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We investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of R 2 . Global existence and uniqueness of strong probabilistic solution is established. In contrast to previous results on this model we show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem. We also give two results on the long time behaviour of the solution. Mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time-independent second grade fluids equations if the deterministic part of the external force does not depend on time. If the deterministic forcing term explicitly depends on time, then the strong probabilistic solution decays exponentially in mean square.

Abstract. The purposes of this paper is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point for this work is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets sometimes complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail, which functions are subject to verify a functional equation often nonlinear. The problem is therefore to give an interpretation of these phenomena by functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, which vary irregularly, undergoing many nonperiodic oscillations. In this work we study the qualitative properties of spaces of such functions, say generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [26] (Homogenization of nonlinear pde's in the Fourier-Stieltjes algebras, SIAM J. Math. Anal., Vol. 41, No. 4, pp. 1589-1620 to know whether there exist or not ergodic algebras that are not subalgebras of the Fourier-Stieltjes algebra, and further we show that the theory of homogenization algebras by Nguetseng in [36] (Homogenization structures and applications I, Zeit. Anal. Anw., 22 (2003) 73-107) cannot handle weakly almost periodic homogenization problems.

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