Preliminaries 1.1 Function spaces 1.2 Basic facts from measure theory 1.3 Markov processes and random dynamical systems Notes and comments 2 Two-dimensional Navier-Stokes equations 2.1 Cauchy problem for the deterministic system 2.2 Stochastic Navier-Stokes equations 2.3 Navier-Stokes equations perturbed by a random kick force 2.4 Navier-Stokes equations perturbed by spatially regular white noise 2.5 Existence of a stationary distribution 2.6 Appendix: some technical proofs Notes and comments 3 Uniqueness of stationary measure and mixing 3.1 Three results on uniqueness and mixing 3.2 Dissipative RDS with bounded kicks 3.3 Navier-Stokes system perturbed by white noise 3.4 Navier-Stokes system with unbounded kicks 3.5 Further results and generalisations 3.6 Appendix: some technical proofs 3.7 Relevance of the results for physics Notes and comments vii www.cambridge.org

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We formulate two results on controllability properties of the 3D NavierStokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qualitative properties of admissible weak solutions for randomly forced NS equations. AMS subject classifications: 35Q30, 60H15, 76D05, 93B05, 93C20Keywords: Approximate controllability, exact controllability in projections, 3D Navier-Stokes system, Agrachev-Sarychev method, stationary solutions, irreducibility. Main resultsLet D ⊂ R 3 be a bounded domain with C 2 -smooth boundary ∂D. Consider 3D Navier-Stokes (NS) equationṡwhere u = (u 1 , u 2 , u 3 ) and p are unknown velocity and pressure fields, ν > 0 is the viscosity, and f (t, x) is an external force. We introduce the spaceswhere n stands for the outward unit normal to ∂D and ·, · denotes the scalar product in R 3 . It is well known (e.g., see [Tem79]) that H is a closed vector 1