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

• Published in: ESAIM: Mathematical Modelling and Numerical Analysis Vol. 33; no. 6; pp. 1091 - 1120
• Main Author: Bernard, Jean Marie
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.11.1999
• Subjects: 35D05; 35Q99; 76A05; energy estimates; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Galerkin method; Mathematical analysis; Mathematics; Non-newtonian fluid flows; Ordinary differential equations; Partial differential equations; Physics; Sciences and techniques of general use; special basis; Solution uniqueness; Euler equation; Sobolev space; Existence of solution; Clausius Duhem inequality; Rivlin Eriksen fluid; Decomposition method; Partial differential equation; Poincaré inequality; Navier Stokes equation; Vorticity; Dirichlet problem; Kinematic viscosity; Transport equation; Fluid mechanics; Equation of motion; Banach space; Incompressible flow; Three dimensional model; Boundary value problem; Weak solution; Rivlin Eriksen tensor; A priori estimation; Galerkin method; Green formula
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an:1999136

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 $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^{1/2}$, studied by Amrouche and Cioranescu, the H1 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. 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 $\vert\alpha_1+\alpha_2\vert\le (24\nu\beta)^{1/2}$, étudié par Amrouche et Cioranescu, la norme H1 de la vitesse n'est pas majorée pour toute donnée. Ce fait, qui conduisait à 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. On établit des résultats de régularité par une méthode similaire à celle de Cioranescu et Girault pour des fluides