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May  2022, 21(5): 1621-1636. doi: 10.3934/cpaa.2022034

Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains

Faculty of Sciences, Beijing University of Technology, PingLeYuan 100, Chaoyang District, Beijing 100124, China

* Corresponding author

Received  December 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: The work is supported by National Natural Science Foundation of China (Nos.11601021, 11831003, 11771031, and 12171111), the Science and Technology Project of Beijing Municipal Education Commission(No.KZ202110005011), and Project for University Key Young Teacher by Education of Henan Province (No.2021GGJS158)

In this paper, we study the asymptotic behavior of the non-autonomous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.

Citation: Shu Wang, Mengmeng Si, Rong Yang. Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1621-1636. doi: 10.3934/cpaa.2022034
References:
[1]

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

[2]

J. M. Ball, Global attractor for damped semilinear wave equations, Discrete Contin. Dyna. Syst., 6 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[4]

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

[5]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[6]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Rel., 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[7]

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

[8]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, Comptes Rendus Mathematique, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

T. CaraballoP. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[10]

J. R. Kang and J. Y. Park, Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay, Acta Math. Sin., 29 (2013), 993-1006.  doi: 10.1007/s10114-013-1392-0.

[11]

K. Kinra and M. T. Mohan, Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise, arXiv: 2105.13770.

[12]

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

[13]

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

[14]

X. Song and Y. Hou, Uniform attractors for a non-autonomous Brinkman-Forchheimer equation, J. Math. Res. Appl., 32(1) (2012), 63-75.  doi: 10.3770/j.issn:2095-2651.2012.01.008.

[15]

X. Song, Pullback $\mathcal{D}$-attractors for a non-autonomous Brinkman-Forcheimer system, J. Math. Res. Appl., 33 (2013), 90-100.  doi: 10.3770/j.issn:2095-2651.2013.01.010.

[16]

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

[17]

D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 68 (2008), 1986-1992.  doi: 10.1016/j.na.2007.01.025.

[18]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Methods Appl. Sci., 31 (2008), 1479-1495.  doi: 10.1002/mma.985.

[19]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electron. J. Differ. Equ., 59 (2012), 1-18.  doi: 10.1016/j.jmaa.2011.11.022.

[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[22]

X. G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1396-1418.  doi: 10.3934/era.2020074.

show all references

References:
[1]

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

[2]

J. M. Ball, Global attractor for damped semilinear wave equations, Discrete Contin. Dyna. Syst., 6 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[4]

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

[5]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[6]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Rel., 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[7]

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

[8]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, Comptes Rendus Mathematique, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

T. CaraballoP. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[10]

J. R. Kang and J. Y. Park, Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay, Acta Math. Sin., 29 (2013), 993-1006.  doi: 10.1007/s10114-013-1392-0.

[11]

K. Kinra and M. T. Mohan, Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise, arXiv: 2105.13770.

[12]

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

[13]

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

[14]

X. Song and Y. Hou, Uniform attractors for a non-autonomous Brinkman-Forchheimer equation, J. Math. Res. Appl., 32(1) (2012), 63-75.  doi: 10.3770/j.issn:2095-2651.2012.01.008.

[15]

X. Song, Pullback $\mathcal{D}$-attractors for a non-autonomous Brinkman-Forcheimer system, J. Math. Res. Appl., 33 (2013), 90-100.  doi: 10.3770/j.issn:2095-2651.2013.01.010.

[16]

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

[17]

D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 68 (2008), 1986-1992.  doi: 10.1016/j.na.2007.01.025.

[18]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Methods Appl. Sci., 31 (2008), 1479-1495.  doi: 10.1002/mma.985.

[19]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electron. J. Differ. Equ., 59 (2012), 1-18.  doi: 10.1016/j.jmaa.2011.11.022.

[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[22]

X. G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1396-1418.  doi: 10.3934/era.2020074.

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