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

Qiangheng Zhang; Yangrong Li

doi:10.3934/cpaa.2021117

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2021, Volume 20, Issue 10: 3515-3537. Doi: 10.3934/cpaa.2021117 open access

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

Qiangheng Zhang and
Yangrong Li^,

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

^* Corresponding author
^* Corresponding author

Received: March 2021

Revised: June 2021

Early access: July 8, 2021

Published online: July 8, 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 35B41, 35R60.

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References

[1]	P. W. Bates, K. 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. Brzézniak, M. 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. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[8]	H. Cui, J. 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. Kloeden, J. 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. Li, X. Yang, X. Li, X. 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. Li, A. 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. Li, L. 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. Markowich, E. 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. Shi, X. 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. Wang, K. 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. Yang, L. Li, X. 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. Yin, A. 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. You, C. 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.