January  2021, 26(1): 693-716. doi: 10.3934/dcdsb.2020189

Random attractors for 2D stochastic micropolar fluid flows on unbounded domains

1. 

School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China

2. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36832, USA

* Corresponding author: Xiaoying Han

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: This work is partially supported by the National Science Foundation of China (Grant No. 61673006)

The asymptotic behavior of a model for 2D incompressible stochastic micropolar fluid flows with rough noise on a Poincaré domain is investigated. First, the existence and uniqueness of solutions to an evolution equation arising from the underlying stochastic micropolar fluid model is established via the Galerkin method and energy method. Then the existence of a random attractor is studied by using the theory of random dynamical systems for which the noise is dealt with by appropriate reproducing kernel Hilbert space.

Citation: Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189
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Z. BrzeźniakM. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probability Theory and Related Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

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Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar

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Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Łukaszewicz and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919.  doi: 10.1016/j.jde.2013.07.043.  Google Scholar

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T. Caraballo, The long-time behaviour of stochastic 2D-Navier-Stokes equations, Probabilistic Methods in Fluids, (2003), 70-83.  doi: 10.1142/9789812703989_0005.  Google Scholar

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T. Caraballo and X. Han, Applied Nonautonomous and Dynamical Systems, SpringerBriefs in Mathematics, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-49247-6.  Google Scholar

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T. Caraballo and J. A. 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

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T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 10 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

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T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

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

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580, Springer, Berlin.  Google Scholar

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J. ChenZ. Chen and B. Dong, Existence of $H^2$-global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl., 322 (2006), 512-522.  doi: 10.1016/j.jmaa.2005.09.011.  Google Scholar

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J. ChenB. Dong and Z. Chen, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619-1635.  doi: 10.1088/0951-7715/20/7/005.  Google Scholar

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H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11, Taylor & Francis, London, 2002.  Google Scholar

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B. Dong and Z. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comp., 182 (2006), 610-620.  doi: 10.1016/j.amc.2006.04.024.  Google Scholar

[21]

B. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[22]

B. DongJ. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.  Google Scholar

[23]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

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

[25]

M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Analysis, 64 (2006), 1100-1118.  doi: 10.1016/j.na.2005.05.057.  Google Scholar

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X. Han and P. E. Kloeden, Random Ordinary Differential Equations and their Numerical Solutions, Springer Nature, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.  Google Scholar

[27]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[28]

P. E. Kloeden and B. Schmalfuss, Asymptotic behavior of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280.  doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar

[29]

J.-L. Lions and E. Magenes, Non-Homegeneous Boundedary Value Problem and Applications, Spring-Verlag, Berlin, Heidelberg, New York, 1972.  Google Scholar

[30]

L. Liu and T. Caraballo, Analysis of a stochastic 2D-Navier-Stokes model with infinite delay, J. Dyn. Diff. Eqns., 31 (2019), 2249-2274.  doi: 10.1007/s10884-018-9703-x.  Google Scholar

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G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

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G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Modelling, 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.  Google Scholar

[33]

G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Analysis, 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.  Google Scholar

[34]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[35]

B. Schmaluss, Attractors for non-autonomous dynamical systems, International Conference on Differential Equations, 99 (2000), 684-689.   Google Scholar

[36]

W. Sun, Micropolar fluid flows with delay on 2D unbounded domains, Journal of Applied Analysis and Computation, 8 (2018), 356-378.  doi: 10.11948/2018.356.  Google Scholar

[37]

W. Sun and Y. Li, Asymptotic behavior of pullback attractors for non-autonomous micropolar fluid flows in 2D unbounded domains, Electronic Journal of Differential Equations, 2018 (2018), 1-21.   Google Scholar

[38]

R. Temam, Navier-Stokes Equations, North-Holland Publish Company, Amsterdam, 1979.  Google Scholar

[39]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disc. Cont. Dyn. Sys., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[40]

L. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777.  doi: 10.1002/mma.1491.  Google Scholar

[41]

C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032.  Google Scholar

[42]

C. ZhaoS. Zhou and X. Lian, $H^1$-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains, Nonlinear Anal.-RWA, 9 (2008), 608-627.  doi: 10.1016/j.nonrwa.2006.12.005.  Google Scholar

[43]

C. ZhaoW. Sun and C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dynamics of Partial Differential Equations, 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.  Google Scholar

[44]

C. Zhao and W. Sun, Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays, Commun. Math. Sci., 15 (2017), 97-121.  doi: 10.4310/CMS.2017.v15.n1.a5.  Google Scholar

[45]

Caidi ZhaoYanjiao Li and Tomás Caraballo, Trajectory rajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011.  Google Scholar

[46]

Caidi ZhaoYanjiao Li and Yanmiao Sang, Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows, Z. Angew. Math. Mech., 100 (2020), e201800197.  doi: 10.1002/zamm.201800197.  Google Scholar

show all references

References:
[1]

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

[2]

Z. BrzeźniakM. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probability Theory and Related Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[3]

Z. Brzeźniak, On Sobolev and Besov spaces regularity of Brownian paths, Stochastics and Stochastics Reports, 56 (1996), 1-15.  doi: 10.1080/17442509608834032.  Google Scholar

[4]

Z. Brzeźniak and S. Peszat, Stochastic two dimensional Euler equations, Ann. Probab., 29 (2001), 1796-1832.  doi: 10.1214/aop/1015345773.  Google Scholar

[5]

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

[6]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar

[7]

Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Łukaszewicz and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919.  doi: 10.1016/j.jde.2013.07.043.  Google Scholar

[8]

T. Caraballo, The long-time behaviour of stochastic 2D-Navier-Stokes equations, Probabilistic Methods in Fluids, (2003), 70-83.  doi: 10.1142/9789812703989_0005.  Google Scholar

[9]

T. Caraballo and X. Han, Applied Nonautonomous and Dynamical Systems, SpringerBriefs in Mathematics, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-49247-6.  Google Scholar

[10]

T. Caraballo and J. A. 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

[11]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 10 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[12]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

[13]

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

[14]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580, Springer, Berlin.  Google Scholar

[15]

J. ChenZ. Chen and B. Dong, Existence of $H^2$-global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl., 322 (2006), 512-522.  doi: 10.1016/j.jmaa.2005.09.011.  Google Scholar

[16]

J. ChenB. Dong and Z. Chen, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619-1635.  doi: 10.1088/0951-7715/20/7/005.  Google Scholar

[17]

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

[18]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11, Taylor & Francis, London, 2002.  Google Scholar

[19] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional System, Cambridge University Press, Cambridge, 1966.  doi: 10.1017/CBO9780511662829.  Google Scholar
[20]

B. Dong and Z. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comp., 182 (2006), 610-620.  doi: 10.1016/j.amc.2006.04.024.  Google Scholar

[21]

B. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[22]

B. DongJ. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.  Google Scholar

[23]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[24]

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

[25]

M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Analysis, 64 (2006), 1100-1118.  doi: 10.1016/j.na.2005.05.057.  Google Scholar

[26]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and their Numerical Solutions, Springer Nature, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.  Google Scholar

[27]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[28]

P. E. Kloeden and B. Schmalfuss, Asymptotic behavior of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280.  doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar

[29]

J.-L. Lions and E. Magenes, Non-Homegeneous Boundedary Value Problem and Applications, Spring-Verlag, Berlin, Heidelberg, New York, 1972.  Google Scholar

[30]

L. Liu and T. Caraballo, Analysis of a stochastic 2D-Navier-Stokes model with infinite delay, J. Dyn. Diff. Eqns., 31 (2019), 2249-2274.  doi: 10.1007/s10884-018-9703-x.  Google Scholar

[31]

G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[32]

G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Modelling, 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.  Google Scholar

[33]

G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Analysis, 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.  Google Scholar

[34]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[35]

B. Schmaluss, Attractors for non-autonomous dynamical systems, International Conference on Differential Equations, 99 (2000), 684-689.   Google Scholar

[36]

W. Sun, Micropolar fluid flows with delay on 2D unbounded domains, Journal of Applied Analysis and Computation, 8 (2018), 356-378.  doi: 10.11948/2018.356.  Google Scholar

[37]

W. Sun and Y. Li, Asymptotic behavior of pullback attractors for non-autonomous micropolar fluid flows in 2D unbounded domains, Electronic Journal of Differential Equations, 2018 (2018), 1-21.   Google Scholar

[38]

R. Temam, Navier-Stokes Equations, North-Holland Publish Company, Amsterdam, 1979.  Google Scholar

[39]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disc. Cont. Dyn. Sys., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[40]

L. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777.  doi: 10.1002/mma.1491.  Google Scholar

[41]

C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032.  Google Scholar

[42]

C. ZhaoS. Zhou and X. Lian, $H^1$-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains, Nonlinear Anal.-RWA, 9 (2008), 608-627.  doi: 10.1016/j.nonrwa.2006.12.005.  Google Scholar

[43]

C. ZhaoW. Sun and C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dynamics of Partial Differential Equations, 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.  Google Scholar

[44]

C. Zhao and W. Sun, Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays, Commun. Math. Sci., 15 (2017), 97-121.  doi: 10.4310/CMS.2017.v15.n1.a5.  Google Scholar

[45]

Caidi ZhaoYanjiao Li and Tomás Caraballo, Trajectory rajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011.  Google Scholar

[46]

Caidi ZhaoYanjiao Li and Yanmiao Sang, Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows, Z. Angew. Math. Mech., 100 (2020), e201800197.  doi: 10.1002/zamm.201800197.  Google Scholar

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