Research output: Contribution to journal › Article

**Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains.** / Brzezniak, Z.; Li, Yuhong.

Research output: Contribution to journal › Article

Brzezniak, Z & Li, Y 2006, 'Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains', *Transactions of the American Mathematical Society*, vol. 358, no. 12, pp. 5587-5629. https://doi.org/10.1090/S0002-9947-06-03923-7

Brzezniak, Z., & Li, Y. (2006). Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains. *Transactions of the American Mathematical Society*, *358*(12), 5587-5629. https://doi.org/10.1090/S0002-9947-06-03923-7

Brzezniak Z, Li Y. Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains. Transactions of the American Mathematical Society. 2006 Jul 24;358(12):5587-5629. https://doi.org/10.1090/S0002-9947-06-03923-7

@article{9c659858f22b4b47b3685ed783179877,

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

abstract = "We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS).We prove that for an AC RDS the O-limit set OB(¿) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an AC RDS in the energy space 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.",

keywords = "stochastic Navier-Stokes equations, unbounded domains, cylindrical white noise, asymptotic compactness, random dynamic systems, absorbing sets, DIFFERENTIAL-EQUATIONS, EVOLUTION-EQUATIONS, INVARIANT-MEASURES, GLOBAL ATTRACTOR, DRIVEN, REGULARITY, DIMENSION, EXISTENCE, SYSTEMS, NOISE",

author = "Z. Brzezniak and Yuhong Li",

year = "2006",

month = "7",

day = "24",

doi = "10.1090/S0002-9947-06-03923-7",

language = "English",

volume = "358",

pages = "5587--5629",

journal = "Transactions of the American Mathematical Society",

issn = "1088-6850",

publisher = "American Mathematical Society",

number = "12",

}

TY - JOUR

T1 - Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains

AU - Brzezniak, Z.

AU - Li, Yuhong

PY - 2006/7/24

Y1 - 2006/7/24

N2 - We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS).We prove that for an AC RDS the O-limit set OB(¿) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an AC RDS in the energy space 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.

AB - We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS).We prove that for an AC RDS the O-limit set OB(¿) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an AC RDS in the energy space 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.

KW - stochastic Navier-Stokes equations

KW - unbounded domains

KW - cylindrical white noise

KW - asymptotic compactness

KW - random dynamic systems

KW - absorbing sets

KW - DIFFERENTIAL-EQUATIONS

KW - EVOLUTION-EQUATIONS

KW - INVARIANT-MEASURES

KW - GLOBAL ATTRACTOR

KW - DRIVEN

KW - REGULARITY

KW - DIMENSION

KW - EXISTENCE

KW - SYSTEMS

KW - NOISE

U2 - 10.1090/S0002-9947-06-03923-7

DO - 10.1090/S0002-9947-06-03923-7

M3 - Article

VL - 358

SP - 5587

EP - 5629

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 1088-6850

IS - 12

ER -