This chapter is mostly devoted to the 3D Navier-Stokes equations with random perturbations. We begin with the problem in thin domains and state a result on the convergence of the unique stationary distribution to a unique measure which is invariant under the flow of the limiting 2D Navier-Stokes system. We next turn to the 3D problem in an arbitrary bounded domain or a torus. We describe two different approaches for constructing Markov processes whose trajectories are concentrated on weak solutions of the Navier-Stokes system and investigate the large-time asymptotics of their trajectories. Finally, we discuss some qualitative properties of solutions in the case of perturbations of low dimension. Almost all the results of this chapter are presented without proofs.

3D Navier-Stokes system in thin domains

In this section, we present a result that justifies the study of 2D Navier-Stokes equations in the context of hydrodynamical turbulence. Namely, we study the 3D Navier-Stokes system in a thin domain and prove that, roughly speaking, if the domain is sufficiently thin, then the problem in question has a unique stationary measure, which attracts exponentially all solutions in a large ball and converges to a limiting measure invariant under the 2D dynamics. Moreover, when the width of the domain shrinks to zero, the law of a 3D solution converges to that of a 2D solution uniformly in time. The accurate formulation of these results requires some preliminaries from the theory of Navier-Stokes equations in thin domains. They are discussed in the first subsection. We next turn to the large-time asymptotics of solutions and the limiting behaviour of stationary measures and solutions.