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

Xin-Guang Yang; Lu Li; Xingjie Yan; Ling Ding; Xin-Guang Yang; Lu Li; Xingjie Yan; Ling Ding

doi:10.3934/era.2020074

Electronic Research Archive

2020, Volume 28, Issue 4: 1395-1418. doi: 10.3934/era.2020074

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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
,
,},
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

Received: 01 March 2020 Revised: 01 June 2020 Published: 31 July 2020
35Q30, 35B40, 35B41, 76D03, 76D05

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.
- 3D Brinkman-Forchheimer equations,
- the generalized Grashof number,
- complete trajectory
Citation: Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay[J]. Electronic Research Archive, 2020, 28(4): 1395-1418. doi: 10.3934/era.2020074

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Abstract

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.

References

[1]	Navier-Stokes equations with delays. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. (2001) 457: 2441-2453.
[2]	Attractors for 2D Navier-Stokes models with delays. J. Differential Equations (2004) 205: 271-297.
[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
[4]	Attractors for a double time-delayed 2D Navier-Stokes model. Disc. Contin. Dyn. Syst. (2014) 34: 4085-4105.
[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.
[6]	Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ). Asymptot. Anal. (2019) 113: 167-194.
[7]	J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[8]	Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete Contin. Dyn. Syst. (2010) 26: 989-1006.
[9]	The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow (1991) 12: 269-272.
[10]	A note on the existence of a global attractor for the Brinkman-Forchheimer equations. Nonlinear Anal. (2009) 70: 2054-2059.
[11]	Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ: Linear Inequalities and Vol. Ⅱ: Nonlinear Inequalities, Birkhäser, Basel/Boston/Berlin, 2016.
[12]	Compact sets in the space $L^p(0, T;B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96.
[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
[14]	(2001) Mathematical Modeling in Continuum Mechanics. Cambridge: Cambridge University Press.
[15]	On the existence of a global attractor for the Brinkman-Forchheimer equations. Nonlinear Anal. (2008) 68: 1986-1992.
[16]	Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation. Math. Meth. Appl. Sci. (2008) 31: 1479-1495.
[17]	Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays. Math. Meth. Appl. Sci. (2020) 43: 1892-1900.
[18]	The Forchheimer equation: A theoretical development. Transp. Porous Media (1996) 25: 27-62.
[19]	Flow in porous media Ⅰ: A theoretical derivation of Darcy's law. Transp. Porous Media (1986) 1: 3-25.
[20]	Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Disc. Contin. Dyn. Syst. (2020) 40: 1493-1515.
[21]	The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Disc. Contin. Dyn. Syst. (2012) 32: 3787-3800.

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