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ISSN 1088-6850(e) ISSN 0002-9947(p)

Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains

Authors: Zdzislaw Brzezniak and Yuhong Li
Journal: Trans. Amer. Math. Soc. 358 (2006), 5587-5629
MSC (2000): Primary 60H15, 35R60; Secondary 37H10, 34F05
Posted: July 24, 2006
MathSciNet review: 2238928
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Abstract: We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the $\Omega$ -limit set $\Omega_B(\omega)$ of any bounded set is nonempty, compact, strictly invariant and attracts the set . We establish that the D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space $\mathrm{H}$ . As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.

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