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

We investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of R²${\mathbb{R}}^2$. 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 behavior 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.