May  2021, 20(5): 1907-1930. doi: 10.3934/cpaa.2021052

Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay

1. 

Faculty of Science, Beijing University of Technology, Ping Le Yuan 100, Chaoyang District, Beijing, 100124, China

2. 

Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

* Corresponding author: Rong Yang

Received  October 2020 Revised  January 2021 Published  May 2021 Early access  March 2021

Fund Project: R. Yang is partially supported by NSFC (No. 11601021). X.-G. Yang is partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039), Incubation Fund from Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003)

This paper is concerned with the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with infinite delay. The well-posedness of weak solution to the 3D Brinkman-Forchheimer flow with infinite delay is investigated in the weighted space $ C_\kappa(H) $ firstly, then the pullback attractors are presented for the process of weak solution. Moreover, the existence of global attractor and the exponential stability analysis of stationary solutions are shown, which is based on the estimate of corresponding steady state equation.

Citation: Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1907-1930. doi: 10.3934/cpaa.2021052
References:
[1]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys., 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.

[2]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Continuous Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. 

[3]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[4]

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.

[5]

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.

[6]

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

[7]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[9]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[10]

J. Garcia-Luengo and P. Marin-Rubio, Attractors for a double time-delayed 2D Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.

[11]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. 

[12]

Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[13]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[14]

J. R. Kang and J. Y. Park, Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay, Acta Math. Sin. (Engl. Ser.), 29 (2013), 993-1006.  doi: 10.1007/s10114-013-1392-0.

[15]

L. LiX. YangX. LiX. 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.

[16]

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

[17]

Y. Liu, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Math. Comput. Modelling, 49 (2009), 1401-1415.  doi: 10.1016/j.mcm.2008.11.010.

[18]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009) 3956–3963. doi: 10.1016/j.na.2009.02.065.

[19]

P. Marín-RubioJ. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030.  doi: 10.1016/j.na.2010.11.008.

[20]

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.

[21]

Y. Ouyang and L. Yan, 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.

[22]

L. E. Payne and B. Straughan, Convergence and continuous dependence for the Brinkman–Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439.  doi: 10.1111/1467-9590.00116.

[23]

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

[24]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Vol. 45, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

D. Ugurlu, 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.

[26]

K. Vafai and S. J. Kim, Fluid mechanics of the interface region between a porous medium and a fluid layer–An exact solution, Int. J. Heat Fluid Flow, 11 (1990), 254-256.  doi: 10.1016/0142-727X(90)90045-D.

[27]

K. Vafai and S. J. Kim, On the limitations of the Brinkman-Forchheimer-extended Darcy equation, Int. J. Heat and Fluid Flow, 16 (1995), 11-15.  doi: 10.1016/0142-727X(94)00002-T.

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

[30]

R. Yang, W. Liu and X.-G. Yang, Asymptotic stability of 3D Brinkman-Forchheimer equation with delay, preprint.

[31]

X.-G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1396-1418.  doi: 10.3934/era.2020074.

show all references

References:
[1]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys., 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.

[2]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Continuous Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. 

[3]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[4]

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.

[5]

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.

[6]

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

[7]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[9]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[10]

J. Garcia-Luengo and P. Marin-Rubio, Attractors for a double time-delayed 2D Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.

[11]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. 

[12]

Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[13]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[14]

J. R. Kang and J. Y. Park, Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay, Acta Math. Sin. (Engl. Ser.), 29 (2013), 993-1006.  doi: 10.1007/s10114-013-1392-0.

[15]

L. LiX. YangX. LiX. 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.

[16]

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

[17]

Y. Liu, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Math. Comput. Modelling, 49 (2009), 1401-1415.  doi: 10.1016/j.mcm.2008.11.010.

[18]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009) 3956–3963. doi: 10.1016/j.na.2009.02.065.

[19]

P. Marín-RubioJ. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030.  doi: 10.1016/j.na.2010.11.008.

[20]

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.

[21]

Y. Ouyang and L. Yan, 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.

[22]

L. E. Payne and B. Straughan, Convergence and continuous dependence for the Brinkman–Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439.  doi: 10.1111/1467-9590.00116.

[23]

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

[24]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Vol. 45, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

D. Ugurlu, 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.

[26]

K. Vafai and S. J. Kim, Fluid mechanics of the interface region between a porous medium and a fluid layer–An exact solution, Int. J. Heat Fluid Flow, 11 (1990), 254-256.  doi: 10.1016/0142-727X(90)90045-D.

[27]

K. Vafai and S. J. Kim, On the limitations of the Brinkman-Forchheimer-extended Darcy equation, Int. J. Heat and Fluid Flow, 16 (1995), 11-15.  doi: 10.1016/0142-727X(94)00002-T.

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

[30]

R. Yang, W. Liu and X.-G. Yang, Asymptotic stability of 3D Brinkman-Forchheimer equation with delay, preprint.

[31]

X.-G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1396-1418.  doi: 10.3934/era.2020074.

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