The dynamics of the three-dimensional Navier-Stokes equations

The analysis of the asymptotic behavior of linear and nonlinear stochastic partial differential equations (SPDEs) is well developed and fruitful. There is one important case, the Navier-Stokes equations in three dimensions, that remains essentially open. ``The stochastic approach is not the real source of difficulties in the study of such equations. Well-posedness of the deterministic equations is a major open problem (see, for example, [C. L. Fefferman, ``Existence and smoothness of the Navier-Stokes equations'', available at \url{www.claymath.org/millennium/Navier-Stokes_Equations}], where the problem is introduced in relation to the Millennium prizes announced by the Clay Institute). In this review we will focus mainly on the ergodicity of the stochastic equations (Section 21.2) and on the existence of the global attractor (Sections 21.3 and 21.4). In both cases, the analysis faces a main open problem, since the very beginning. No natural way is known in order to define the principal objects, such as dynamical systems, flows, invariant measures, and attractors, that are the subject of the study. We shall see how the selected authors have dealt with these difficulties. Every section has its own short introduction to the problem, and we refer to those for the understanding of each single subject. The review given here is by no means complete. There are several other possible ideas that can be applied to the problem, such as the statistical approach in [M. I. Višik and A. V. Fursikov, Mathematical problems of statistical hydromechanics (Russian), ``Nauka'', Moscow, 1980; MR0591678 (82g:35095); English translation, Kluwer, Dordrecht, 1988; Zbl 0688.35077], the abstract limit approach of Foiaş, the set-valued trajectories in [A. V. Babin and M. I. Vishik, Attractors of evolution equations, Translated and revised from the 1989 Russian original by Babin, North-Holland, Amsterdam, 1992; MR1156492 (93d:58090)], or the nonstandard analysis approach of M. Capiński and N. J. Cutland [Proc. Roy. Soc. London Ser. A 453 (1997), no. 1966, 2413--2426; MR1480123 (98i:35140)], and many others. Lecture Notes in Pure and Applied Mathematics Volume: 245