Pullback attractors and statistical solutions for 2-D Navier-Stokes equations

In this paper we investigate some relations among the notions of pullback attractor, time-average measure and statistical solution.
    Using time-averages and Banach generalized limits we construct a family of probability measures $\{\mu_t\}_{t\in \IR}$ on the pullback attractor $\{A(t)\}_{t\in \R}$ of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures satisfy supp$\mu_t \subset A(t)$ for all $t\in \R$ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure $\mu$ with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.