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 \μₜ∈ on the pullback attractor (t)∈ of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures satisfy suppμₜ ⊂ A(t) for all t∈ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure μ with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.