| Abstract: |
| In this talk we establish two results: One is the strong Feller property for the Markov semigroups associated to the two or three dimensional Navier-Stokes equations driven by space-time white noise using the theory of regularity structures introduced by Martin Hairer in \cite{Hai14}. In the 2D case this implies ergodicity and global well-posedness of the Navier-Stokes equations driven by space-time white noise starting from every initial point in $(\mathcal{C}^\eta)^2$ for $\eta\in (-\kappa,0)$ for $\kappa$ small enough.
The other is the existence and uniqueness of the global solutions to the stochastic Navier-Stokes equations in 3D case for the small initial data independent of time, with the stochastic integration being understood in the sense of the integration of controlled rough path which can be viewed as a stochastic version of the Kato-Fujita result. |
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