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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, 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.  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), 801-807. 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, AMS, Providence, RI, 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), 331-350. doi: 10.1017/S0022112097005843.  Google Scholar

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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.  Google Scholar

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T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. 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), 777-786. 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, arXiv:1101.4070. Google Scholar

[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.  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), 701-719. 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), 2054-2059. 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), 2173-2190. 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), 419-439. 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), 71-85. 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, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[17]

B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[19]

S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62. 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), 1479-1495. 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), 12 pp. 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), 394-425. 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, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  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), 801-807. 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, AMS, Providence, RI, 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), 331-350. 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), 3195-3202. 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), 191-206. 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), 777-786. 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, arXiv:1101.4070. Google Scholar

[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.  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), 701-719. 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), 2054-2059. 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), 2173-2190. 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), 419-439. 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), 71-85. 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, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[17]

B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[19]

S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62. 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), 1479-1495. 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), 12 pp. 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), 394-425. doi: 10.1016/j.jde.2007.04.001.  Google Scholar

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