This article studies the solutions in L 2 of a steady transport equation with a divergence-free driving velocity that is H 1 , in a Lipschitz domain of IR d. Since the velocity is assumed fully non-homogeneous on the boundary, existence and uniqueness of solution require a boundary condition. A new Green's formula allows us to define the normal component of zu on the boundary, where z denotes the stress and u the velocity. A substantial part of the article is devoted to properties of a truncature operator in the space where z and u. ∇z are L 2. By means of these properties, which allows us to prove density results, and using in addition a non-bounded linear operator from L 2 to L 2 , we establish existence and uniqueness of the solution for the transport equation with a boundary condition on the open part where the normal component of u is strictly negative.
SUMMARYThis paper is devoted to Stokes and Navier-Stokes problems with non-standard boundary conditions: we consider, in particular, the case where the pressure is given on a part of the boundary. These problems were studied by BÃ egue, Conca, Murat and Pironneau. They proved the existence of variational solutions, indicating that these were solutions of the initial non-standard problems, if they are regular enough, but without specifying the conditions on the data which would imply this regularity. In this paper, ÿrst we show that the variational solutions, on supposing pressure on the boundary 2 of regularity H 1=2 instead of H −1=2 , have their Laplacians in L 2 and, therefore, are solutions of non-standard Stokes problem. Next, we give a result of regularity H 2 , which we generalize, obtaining regularities W m; r , m ∈ N; m¿2; r¿2. Finally, by a ÿxed-point argument, we prove analogous results for the Navier-Stokes problem, in the case where the viscosity is large compared to the data.
Abstract. This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where |α1 + α2| ≤ (24νβ) 1/2 , studied by Amrouche and Cioranescu, the H 1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to the general case of third grade, complicates substantially the proof of the existence of the solution. We also prove further regularity results by a method similar to that of Cioranescu and Girault for second grade fluids. This extension to the third grade fluids is not straightforward, because of a transport equation which is much more complex.Résumé. Dans cet article, on montre que la méthode de décomposition avec base spéciale introduite par Cioranescu et Ouazar, permet de démontrer l'existence globale en temps de la solution faible pour les fluides de grade trois, en dimension trois, avec des données petites. Contrairement au cas particulier où |α1 + α2| ≤ (24νβ) 1/2 ,étudié par Amrouche et Cioranescu, la norme H 1 de la vitesse n'est pas majorée pour toute donnée. Ce fait, qui conduisaità penser, en contradiction avec cet article, que la méthode de décomposition ne pouvait pas s'appliquer au cas général du grade trois, complique substantiellement la démonstration d'existence de la solution. Onétablit des résultats de régularité par une méthode similaireà celle de Cioranescu et Girault pour des fluides
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