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

Anhui Gu; Boling Guo; Bixiang Wang

doi:10.3934/dcdsb.2020020

Article Contents

2020, Volume 25, Issue 7: 2495-2532. Doi: 10.3934/dcdsb.2020020

This issue Previous Article Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise Next Article Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation

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
^* Corresponding author: Anhui Gu

Received: May 31, 2019

Early access: April 2020

Published: July 2020

This work is supported by NSF of Chongqing grant cstc2018jcyjA0897

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[3]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[4]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[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.
[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.
[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.
[8]	P. W. Bates, K. 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.
[9]	W.-J. Beyn, B. Gess, P. 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.
[10]	Z. Brzeźniak, M. Capiński and F. Flandoli, Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1 (1991), 41-59. doi: 10.1142/S0218202591000046.
[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.
[12]	T. Caraballo, G. 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.
[13]	T. Caraballo, G. 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.
[14]	T. Caraballo, M. J. Garrido-Atienza, B. 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.
[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.
[16]	T. Caraballo, M. J. Garrido-Atienza, B. 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.
[17]	T. Caraballo, M. 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.
[18]	T. Caraballo, J. A. Langa, V. 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.
[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.
[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.
[21]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[22]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[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.
[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.
[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.
[26]	M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.
[27]	M. J. Garrido-Atienza, B. 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.
[28]	W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014.
[29]	B. Gess, W. 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.
[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.
[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.
[32]	A. H. Gu, K. 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.
[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.
[34]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
[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.
[36]	P. Häunggi and P. Jung, Colored Noise in Dynamical Systems, Advances in Chemical Physics, Volume 89, John Wiley & Sons, Inc., Hoboken, NJ 1994.
[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.
[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.
[39]	T. Jiang, X. 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.
[40]	N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981.
[41]	D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979.
[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.
[43]	M. M. Klosek-Dygas, B. J. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441. doi: 10.1137/0148023.
[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.
[45]	L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511984730.
[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.
[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.
[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.
[49]	J. Shen, K. 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.
[50]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979
[51]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[52]	G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841.
[53]	B. X. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012.
[54]	B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb R^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[55]	B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[56]	B. X. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electronic J. Differential Equations, 2012 (2012), No. 59, 18 pp.
[57]	B. X. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.
[58]	B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.
[59]	M. C. Wang and G. E. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323.
[60]	E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564. doi: 10.1214/aoms/1177699916.
[61]	E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5.