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Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. (English) Zbl 1024.76012
The non-stationary incompressible Navier-Stokes problem in two spatial dimensions with periodic boundary conditions is considered in vorticity formulation. The authors study Galerkin approximations for the case of a degenerate large-scale stochastic force. So, the right-hand side incorporates derivatives of independent Wiener processes. The aim is the verification of assumption of ergodicity that serves as a basic assumption in the theory of turbulence. The minimal sufficient condition for existence of a unique invariant measure is proved by establishing a priori bounds on the enstrophy, and by compactness arguments. In an appendix, the exponential convergence towards the unique invariant measure is shown for finite-dimensional diffusion processes that satisfy the hypoellipticity condition. The results remain true even if an additional frictional damping occurs.

MSC:
76D06 Statistical solutions of Navier-Stokes and related equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37H99 Random dynamical systems
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
60J60 Diffusion processes
76D05 Navier-Stokes equations for incompressible viscous fluids
76M35 Stochastic analysis applied to problems in fluid mechanics
35Q30 Navier-Stokes equations
76F55 Statistical turbulence modeling
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