American Institute of Mathematical Sciences

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December  2020, 28(4): 1395-1418. doi: 10.3934/era.2020074

The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay

Xin-Guang Yang 1,2, , Lu Li 1,2, , Xingjie Yan 3,, and  Ling Ding 4,

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2. 

Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie-Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France
3. 

Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, China
4. 

School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China

* Corresponding author: Xingjie Yan

Received  March 2020 Revised  June 2020 Published  July 2020

Fund Project: Research partly supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039)

This paper concerns the stability of pullback attractors for 3D Brinkman-Forchheimer equation with delays. By some regular estimates and the variable index to deal with the delay term, we get the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers.

Keywords: 3D Brinkman-Forchheimer equations, the generalized Grashof number, complete trajectory.
Mathematics Subject Classification: 35Q30, 35B40, 35B41, 76D03, 76D05.
Citation: Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
References:
[1]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.  Google Scholar

[2]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

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J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D Navier-Stokes model, Disc. Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.  Google Scholar

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D. Li, Q. Liu and X. Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations, Accepted, 2020. Google Scholar

[6]

L. Li, X.-G. Yang, X. Li, X. Yan and Y. Lu, Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ), Asymptot. Anal., 113 (2019), 167-194.  doi: 10.3233/ASY-181512.  Google Scholar

[7]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[9]

D. A. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow, 12 (1991), 269-272.  doi: 10.1016/0142-727X(91)90062-Z.  Google Scholar

[10]

Y. Ouyang and L. Yang, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 70 (2009), 2054-2059.  doi: 10.1016/j.na.2008.02.121.  Google Scholar

[11]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ: Linear Inequalities and Vol. Ⅱ: Nonlinear Inequalities, Birkhäser, Basel/Boston/Berlin, 2016.  Google Scholar

[12]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[13]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[14] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, 2001.   Google Scholar
[15]

D. Uǧurlu, 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.  Google Scholar

[16]

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

[17]

Y. Wang, X. Yang and Y. Lu, Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays, Math. Meth. Appl. Sci., 43 (2020), 1892-1900.   Google Scholar

[18]

S. Whitaker, The Forchheimer equation: A theoretical development, Transp. Porous Media, 25 (1996), 27-62.  doi: 10.1007/BF00141261.  Google Scholar

[19]

S. Whitaker, Flow in porous media Ⅰ: A theoretical derivation of Darcy's law, Transp. Porous Media, 1 (1986), 3-25.  doi: 10.1007/BF01036523.  Google Scholar

[20]

X.-G. Yang, J. Zhang and S. Wang, Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay, Disc. Contin. Dyn. Syst., 40 (2020), 1493-1515.  doi: 10.3934/dcds.2020084.  Google Scholar

[21]

Y. You, C. Zhao and S. Zhou, The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Disc. Contin. Dyn. Syst., 32 (2012), 3787-3800.  doi: 10.3934/dcds.2012.32.3787.  Google Scholar

show all references

References:
[1]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.  Google Scholar

[2]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[4]

J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D Navier-Stokes model, Disc. Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.  Google Scholar

[5]

D. Li, Q. Liu and X. Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations, Accepted, 2020. Google Scholar

[6]

L. Li, X.-G. Yang, X. Li, X. Yan and Y. Lu, Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ), Asymptot. Anal., 113 (2019), 167-194.  doi: 10.3233/ASY-181512.  Google Scholar

[7]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[9]

D. A. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow, 12 (1991), 269-272.  doi: 10.1016/0142-727X(91)90062-Z.  Google Scholar

[10]

Y. Ouyang and L. Yang, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 70 (2009), 2054-2059.  doi: 10.1016/j.na.2008.02.121.  Google Scholar

[11]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ: Linear Inequalities and Vol. Ⅱ: Nonlinear Inequalities, Birkhäser, Basel/Boston/Berlin, 2016.  Google Scholar

[12]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[13]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[14] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, 2001.   Google Scholar
[15]

D. Uǧurlu, 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.  Google Scholar

[16]

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

[17]

Y. Wang, X. Yang and Y. Lu, Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays, Math. Meth. Appl. Sci., 43 (2020), 1892-1900.   Google Scholar

[18]

S. Whitaker, The Forchheimer equation: A theoretical development, Transp. Porous Media, 25 (1996), 27-62.  doi: 10.1007/BF00141261.  Google Scholar

[19]

S. Whitaker, Flow in porous media Ⅰ: A theoretical derivation of Darcy's law, Transp. Porous Media, 1 (1986), 3-25.  doi: 10.1007/BF01036523.  Google Scholar

[20]

X.-G. Yang, J. Zhang and S. Wang, Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay, Disc. Contin. Dyn. Syst., 40 (2020), 1493-1515.  doi: 10.3934/dcds.2020084.  Google Scholar

[21]

Y. You, C. Zhao and S. Zhou, The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Disc. Contin. Dyn. Syst., 32 (2012), 3787-3800.  doi: 10.3934/dcds.2012.32.3787.  Google Scholar

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