doi: 10.3934/eect.2021061
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Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

*Corresponding author: Manil T. Mohan

Received  May 2021 Revised  November 2021 Early access December 2021

Fund Project: The second author is supported by DST INSPIRE Faculty Award (IFA17-MA110)

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an
$ n $
-dimensional torus (
$ n = 2, 3 $
):
$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $
where
$ r\geq1 $
. We prove that the global attractor of the above system is singleton under small forcing intensity (
$ r\geq 1 $
for
$ n = 2 $
and
$ r\geq 3 $
for
$ n = 3 $
with
$ 2\beta\mu\geq 1 $
for
$ r = n = 3 $
). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all
$ 1\leq r<\infty $
, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for
$ 3\leq r<\infty $
(
$ 2\beta\mu\geq 1 $
for
$ r = 3 $
), when the coefficient of random perturbation converges to zero.
Citation: Kush Kinra, Manil T. Mohan. Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, doi: 10.3934/eect.2021061
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[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, 1992.

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J. Babutzka and P. C. Kunstmann, $L^q$-Helmholtz decomposition on periodic domains and applications to Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), 1093-1121.  doi: 10.1007/s00021-017-0356-z.

[5]

P. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

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

[7]

Z. BrzeźniakB. Goldys and Q. T. Le Gia, Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.

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T. CaraballoH. CrauelJ. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.

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T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.

[10]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824.  doi: 10.1016/j.na.2009.01.016.

[11]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.

[12]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[13]

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

[14]

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

[15]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[16]

H. Cui and P. E. Kloeden, Convergence rate of random attractors for 2D Navier-Stokes equation towards the deterministic singleton attractor, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, (2021), 233–255. doi: 10.1007/978-3-030-50302-4_10.

[17]

X. Fan, Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.

[18]

C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, https://arXiv.org/pdf/1904.03337.pdf.

[19]

X. Feng and B. You, Random attractors for the two-dimensional stochastic g-Navier-Stokes equations, Stochastics, 92 (2020), 613-626.  doi: 10.1080/17442508.2019.1642340.

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F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier- Stokes equation with multiplicative white noise, Stochastics Stochastic Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

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C. FoiasO. P. ManleyR. Temam and Y. M. Tréve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.

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D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. 

[23]

B. GessW. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 269 (2020), 3414-3455.  doi: 10.1016/j.jde.2020.03.002.

[24]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.

[25]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.

[26]

X. Jia and X. Ding, Random attractors for stochastic retarded 2D-Navier-Stokes equations with additive noise, J. Funct. Spaces, 2018 (2018), Art. ID 3105239, 14pp. doi: 10.1155/2018/3105239.

[27]

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.

[28]

K. Kinra and M. T. Mohan, Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2010.08753.pdf.

[29]

K. Kinra and M. T. Mohan, Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains, Submitted, https://arXiv.org/pdf/2011.06206.pdf.

[30]

K. Kinra and M. T. Mohan, ${\mathbb{H}}^1$-random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2111.07841.pdf.

[31]

K. Kinra and M. T. Mohan, Large time behavior of the deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domains, J. Dynamics and Differential Equations, 2021 doi: 10.1007/s10884-021-10073-7.

[32]

K. Kinra and M. T. Mohan, Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations, Submitted, https://arXiv.org/pdf/2012.12605.pdf.

[33]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.

[34]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141.  doi: 10.1017/S0017089503001605.

[35]

L. Liu and X. Fu, Existence and upper semicontinuity of $(L^2, L^q)$ pullback attractors for a stochastic p-laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-474.  doi: 10.3934/cpaa.2017023.

[36]

H. Liu and H. Gao, Stochastic 3D Navier-Stokes equations with nonlinear damping: Martingale solution, strong solution and small time LDP, Interdiscip. Math. Sci., World Sci. Publ., 20 (2019), 9-36. 

[37]

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.

[38]

M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.

[39]

M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations, Submitted, 2020, https://arXiv.org/abs/2007.09376.

[40]

M. T. Mohan, Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity, Submitted, https://arXiv.org/abs/2010.12814.

[41]

M. T. Mohan, The ${\mathbb{H}}^1$-compact global attractor for the two dimentional convective Brinkman-Forchheimer equations in unbounded domains, J. Dynamical and Control Systems, 2021. doi: 10.1007/s10883-021-09545-2.

[42]

M. T. Mohan, Well-posedness and asymptotic behavior of stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise, Stoch PDE: Anal Comp, (2021). doi: 10.1007/s40072-021-00207-9.

[43]

M. T. Mohan, ${\mathbb{L}}^p$-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations, Anal. Math. Phys., 11 (2021), Ar. No.: 164, 33pp. doi: 10.1007/s13324-021-00595-0.

[44]

M. T. Mohan, Martingale solutions of two and three dimensional stochastic convective Brinkman-Forchheimer equations forced by Lévy noise, Submitted, https://arXiv.org/pdf/2109.05510.pdf.

[45]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.

[46]

J. C. Robinson, Attractors and finite-dimensional behavior in the 2D Navier-Stokes equations, ISRN Math. Anal., 2013 (2013), Article ID 291823, 29pp. doi: 10.1155/2013/291823.

[47]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139095143.

[48]

J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159-178.  doi: 10.4171/RSMUP/131-9.

[49]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: Existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.  doi: 10.1142/S0219025709003859.

[50]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[51]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[52]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, (2009), 1–18.

[53]

J. Wang, C. Li, L. Yang and M. Jia, Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys., 62 (2021), 111507, 31pp. doi: 10.1063/5.0039187.

[54]

B. You, The existence of a random attractor for the three dimensional damped Navier-Stokes equations with additive noise, Stoch. Anal. Appl., 35 (2017), 691-700.  doi: 10.1080/07362994.2017.1311794.

[55]

S. Zhou and Z. Wang, Upper-semicontinuity of attractors for reaction-diffusion equation and damped wave equation in $\mathbb{R}^n$ perturbed by small multiplicative noises, Dynam. Systems Appl., 22 (2013), 15-31. 

show all references

References:
[1]

S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825.  doi: 10.1080/00036811.2010.495341.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, 1992.

[4]

J. Babutzka and P. C. Kunstmann, $L^q$-Helmholtz decomposition on periodic domains and applications to Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), 1093-1121.  doi: 10.1007/s00021-017-0356-z.

[5]

P. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[6]

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

[7]

Z. BrzeźniakB. Goldys and Q. T. Le Gia, Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.

[8]

T. CaraballoH. CrauelJ. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.

[9]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.

[10]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824.  doi: 10.1016/j.na.2009.01.016.

[11]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.

[12]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[13]

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

[14]

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

[15]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[16]

H. Cui and P. E. Kloeden, Convergence rate of random attractors for 2D Navier-Stokes equation towards the deterministic singleton attractor, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, (2021), 233–255. doi: 10.1007/978-3-030-50302-4_10.

[17]

X. Fan, Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.

[18]

C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, https://arXiv.org/pdf/1904.03337.pdf.

[19]

X. Feng and B. You, Random attractors for the two-dimensional stochastic g-Navier-Stokes equations, Stochastics, 92 (2020), 613-626.  doi: 10.1080/17442508.2019.1642340.

[20]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier- Stokes equation with multiplicative white noise, Stochastics Stochastic Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[21]

C. FoiasO. P. ManleyR. Temam and Y. M. Tréve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.

[22]

D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. 

[23]

B. GessW. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 269 (2020), 3414-3455.  doi: 10.1016/j.jde.2020.03.002.

[24]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.

[25]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.

[26]

X. Jia and X. Ding, Random attractors for stochastic retarded 2D-Navier-Stokes equations with additive noise, J. Funct. Spaces, 2018 (2018), Art. ID 3105239, 14pp. doi: 10.1155/2018/3105239.

[27]

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.

[28]

K. Kinra and M. T. Mohan, Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2010.08753.pdf.

[29]

K. Kinra and M. T. Mohan, Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains, Submitted, https://arXiv.org/pdf/2011.06206.pdf.

[30]

K. Kinra and M. T. Mohan, ${\mathbb{H}}^1$-random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2111.07841.pdf.

[31]

K. Kinra and M. T. Mohan, Large time behavior of the deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domains, J. Dynamics and Differential Equations, 2021 doi: 10.1007/s10884-021-10073-7.

[32]

K. Kinra and M. T. Mohan, Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations, Submitted, https://arXiv.org/pdf/2012.12605.pdf.

[33]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.

[34]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141.  doi: 10.1017/S0017089503001605.

[35]

L. Liu and X. Fu, Existence and upper semicontinuity of $(L^2, L^q)$ pullback attractors for a stochastic p-laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-474.  doi: 10.3934/cpaa.2017023.

[36]

H. Liu and H. Gao, Stochastic 3D Navier-Stokes equations with nonlinear damping: Martingale solution, strong solution and small time LDP, Interdiscip. Math. Sci., World Sci. Publ., 20 (2019), 9-36. 

[37]

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.

[38]

M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.

[39]

M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations, Submitted, 2020, https://arXiv.org/abs/2007.09376.

[40]

M. T. Mohan, Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity, Submitted, https://arXiv.org/abs/2010.12814.

[41]

M. T. Mohan, The ${\mathbb{H}}^1$-compact global attractor for the two dimentional convective Brinkman-Forchheimer equations in unbounded domains, J. Dynamical and Control Systems, 2021. doi: 10.1007/s10883-021-09545-2.

[42]

M. T. Mohan, Well-posedness and asymptotic behavior of stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise, Stoch PDE: Anal Comp, (2021). doi: 10.1007/s40072-021-00207-9.

[43]

M. T. Mohan, ${\mathbb{L}}^p$-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations, Anal. Math. Phys., 11 (2021), Ar. No.: 164, 33pp. doi: 10.1007/s13324-021-00595-0.

[44]

M. T. Mohan, Martingale solutions of two and three dimensional stochastic convective Brinkman-Forchheimer equations forced by Lévy noise, Submitted, https://arXiv.org/pdf/2109.05510.pdf.

[45]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.

[46]

J. C. Robinson, Attractors and finite-dimensional behavior in the 2D Navier-Stokes equations, ISRN Math. Anal., 2013 (2013), Article ID 291823, 29pp. doi: 10.1155/2013/291823.

[47]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139095143.

[48]

J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159-178.  doi: 10.4171/RSMUP/131-9.

[49]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: Existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.  doi: 10.1142/S0219025709003859.

[50]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[51]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[52]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, (2009), 1–18.

[53]

J. Wang, C. Li, L. Yang and M. Jia, Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys., 62 (2021), 111507, 31pp. doi: 10.1063/5.0039187.

[54]

B. You, The existence of a random attractor for the three dimensional damped Navier-Stokes equations with additive noise, Stoch. Anal. Appl., 35 (2017), 691-700.  doi: 10.1080/07362994.2017.1311794.

[55]

S. Zhou and Z. Wang, Upper-semicontinuity of attractors for reaction-diffusion equation and damped wave equation in $\mathbb{R}^n$ perturbed by small multiplicative noises, Dynam. Systems Appl., 22 (2013), 15-31. 

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