doi: 10.3934/dcdsb.2020020

Long term behavior of random Navier-Stokes equations driven by colored noise

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Anhui Gu

Received  June 2019 Published  December 2019

Fund Project: This work is supported by NSF of Chongqing grant cstc2018jcyjA0897.

This paper is devoted to the study of long term behavior of the two-dimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.

Citation: Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020020
References:
[1]

P. Acquistapace and B. Terreni, An approach to Itö linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

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J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

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J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

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M. J. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.  Google Scholar

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B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar

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A. H. GuK. N. Lu and B. X. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

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A. H. Gu and B. X. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

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show all references

References:
[1]

P. Acquistapace and B. Terreni, An approach to Itö linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

[2]

V. S. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Second edition, Springer Series in Synergetics, Springer, Berlin, 2007.  Google Scholar

[3]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[4]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[8]

P. W. BatesK. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[9]

W.-J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.  Google Scholar

[10]

Z. BrzeźniakM. Capiński and F. Flandoli, Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1 (1991), 41-59.  doi: 10.1142/S0218202591000046.  Google Scholar

[11]

M. Capiński and N. J. Cutland, Existence of global stochastic flow and attractors for Navier-Stokes equations, Probab. Theory Relat. Fields, 115 (1999), 121-151.  doi: 10.1007/s004400050238.  Google Scholar

[12]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[13]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2$D$-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

[14]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[15]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.   Google Scholar

[16]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[17]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar

[18]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[19]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[20]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.  Google Scholar

[21]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[22]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[23]

J. Q. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar

[24]

F. Flandoli and B. Schmalfuss, Random attractors for the 3$D$ stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[25]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[26]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

[27]

M. J. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.  Google Scholar

[28] W. GerstnerW. M. KistlerR. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014.   Google Scholar
[29]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[30]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Eqns., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar

[31]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar

[32]

A. H. GuK. N. Lu and B. X. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[33]

A. H. Gu and B. X. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[34]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar

[35]

P. Hänggi, Colored noise in dynamical systems: A functional calculus approach, Noise in Nonlinear Dynamical Systems, Cambridge University Press, 1 (1989), 307328, 9 pp. Google Scholar

[36]

P. Häunggi and P. Jung, Colored Noise in Dynamical Systems, Advances in Chemical Physics, Volume 89, John Wiley & Sons, Inc., Hoboken, NJ 1994. Google Scholar

[37]

J. H. Huang and W. X. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

[38]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[39]

T. JiangX. M. Liu and J. Q. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[40]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981.  Google Scholar

[41]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.  Google Scholar

[42]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[43]

M. M. Klosek-DygasB. J. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441.  doi: 10.1137/0148023.  Google Scholar

[44]

K. N. Lu and B. X. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2017), 1341–1371, https://doi.org/10.1007/s10884-017-9626-y. doi: 10.1007/s10884-017-9626-y.  Google Scholar

[45] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511984730.  Google Scholar
[46]

R. Rosa, The global attractor for the 2$D$ Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[47]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185–192. Google Scholar

[48]

G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[49]

J. ShenK. N. Lu and W. N. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.  Google Scholar

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