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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: 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 $ B$ is nonempty, compact, strictly invariant and attracts the set $ B$. We establish that the $ 2$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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Additional Information

Zdzislaw Brzezniak
Affiliation: Department of Mathematics, The University of Hull, Hull, HU6 7RX, United Kingdom
Address at time of publication: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Email: zb500@york.ac.uk

Yuhong Li
Affiliation: Department of Mathematics, The University of Hull, Hull, HU6 7RX, United Kingdom
Address at time of publication: School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
Email: chuchuemma@163.com

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03923-7
PII: S 0002-9947(06)03923-7
Keywords: Stochastic Navier-Stokes equations, unbounded domains, cylindrical white noise, asymptotic compactness, random dynamic systems, absorbing sets
Received by editor(s): June 6, 2004
Received by editor(s) in revised form: December 8, 2004
Published electronically: July 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.