Weak and classical solutions of equations of motion for third grade fluids

Abstract

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

Asymptotic profiles for the third grade fluids equations

“…The aim of this section is to show that the inequality (1.10) of Theorem 1.1 holds for the regularized solutions of the system (4.1), provided that the condition (1.9) is satisfied by W 0 . To this end, we consider a fixed constant θ such that 0 < θ < 1 which is twice the rate of convergence of W to ηG in H 2 (2). In fact, we will show that, under the assumption (1.9), the decaying of f to 0 in H 2 (2) is equivalent to e − θτ 2 .…”

“…We note v ε (x) = w ε (x/γ). The system (3.1) provides a new system in v ε , that we will solve in H 2 (2).…”

Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior

“…They proved global existence and uniqueness without restriction on the initial data for the two dimensional case. Bernard in [5], Cioranescu and Girault in [16] extended their results to the three dimensional case; global existence was obtained with some reasonable restrictions on the initial data.…”

“…for any smooth (solenoidal) functions Φ, v and w. Now we recall the following inequalities whose proof can be found in [5], [17] (see also [40])…”

Non‐standard Stokes and Navier–Stokes problems: existence and regularity in stationary case

“…But, to prove that these variational solutions verify all the boundary conditions of the initial problem, precisely, the condition on the pressure and that on the curl of the velocity, they suppose that they have their Laplacians in L 2 , which is a condition on the solution and not on the data. In the same way, considering the Navier-Stokes problem − u + u · ∇u + ∇p = f in (6) div u = 0 in (7) u = u 0 on 1 (8) u × n = a × n; p+ (1=2)|u| 2 = p 0 on 2 (9) u · n = b · n; curl u × n = h × n on 3 (10) they prove that all the boundary conditions are veriÿed by a variational solution u by supposing that u and curl u × u belong to L 2 ( ) 3 , but without specifying the conditions on the data which would imply this regularity. The purpose of the present paper is, ÿrst, to show that, if we suppose the data of the pressure on the boundary as being a little more regular, precisely H 1=2 instead of H −1=2 , we can prove that any variational solution u of the Stokes problem veriÿes u ∈L 2 and, therefore, the boundary conditions of the initial problem.…”