Recent results on mathematical and statistical hydrodynamics

This paper is a survey of recent results of the authors and their collaborators on stochastic partial differential equations in hydrodynamics. We discuss the stochastic Burgers equation, the stochastic Navier-Stokes equation, and the stochastic passive scalar transport equation. In contrast to previous publications on this subject (see, for example, [25], which is mainly devoted to existence problems for stochastic dynamics), the work surveyed here emphasizes qualitative properties of solutions, including the existence and uniqueness of an invariant measure under certain physical assumptions, the asymptotic behaviour of the statistics of these solutions, and so on. We also discuss new investigations concerning the deterministic Navier-Stokes equation.