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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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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