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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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