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

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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

Abstract
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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 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.

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.
[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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