Ergodicity for the 3D stochastic Navier-Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier-Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u(m) of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier-Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.