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**Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing.**
*(English)*
Zbl 1130.37038

Summary: The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in \(\L^2_0(\mathbb T^2)\). Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a Hörmander-type condition. This requires some interesting nonadapted stochastic analysis.

### MSC:

37L55 | Infinite-dimensional random dynamical systems; stochastic equations |

37A25 | Ergodicity, mixing, rates of mixing |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

35Q30 | Navier-Stokes equations |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M35 | Stochastic analysis applied to problems in fluid mechanics |