Pullback Attractors and Statistical Solutions

Abstract

This chapter is devoted to constructions of invariant measures and statistical solutions for non-autonomous Navier–Stokes equations in bounded and certain unbounded domains in $\mathbb{R}^{2}$.After introducing some basic notions and results concerning attractors in the context of the Navier–Stokes equations, we construct the family of probability measures $\{\mu _{t}\}_{t\in \mathbb{R}}$ and prove the relations $\mu _{t}(E) =\mu _{\tau }(U(t,\tau )^{-1}E)$ for $t,\tau \in \mathbb{R}$, t ≥ τ and Borel sets E in H. Then we prove the Liouville and energy equations. Finally, we consider statistical solutions of the Navier–Stokes equations supported on the pullback attractor.