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

By Zdzisław Brzeźniak, Yuhong Li. 2006

154 citations recorded by the source corpus.

We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the Ω-limit set Ω B (ω) 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 Poincare 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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/46ea1074a3307ded8657ac787bd035f5e6b58790

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In Computational Theory and Mathematics

1. Michael Röckner
2. Vladimir Bogachev
3. Bixiang Wang
4. Anhui Gu
5. Nicolai V. Krylov
6. Yangrong Li
7. El.zbieta Motyl
8. Jia Li

In Control and Systems Engineering

1. Bixiang Wang
2. Anhui Gu
3. El.zbieta Motyl
4. Michael Röckner
5. Yangrong Li
6. Huaizhong Zhao
7. Jia Li
8. Vladimir Bogachev

In Finance

1. Michael Röckner
2. El.zbieta Motyl
3. Nicolai V. Krylov
4. Bixiang Wang
5. Zdzisław Brzeźniak
6. Vladimir Bogachev
7. Anhui Gu
8. Huaizhong Zhao

Citations in the corpus

• Elliptic and parabolic equations for measures (2009, background): “, the book [67]), especially connected with stochastic partial differential equations (see [11], [52], [57], [58], [62], [63], [70], [74], [79], [82], [90], [91], [128], [150], [163]), infinite gradient systems, Gibbs measures, and stochastic quantization (see [5], [6], [56], [83], [85], [86], [96],”
• Random attractors for a class of stochastic partial differential equations driven by general additive noise (2010, unknown): “46 41 v2 [ m at AMS Subject Classification: 35B41, 60H15, 37L30, 35B40 Keywords: Random attractor; Lévy noise; fractional Brownian motion; stochastic evolution equations; porous media equations; p-Laplace equation; reaction-diffusion equations.”
• Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains (2012, unknown): “[8, 11, 12] and [9].”
• Asymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R^3 (2008, background): “The asymptotic compactness of the stochastic Navier-Stokes equations on unbounded domains was proved in [8].”
• Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains (2012, background): “Our results generalize the corresponding existence results from [15], [9] [21], and [31].”
• Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients (2006, background): “In this connection, we mention that in recent years, substantial results on the existence and uniqueness of invariant measures for SPDEs and weak convergence of the law of the solutions as time tends to infinity have been proved for many important SPDEs ([5], [6], [8], [12], [13] to name but a few).”
• Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations (2015, background): “This concept generalizes some common concepts of random attractors (see [8,14,15,17]) and tempered random attractors (see [3–5,7,30,31,34]).”
• Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$ (2008, background): “The asymptotic compactness of the stochastic Navier-Stokes equations on unbounded domains was proved in [8].”