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Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. (English) Zbl 1196.60118
The authors study a stochastic tamed 3D Navier-Stokes equation, where an additional nonlinear term of the type \(g(|u|^2)u\) is added. The equation is considered in the whole space and for periodic boundary conditions with quite general multiplicative and additive noise.
The first main result is the existence of a unique strong solution. Then the Feller property is studied and in the case of periodic boundary conditions the existence of invariant measures for the corresponding Feller semigroup is verified.
The uniqueness of these invariant measures in the case of periodic boundary conditions and degenerated additive noise is proved, using the notion of asymptotic strong Feller property introduced by M. Hairer and J. C. Mattingly [Ann. Math. (2) 164, No. 3, 993–1032 (2006; Zbl 1130.37038)].

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37A25 Ergodicity, mixing, rates of mixing
37L55 Infinite-dimensional random dynamical systems; stochastic equations
35R60 PDEs with randomness, stochastic partial differential equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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