term was h ∈ L p (Ω) (here of h ∈ L r (Ω)). Moreover, our solution is obtained in the space T p,r (Ω), which has been clearly characterized, contrary to the space W 1,p (Ω) appearing in [8], which was not characterized, completely abstract and obtained as the closure of W 1,p (Ω) for the norm…”

Section: Definition 32 (Very Weak Solution For the Stokes Problem)mentioning

confidence: 99%

“…The notion of very weak solutions (u, q) ∈ L p (Ω)×W −1,p (Ω) for the stationary Stokes or Navier-Stokes equations, corresponding to very irregular data, has been developed in the last years by Giga [9] (and also by Lions-Magenes [11] for the Laplace's equation, in a domain Ω of class C ∞ ), Amrouche-Girault [1] (in a domain Ω of class C 1,1 ) and more recently by Galdi-Simader-Sohr [8], Farwig-Galdi-Sohr [7] (in a domain Ω of class C 2,1 , see also Schumacher [14]) and finally by Kim [10] (in a domain Ω of class C 2 with connected boundary). In this context, the boundary condition is chosen in L p (Γ) (see Brown-Shen [3], Conca [5], Fabes-Kenig-Verchota [6], Moussaoui [12], Shen [15], Savaré [13]) or more generally in W −1/p,p (Γ).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Definition 32 (Very Weak Solution For the Stokes Problem)mentioning

confidence: 81%

Section: Definition 32 (Very Weak Solution For the Stokes Problem)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Very weak solutions for the stationary Stokes equations

Amrouche¹,

Rodríguez-Bellido²

2010

Comptes Rendus. Mathématique

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“…More precisely, one can permit domains with a boundary regularity that is of one order lower than in the former results. Using this it is possible to show that the results on very weak solutions to the Navier-Stokes equations by Galdi, Simader and Sohr in [10] and by Farwig, Galdi and Sohr in [5] hold not only in C 2,1 -domains but, more generally, in C 1,1 -domains. This can be seen in [14] where a weighted approach to this problem is given.…”

Section: Introductionmentioning

confidence: 88%