The tempered pullback dynamics of the 3D Brinkman–Forchheimer equation with variable delay has been studied in this paper. With the different universes which has some topology property, the existence of minimal and unique family of pullback attractors were obtained. Moreover, the convergence of pullback attractors for the 3D Brinkman–Forchheimer equation as delay term vanishes is also been proved, i.e., the upper semi-continuity of attractors.
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.
2.
T.Caraballo and J.Real, Attractors for 2D Navier–Stokes models with delays, J. Diff. Equ.205 (2004), 271–297. doi:10.1016/j.jde.2004.04.012.
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.
4.
A.O.Celebi, V.K.Kalantarov and D.Ugurlu, Continuous dependence for the covective Brinkman–Forchheimer equations, Appl. Anal.84(9) (2005), 877–888. doi:10.1080/00036810500148911.
5.
A.O.Celebi, V.K.Kalantarov and D.Ugurlu, On continuous dependence on coefficients of the Brinkman–Forchheimer equations, Appl. Math. Lett.19 (2006), 801–807. doi:10.1016/j.aml.2005.11.002.
6.
H.Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856.
7.
J.K.Djoko and M.Mbehoua, A global attractor for the Brinkman–Forchheimer equations with nonlinear slip boundary conditions of friction type, Technical Report, UPWT, 2013/Aprial/21, preprint.
8.
R.C.Gilver and S.A.Altobelli, A determination of effective viscosity for the Brinkman–Forchheimer flow model, J. Fluid Mech.370 (1994), 258–355.
9.
V.K.Kalantarov and S.Zelik, Smooth attractors for the Brinkman–Forchheimer equations with fast growing nonlinearities, arXiv.org/abs/1101.4070v1, [Math. AP], 21, Jan. 2011.
10.
J.R.Kang and J.Y.Park, Uniform attractors for non-autonomous Brinkman–Forchheimer equations with delay, Acta Math. Sin.29(5) (2013), 993–1006. doi:10.1007/s10114-013-1392-0.
11.
P.E.Kloeden, Upper semi-continuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc.73 (2006), 299–306. doi:10.1017/S0004972700038880.
12.
J.-L.Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
13.
L.Liu and X.Fu, Existence and upper semi-continuity of pullback attractors of a p-Laplacian equation with delay, J. Math. Physics58 (2017), 082702. doi:10.1063/1.5017932.
14.
Y.Liu, Convergence and continuous dependence for the Brinkman–Forchheimer equations, Math. Computer Model.49 (2009), 1401–1415. doi:10.1016/j.mcm.2008.11.010.
15.
P.Marín-Rubio and J.Real, Pullback attractors for 2D-Navier–Stokes equations with delays in continuous and sub-linear operators, Disc. Cont. Dyn. Syst.26 (2010), 989–1006. doi:10.3934/dcds.2010.26.989.
16.
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 Flow12 (1991), 269–272. doi:10.1016/0142-727X(91)90062-Z.
17.
Y.Ouyang and L.Yang, A note on the existence of a global attractor for the Brinkman–Forchheimer equations, Nonlinear Anal. TMA70 (2009), 2054–2059. doi:10.1016/j.na.2008.02.121.
18.
L.E.Payne, J.C.Song and B.Straughan, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, Proc. R. Soc. Lond. A45S (1999), 2173–2190. doi:10.1098/rspa.1999.0398.
19.
L.E.Payne and B.Straughan, Convergence and continuous dependence for the Brinkman–Forchheimer equations, Studies Appl. Math.102 (1999), 419–439. doi:10.1111/1467-9590.00116.
20.
Y.Qin and P.N.Kaloni, Spatial decay estimates for plane flow in the Brinkman–Forchheimer model, Quart. Appl. Math.87 (1998), 56–71.
21.
X.Song, Pullback -attractors for a non-autonomous Brinkman–Forcheimer system, J. Math. Research Appl.33(1) (2013), 90–100.
22.
X.Song and Y.Hou, Uniform attractors for a non-autonomous Brinkman–Forchheimer equation, J. Math. Research Appl.32(1) (2012), 63–75.
23.
R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1997.
24.
D.Uğurlu, On the existence of a global attractor for the Brinkman–Forchheimer equations, Nonlinear Anal. TMA68 (2008), 1986–1992. doi:10.1016/j.na.2007.01.025.
25.
K.Vafai and S.Kim, Fluid mechanics of the interface region between a porous medium and a fluid layer – An exact solution, Int. J. Heat Fluid Flow11(a) (1990), 254–256.
26.
K.Vafai and S.J.Kim, On the limitations of the Brinkman–Forchheimer-extended Darcy equation, Int. J. Heat and Fluid Flow16(1) (1995), 11–15. doi:10.1016/0142-727X(94)00002-T.
27.
B.Wang and B.Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differ. Equations (2013), 25.
28.
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.
29.
S.Whitaker, Flow in porous media I: A theoretical derivation of Darcy’s law, Transport in Porous Media1 (1986), 3–25. doi:10.1007/BF01036523.
30.
S.Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media25 (1996), 27–62. doi:10.1007/BF00141261.
31.
Y.You, C.Zhao and S.Zhou, The existence of uniform attractors for 3D Brinkman–Forchheimer equations, Disc. Cont. Dyn. Syst.32(10) (2012), 3787–3800. doi:10.3934/dcds.2012.32.3787.
32.
C.Zhao and Y.You, Approximation of the incompressible convective Brinkman–Forchheimer equations, J. Evol. Equ.12 (2012), 767–788. doi:10.1007/s00028-012-0153-3.
