Abstract:
We investigate a stochastic evolution equation for the motion of a second grade
fluid filling a bounded domain of R2. 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.
