Section: New Results

Ergodicity results for the stochastic Navier-Stokes equations: an introduction

In the chapter [36] , we give an overview of the results on ergodicity for the stochastic Navier-Stokes equations. We first explain the basis on SPDEs and on the concept of invariant measures and ergodicity. Then, in the 2D case, we introduce progressively the various methods, finishing with a celebrated result due to M. Hairer and J. Mattingly on ergodicity with very degenerated noises. In the 3D case, the theory is much less complete. Nonetheles, we show that it is possible to construct Markov evolutions and, under some non degenary assumptions on the noise, to obtain ergodicity.