• Previous Article
    Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment
  • CPAA Home
  • This Issue
  • Next Article
    Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics
October  2021, 20(10): 3515-3537. doi: 10.3934/cpaa.2021117

Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  March 2021 Revised  June 2021 Published  October 2021 Early access  July 2021

Fund Project: This work was supported by the Natural Science Foundation of China Grant 11571283

We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.

Citation: Qiangheng Zhang, Yangrong Li. Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3515-3537. doi: 10.3934/cpaa.2021117
References:
[1]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[2]

Z. BrzézniakM. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[3]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.  doi: 10.1016/S0166-218X(03)00183-5.  Google Scholar

[4]

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

[5]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1142/S0219493706001621.  Google Scholar

[6]

A. N. Carvalho, J. A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, vol.182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[8]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dyn. Differ. Equ., 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[9]

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.  doi: 10.1017/s0022112094003368.  Google Scholar

[10]

V. 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.  Google Scholar

[11]

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

[12]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[13]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[14]

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.  Google Scholar

[15]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[16]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[17]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[18]

P. A. MarkowichE. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy Model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.  Google Scholar

[19]

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

[20]

L. ShiX. Wang and D. Li, Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains, Commun. Pure Appl. Anal., 19 (2020), 5367-5386.  doi: 10.3934/cpaa.2020242.  Google Scholar

[21]

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

[22]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[23]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[24]

X. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2021), 1395-1418.  doi: 10.3934/era.2020074.  Google Scholar

[25]

J. YinA. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dyn. Partial Differ. Equ., 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4.  Google Scholar

[26]

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

show all references

References:
[1]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[2]

Z. BrzézniakM. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[3]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.  doi: 10.1016/S0166-218X(03)00183-5.  Google Scholar

[4]

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

[5]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1142/S0219493706001621.  Google Scholar

[6]

A. N. Carvalho, J. A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, vol.182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[8]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dyn. Differ. Equ., 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[9]

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.  doi: 10.1017/s0022112094003368.  Google Scholar

[10]

V. 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.  Google Scholar

[11]

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

[12]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[13]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[14]

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.  Google Scholar

[15]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[16]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[17]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[18]

P. A. MarkowichE. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy Model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.  Google Scholar

[19]

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

[20]

L. ShiX. Wang and D. Li, Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains, Commun. Pure Appl. Anal., 19 (2020), 5367-5386.  doi: 10.3934/cpaa.2020242.  Google Scholar

[21]

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

[22]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[23]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[24]

X. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2021), 1395-1418.  doi: 10.3934/era.2020074.  Google Scholar

[25]

J. YinA. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dyn. Partial Differ. Equ., 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4.  Google Scholar

[26]

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

[1]

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

[2]

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

[3]

Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037

[4]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[5]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326

[6]

Manil T. Mohan. Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021020

[7]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[8]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861

[9]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[10]

Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3393-3436. doi: 10.3934/dcdsb.2020067

[11]

Mahmoud Abouagwa, Ji Li. G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1583-1606. doi: 10.3934/dcdsb.2019241

[12]

Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259

[13]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032

[14]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062

[15]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[16]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

[17]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[18]

Tomás Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, José Valero. Non--autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 415-443. doi: 10.3934/dcds.2008.21.415

[19]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589

[20]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (81)
  • HTML views (165)
  • Cited by (0)

Other articles
by authors

[Back to Top]