American Institute of Mathematical Sciences

Advanced
October  2012, 32(10): 3787-3800. doi: 10.3934/dcds.2012.32.3787

The existence of uniform attractors for 3D Brinkman-Forchheimer equations

Yuncheng You 1, , Caidi Zhao 2, and  Shengfan Zhou 3,

1. 

Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China
2. 

Department of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
3. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Received  April 2011 Revised  April 2012 Published  May 2012

The longtime dynamics of the three dimensional (3D) Brinkman-Forchheimer equations with time-dependent forcing term is investigated. It is proved that there exists a uniform attractor for this nonautonomous 3D Brinkman-Forchheimer equations in the space $\mathbb{H}^1(\Omega)$. When the Darcy coefficient $\alpha$ is properly large and $L^2_b$-norm of the forcing term is properly small, it is shown that there exists a unique bounded and asymptotically stable solution with interesting corollaries.
Keywords: uniform attractor., Brinkman-Forchheimer equation, asymptotic dynamics.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35Q35; Secondary: 46E25, 20C2.
Citation: Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar

[2]

O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations,, Applied Mathematics Letters, 19 (2006), 801.  doi: 10.1016/j.aml.2005.11.002.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", AMS Colloquium Publications, 49 (2002).   Google Scholar

[4]

M. Firdaouss, J.-L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers,, Journal of Fluid Mechanics, 343 (1997), 331.  doi: 10.1017/S0022112097005843.  Google Scholar

[5]

F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3195.  doi: 10.1098/rspa.2003.1169.  Google Scholar

[6]

T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions,, Transport in Porous Media, 29 (1997), 191.  doi: 10.1023/A:1006533931383.  Google Scholar

[7]

N. Ju, Existence of global attractor for the three-dimensional modified Navier-Stokes equations,, Nonlinearity, 14 (2001), 777.  doi: 10.1088/0951-7715/14/4/306.  Google Scholar

[8]

V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities,, preprint, (2011).   Google Scholar

[9]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196.  doi: 10.1016/j.jde.2006.07.009.  Google Scholar

[10]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Disc. Cont. Dyn. Syst., 13 (2005), 701.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[11]

Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations,, Nonlinear Analysis, 70 (2009), 2054.  doi: 10.1016/j.na.2008.02.121.  Google Scholar

[12]

L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2173.  doi: 10.1098/rspa.1999.0398.  Google Scholar

[13]

L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations,, Studies in Applied Mathematics, 102 (1999), 419.  doi: 10.1111/1467-9590.00116.  Google Scholar

[14]

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

[15]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar

[16]

A. Shenoy, Non-Newtonian fluid heat transfer in porous media,, Adv. Heat transfer, 24 (1994), 101.  doi: 10.1016/S0065-2717(08)70233-8.  Google Scholar

[17]

B. Straughan, "Stability and Wave Motion in Porous Media,", Applied Mathematical Sciences, 165 (2008).   Google Scholar

[18]

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

[19]

S. Whitaker, The Forchheimer equation: A theoretical development,, Transport in Porous Media, 25 (1996), 27.  doi: 10.1007/BF00141261.  Google Scholar

[20]

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

[21]

C. Zhao and S. Zhou, L$^2$-compact uniform attractors for a non-autonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2709845.  Google Scholar

[22]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid,, J. Differential Equations, 238 (2007), 394.  doi: 10.1016/j.jde.2007.04.001.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar

[2]

O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations,, Applied Mathematics Letters, 19 (2006), 801.  doi: 10.1016/j.aml.2005.11.002.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", AMS Colloquium Publications, 49 (2002).   Google Scholar

[4]

M. Firdaouss, J.-L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers,, Journal of Fluid Mechanics, 343 (1997), 331.  doi: 10.1017/S0022112097005843.  Google Scholar

[5]

F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3195.  doi: 10.1098/rspa.2003.1169.  Google Scholar

[6]

T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions,, Transport in Porous Media, 29 (1997), 191.  doi: 10.1023/A:1006533931383.  Google Scholar

[7]

N. Ju, Existence of global attractor for the three-dimensional modified Navier-Stokes equations,, Nonlinearity, 14 (2001), 777.  doi: 10.1088/0951-7715/14/4/306.  Google Scholar

[8]

V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities,, preprint, (2011).   Google Scholar

[9]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196.  doi: 10.1016/j.jde.2006.07.009.  Google Scholar

[10]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Disc. Cont. Dyn. Syst., 13 (2005), 701.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[11]

Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations,, Nonlinear Analysis, 70 (2009), 2054.  doi: 10.1016/j.na.2008.02.121.  Google Scholar

[12]

L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2173.  doi: 10.1098/rspa.1999.0398.  Google Scholar

[13]

L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations,, Studies in Applied Mathematics, 102 (1999), 419.  doi: 10.1111/1467-9590.00116.  Google Scholar

[14]

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

[15]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar

[16]

A. Shenoy, Non-Newtonian fluid heat transfer in porous media,, Adv. Heat transfer, 24 (1994), 101.  doi: 10.1016/S0065-2717(08)70233-8.  Google Scholar

[17]

B. Straughan, "Stability and Wave Motion in Porous Media,", Applied Mathematical Sciences, 165 (2008).   Google Scholar

[18]

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

[19]

S. Whitaker, The Forchheimer equation: A theoretical development,, Transport in Porous Media, 25 (1996), 27.  doi: 10.1007/BF00141261.  Google Scholar

[20]

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

[21]

C. Zhao and S. Zhou, L$^2$-compact uniform attractors for a non-autonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2709845.  Google Scholar

[22]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid,, J. Differential Equations, 238 (2007), 394.  doi: 10.1016/j.jde.2007.04.001.  Google Scholar

[1]

Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037
[2]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[3]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305
[4]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[5]

David Rossmanith, Ashok Puri. Recasting a Brinkman-based acoustic model as the damped Burgers equation. Evolution Equations & Control Theory, 2016, 5 (3) : 463-474. doi: 10.3934/eect.2016014
[6]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019229
[7]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
[8]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[9]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[10]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
[11]

Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[12]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[13]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
[14]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[15]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695
[16]

Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70.

[17]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[18]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080
[19]

Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590
[20]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences