Qualitative properties for the 2- D $D$ nonautonomous stochastic Navier–Stokes equations

We establish the pullback asymptotic compact of the family probability measures with respect to probability distributions of the solutions of the 2-$D$ nonautonomous stochastic Navier–Stokes equations, and prove the existence and uniqueness of a pullback measure attractor. The structures of pullback measure attractors is characterized by complete solutions, which is an extension of the notation of evolution systems of measures introduced and developed by Da Prato and Röckner in [Rendiconti Lincei-Matematica e Applicazioni 17 (2006), no. 4, 397–403] and [Seminar on Stochastic Analysis, Random Fields and Applications, 115–122, Springer, 2007]. Moreover, for stochastic systems containing periodic deterministic forcing terms, we show the pullback measure attractors are also periodic under certain conditions.