## Ergodicity for the 3D stochastic Navier-Stokes equations.(English)Zbl 1109.60047

Summary: We consider the Kolmogorov equation associated with the stochastic Navier-Stokes equations in 3D and 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.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q30 Navier-Stokes equations 35R60 PDEs with randomness, stochastic partial differential equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76M35 Stochastic analysis applied to problems in fluid mechanics
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