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A Siegel theorem for dynamical systems under random perturbations
Pullback attractors and statistical solutions for 2D NavierStokes equations
1.  University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02097 Warsaw, Poland 
Using timeaverages and Banach generalized limits we construct a family of probability measures $\{\mu_t\}_{t\in \IR}$ on the pullback attractor $\{A(t)\}_{t\in \R}$ of the dynamical system associated with a twodimensional nonautonomous NavierStokes flow in a bounded domain. The measures satisfy supp$\mu_t \subset A(t)$ for all $t\in \R$ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some timeaverage measure $\mu$ with support included in the global attractor and being a stationary statistical solution of the NavierStokes flow.
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