# American Institute of Mathematical Sciences

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

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

 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.
Citation: Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems, 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, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. [2] O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Applied Mathematics Letters, 19 (2006), 801-807. doi: 10.1016/j.aml.2005.11.002. [3] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. [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), 331-350. doi: 10.1017/S0022112097005843. [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), 3195-3202. doi: 10.1098/rspa.2003.1169. [6] T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. doi: 10.1023/A:1006533931383. [7] N. Ju, Existence of global attractor for the three-dimensional modified Navier-Stokes equations, Nonlinearity, 14 (2001), 777-786. doi: 10.1088/0951-7715/14/4/306. [8] V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities, preprint, 2011, arXiv:1101.4070. [9] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [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), 701-719. doi: 10.3934/dcds.2005.13.701. [11] Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059. doi: 10.1016/j.na.2008.02.121. [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), 2173-2190. doi: 10.1098/rspa.1999.0398. [13] L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439. doi: 10.1111/1467-9590.00116. [14] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [16] A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat transfer, 24 (1994), 101-190. doi: 10.1016/S0065-2717(08)70233-8. [17] B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008. [18] D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992. doi: 10.1016/j.na.2007.01.025. [19] S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62. doi: 10.1007/BF00141261. [20] B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495. doi: 10.1002/mma.985. [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), 12 pp. doi: 10.1063/1.2709845. [22] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.

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##### References:
 [1] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. [2] O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Applied Mathematics Letters, 19 (2006), 801-807. doi: 10.1016/j.aml.2005.11.002. [3] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. [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), 331-350. doi: 10.1017/S0022112097005843. [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), 3195-3202. doi: 10.1098/rspa.2003.1169. [6] T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. doi: 10.1023/A:1006533931383. [7] N. Ju, Existence of global attractor for the three-dimensional modified Navier-Stokes equations, Nonlinearity, 14 (2001), 777-786. doi: 10.1088/0951-7715/14/4/306. [8] V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities, preprint, 2011, arXiv:1101.4070. [9] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [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), 701-719. doi: 10.3934/dcds.2005.13.701. [11] Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059. doi: 10.1016/j.na.2008.02.121. [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), 2173-2190. doi: 10.1098/rspa.1999.0398. [13] L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439. doi: 10.1111/1467-9590.00116. [14] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [16] A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat transfer, 24 (1994), 101-190. doi: 10.1016/S0065-2717(08)70233-8. [17] B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008. [18] D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992. doi: 10.1016/j.na.2007.01.025. [19] S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62. doi: 10.1007/BF00141261. [20] B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495. doi: 10.1002/mma.985. [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), 12 pp. doi: 10.1063/1.2709845. [22] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.
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