The concept of very weak solution introduced by Giga (1981) 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 rigorously 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 of and can be extended to the Laplace's equation and to other dimensions.
Le concept de solution très faible introduit par Giga (1981) pour les équations stationnaires de Stokes a été beaucoup étudié ces dernières années pour les équations stationnaires de Navier–Stokes. Nous donnons ici une nouvelle preuve plus simple de l'existence de solution très faible pour les équations 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 de et peuvent être étendus à l'équation de Laplace ainsi qu'aux autres dimensions.
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Chérif Amrouche 1; María Ángeles Rodríguez-Bellido 2
@article{CRMATH_2010__348_3-4_223_0, author = {Ch\'erif Amrouche and Mar{\'\i}a \'Angeles Rodr{\'\i}guez-Bellido}, title = {Very weak solutions for the stationary {Stokes} equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {223--228}, publisher = {Elsevier}, volume = {348}, number = {3-4}, year = {2010}, doi = {10.1016/j.crma.2009.12.020}, language = {en}, }
Chérif Amrouche; María Ángeles Rodríguez-Bellido. Very weak solutions for the stationary Stokes equations. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 223-228. doi : 10.1016/j.crma.2009.12.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.020/
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