Attractors, time-averaged measures and statistical solutions for 2D nonautonomous Navier-Stokes equations

Using time-averages and Banach generalized limits we construct a family of probability measures on the pullback attractor of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures are supported on the corresponding sections of the attractor for all times and satisfy the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.