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Saddle Singularities in Integrable Hamiltonian Systems: Examples and Algorithms Read chapter Read first chapter

2021 | OriginalPaper | Chapter

10. Convergence Rate of Random Attractors for 2D Navier–Stokes Equation Towards the Deterministic Singleton Attractor

Authors : Hongyong Cui, Peter E. Kloeden

Published in: Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics

Publisher: Springer International Publishing

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Abstract

In this paper we study the long-time behavior of a 2D Navier–Stokes equation. It is shown that under small forcing intensity the global attractor of the equation is a singleton. When endowed with additive or multiplicative white noise no sufficient evidence was found that the random attractor keeps the singleton structure, but the estimate of the convergence rate of the random attractor towards the deterministic singleton attractor as stochastic perturbation vanishes is obtained.

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previous chapter Mixed Dirichlet-Transmission Problems in Non-smooth Domains
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Literature
1.
2.
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland (1992)
3.
Bates, P.W., Lu, K., Wang, B.: Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009)MathSciNetCrossRef
4.
Caraballo, T., Crauel, H., Langa, J.A., Robinson, J.C.: The effect of noise on the “chafee-infante” equation: a nonlinear case study. Proc. Am. Math. Soc. 135, 373–382 (2007)MathSciNetCrossRef
5.
Caraballo, T., Langa, J.A.: On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn. Contin. Discret. Impuls. Syst. Ser. A 10, 491–514 (2003)MathSciNetMATH
6.
Caraballo, T., Langa, J.A., Robinson, J.C.: Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun. Partial Differ. Equ. 23, 1557–1581 (1998)MathSciNetCrossRef
7.
Carvalho, A.N., Langa, J.A.: Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds. J. Differ. Equ. 233, 622–653 (2007)MathSciNetCrossRef
8.
Carvalho, A.N., Langa, J.A., Robinson, J.C.: On the continuity of pullback attractors for evolution processes. Nonlinear Anal.: Theory Methods Appl. 71, 1812–1824 (2009)MathSciNetCrossRef
9.
Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, vol. 182. Springer, Berlin (2013)
10.
Carvalho, A.N., Langa, J.A., Robinson, J.C., Suárez, A.: Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system. J. Differ. Equ. 236, 570–603 (2007)MathSciNetCrossRef
11.
Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics, vol. 49. American Mathematical Society, Providence (2002)MATH
12.
Chueshov, I.: Monotone Random Systems Theory and Applications, vol. 1779, Springer Science & Business Media, Berlin (2002)
13.
Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)MathSciNetCrossRef
14.
Cui, H., Freitas, M.M., Langa, J.A.: On random cocycle attractors with autonomous attraction universes. Discret. Contin. Dyn. Syst. - Ser. B 22, 3379–3407 (2017)MathSciNetMATH
15.
Cui, H., Langa, J.A.: Uniform attractors for non-autonomous random dynamical systems. J. Differ. Equ. 263, 1225–1268 (2017)MathSciNetCrossRef
16.
Fan, X.: Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise. Stoch. Anal. Appl. 24, 767–793 (2006)MathSciNetCrossRef
17.
Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stoch. Stoch. Rep. 59, 21–45 (1996)CrossRef
18.
Foias, C., Manley, O., Temam, R., Treve, Y.: Asymptotic analysis of the Navier-Stokes equations. Phys. D: Nonlinear Phenom. 9, 157–188 (1983)MathSciNetCrossRef
19.
Hale, J.K., Raugel, G.: Lower semicontinuity of attractors of gradient systems and applications. Ann. Mat. Pura ed Appl. 154, 281–326 (1989)MathSciNetCrossRef
20.
Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems, vol. 176. American Mathematical Society, Providence (2011)CrossRef
21.
Langa, J.A., Robinson, J.C., Suárez, A., Vidal-López, A.: The stability of attractors for non-autonomous perturbations of gradient-like systems. J. Differ. Equ. 234, 607–625 (2007)MathSciNetCrossRef
22.
Li, D., Kloeden, P.: Equi-attraction and the continuous dependence of attractors on parameters. Glasg. Math. J. 46, 131–141 (2004)MathSciNetCrossRef
23.
Li, D., Kloeden, P.: Equi-attraction and the continuous dependence of pullback attractors on parameters. Stoch. Dyn. 4, 373–384 (2004)MathSciNetCrossRef
24.
Li, D., Kloeden, P.: Equi-attraction and continuous dependence of strong attractors of set-valued dynamical systems on parameters. Set-Valued Anal. 13, 405–416 (2005)MathSciNetCrossRef
25.
Li, Y., Gu, A., Li, J.: Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. J. Differ. Equ. 258, 504–534 (2015)MathSciNetCrossRef
26.
Ochs, G.: Weak Random Attractors, Citeseer (1999)
27.
Robinson, J.C.: Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28. Cambridge University Press, Cambridge (2001)
28.
Robinson, J.C.: Stability of random attractors under perturbation and approximation. J. Differ. Equ. 186, 652–669 (2002)MathSciNetCrossRef
29.
Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)CrossRef
30.
Wang, B.: Upper semicontinuity of random attractors for non-compact random dynamical systems. Electron. J. Differ. Equ. 2009, 1–18 (2009)MathSciNet
Metadata
Title
Convergence Rate of Random Attractors for 2D Navier–Stokes Equation Towards the Deterministic Singleton Attractor
Authors
Hongyong Cui
Peter E. Kloeden
Copyright Year
2021
Book
Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics

Print ISBN: 978-3-030-50301-7

Electronic ISBN: 978-3-030-50302-4

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-50302-4_10

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