Random attractors for stochastic 2D hydrodynamical type systems

Keywords: Random attractors; stochastic 2D hydrodynamical type systems; upper semicontinuity; energy equation arguments

Abstract

We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray $\alpha$-model.
We prove the existence of random attractors for the associated continuous random dynamical systems.
Then we establish the upper semicontinuity of the random attractors as the parameter tends to zero.

References

Arnold, Ludwig. Random dynamical systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+586 pp. ISBN: 3-540-63758-3 MR1723992, DOI 10.1007/978-3-662-12878-7

Anh, Cung The; Bao, Tang Quoc; Thanh, Nguyen Van. Regularity of random attractors for stochastic semilinear degenerate parabolic equations. Electron. J. Differential Equations 2012, No. 207, 22 pp. MR3001693

Anh, Cung The; Da, Nguyen Tien. The exponential behaviour and stabilizability of stochastic 2D hydrodynamical type systems. Stochastics 89 (2017), no. 3-4, 593--618. MR3607743, DOI 10.1080/17442508.2016.1269767

Bai, Lihong; Zhang, Fang-hong. Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains. Results Math. 69 (2016), no. 1-2, 129--160. MR3449359, DOI 10.1007/s00025-015-0505-8

Ball, J. M. Global attractors for damped semilinear wave equations. Partial differential equations and applications. Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31--52. MR2026182, DOI 10.3934/dcds.2004.10.31

Tang, Quoc Bao. Dynamics of stochastic three dimensional Navier-Stokes-Voigt equations on unbounded domains. J. Math. Anal. Appl. 419 (2014), no. 1, 583--605. MR3217168, DOI 10.1016/j.jmaa.2014.05.003

Bates, Peter W.; Lu, Kening; Wang, Bixiang. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differential Equations 246 (2009), no. 2, 845--869. MR2468738, DOI 10.1016/j.jde.2008.05.017

Brzeźniak, Z.; Caraballo, T.; Langa, J. A.; Li, Y.; Łukaszewicz, G.; Real, J. Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains. J. Differential Equations 255 (2013), no. 11, 3897--3919. MR3097241, DOI 10.1016/j.jde.2013.07.043

Caraballo, Tomás; Langa, José A.; Robinson, James C. Upper semicontinuity of attractors for small random perturbations of dynamical systems. Comm. Partial Differential Equations 23 (1998), no. 9-10, 1557--1581. MR1641776, DOI 10.1080/03605309808821394

Caraballo, T.; Łukaszewicz, G.; Real, J. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal. 64 (2006), no. 3, 484--498. MR2191992, DOI 10.1016/j.na.2005.03.111

Chueshov, Igor. Dynamics of quasi-stable dissipative systems. Universitext. Springer, Cham, 2015. xvii+390 pp. ISBN: 978-3-310-22902-7; 978-3-319-22903-4 MR3408002, DOI 10.1007/978-3-319-22903-4

Chueshov, Igor; Millet, Annie. Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61 (2010), no. 3, 379--420. MR2609596, DOI 10.1007/s00245-009-9091-z

Chueshov, Igor; Millet, Annie. Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem. Stoch. Anal. Appl. 29 (2011), no. 4, 570--611. MR2812518, DOI 10.1080/07362994.2011.581081

Crauel, Hans; Debussche, Arnaud; Flandoli, Franco. Random attractors. J. Dynam. Differential Equations 9 (1997), no. 2, 307--341. MR1451294, DOI 10.1007/BF02219225

Crauel, Hans; Flandoli, Franco. Attractors for random dynamical systems. Probab. Theory Related Fields 100 (1994), no. 3, 365--393. MR1305587, DOI 10.1007/BF01193705

Crauel, Hans; Kloeden, Peter E. Nonautonomous and random attractors. Jahresber. Dtsch. Math.-Ver. 117 (2015), no. 3, 173--206. MR3372173, DOI 10.1365/s13291-015-0115-0

Fan, Xiaoming. Random attractors for damped stochastic wave equations with multiplicative noise. Internat. J. Math. 19 (2008), no. 4, 421--437. MR2416723, DOI 10.1142/S0129167X08004741

Flandoli, Franco; Schmalfuss, Björn. Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise. Stochastics Stochastics Rep. 59 (1996), no. 1-2, 21--45. MR1427258, DOI 10.1080/17442509608834083

Guo, Chunxiao; Guo, Boling; Guo, Yanfeng. Random attractors of stochastic non-Newtonian fluids on unbounded domain. Stoch. Dyn. 14 (2014), no. 1, 1350008, 18 pp. MR3159462, DOI 10.1142/S0219493713500081

Li, Yangrong; Guo, Boling. Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. J. Differential Equations 245 (2008), no. 7, 1775--1800. MR2433486, DOI 10.1016/j.jde.2008.06.031

Rosa, Ricardo. The global attractor for the $2$D Navier-Stokes flow on some unbounded domains. Nonlinear Anal. 32 (1998), no. 1, 71--85. MR1491614, DOI 10.1016/S0362-546X(97)00453-7

Temam, Roger. Infinite-dimensional dynamical systems in mechanics and physics. Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. {rm xxii}+648 pp. ISBN: 0-387-94866-X MR1441312, DOI 10.1007/978-1-4612-0645-3

Wang, Bixiang. Upper semicontinuity of random attractors for non-compact random dynamical systems. Electron. J. Differential Equations 2009, No. 139, 18 pp. MR2558811,

Wang, Bixiang. Asymptotic behavior of stochastic wave equations with critical exponents on $Bbb R^3$. Trans. Amer. Math. Soc. 363 (2011), no. 7, 3639--3663. MR2775822, DOI 10.1090/S0002-9947-2011-05247-5

Yang, Meihua; Kloeden, P. E. Random attractors for stochastic semi-linear degenerate parabolic equations. Nonlinear Anal. Real World Appl. 12 (2011), no. 5, 2811--2821. MR2813224, DOI 10.1016/j.nonrwa.2011.04.007

You, Bo; Li, Fang. Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise. Stoch. Anal. Appl. 34 (2016), no. 2, 278--292. MR3462137, DOI =10.1080/07362994.2015.1126184

Zhao, Wenqiang. $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise. Nonlinear Anal. 84 (2013), 61--72. MR3034571, DOI 10.1016/j.na.2013.01.014

Zhou, Shengfan; Yin, Fuqi; Ouyang, Zigen. Random attractor for damped nonlinear wave equations with white noise. SIAM J. Appl. Dyn. Syst. 4 (2005), no. 4, 883--903. MR2179491, DOI 10.1137/050623097

Published
25.12.2019
How to Cite
CungT. A., and Nguyen T. D. “Random Attractors for Stochastic 2D Hydrodynamical Type Systems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 12, Dec. 2019, pp. 1647-66, https://umj.imath.kiev.ua/index.php/umj/article/view/361.
Section
Research articles