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June  2020, 19(6): 3137-3157. doi: 10.3934/cpaa.2020136

Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation

1. 

School of Mathematics & Computer Science, , Shangrao Normal University, Shangrao 334001, China

2. 

School of Locomotive and Rolling Stock Engineering, Dalian Jiaotong University, Dalian 116028, China

* Corresponding author

Received  June 2019 Revised  December 2019 Published  March 2020

Fund Project: The first author is supported by Science and Technology Foundation of Jiangxi Education Department grant GJJ190880

In this paper, we study the backward compactness of random attractors, which describes the compactness of the union $ \cup_{s\leq\tau}\mathcal A(s,\omega) $ of random attractor sections over past times, $ \tau\in\mathbb R $. In particular, we prove the backward compactness and the regularity of random attractors for stochastic $ g $-Navier-Stokes equations under the condition that the force is backward tempered and backward limiting. The attraction universe in consideration is non-autonomous and consists of backward tempered sets.

Citation: Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136
References:
[1]

C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous $g$-Navier-Stokes equations, Ann. Polon. Math., 103 (2012), 277-302.  doi: 10.4064/ap103-3-5.  Google Scholar

[2]

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H. O. Bae and J. Roh, Existence of solutions of the $g$-Navier-Stokes equations, Taiwan. J. Math., 8 (2004), 85-102.  doi: 10.11650/twjm/1500558459.  Google Scholar

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H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 38 (2018), 187-208.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

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H. CuiJ. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.  Google Scholar

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

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J. Jiang and Y. Hou, The global attractor of $g$-Navier-Stokes equations with linear dampness on $\mathbb R^2$, Appl. Math. Comput., 215 (2009), 1068-1076.  doi: 10.1016/j.amc.2009.06.035.  Google Scholar

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J. Jiang and Y. Hou, Pullback attractor of 2D non-autonomous $g$-Navier-Stokes equations on some bounded domains, Appl. Math. Mech. (English Ed.), 31 (2010), 697-708.  doi: 10.1007/s10483-010-1304-x.  Google Scholar

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P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

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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.  Google Scholar

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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

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Y. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar

[17]

Y. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous Sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[18]

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

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[20]

G. Raugel and G. R. Sell, Navier-Stokes Equations in Thin 3D Domains Ⅲ: Existence of a Global Attractor, Turbulence in Fluid Flows, 55 (1993), 137-163.  doi: 10.1007/978-1-4612-4346-5_9.  Google Scholar

[21]

J. Roh, $g$-Navier-Stokes Equations, Thesis, University of Minnesota, 2001. Google Scholar

[22]

J. Roh, Dynamics of the $g$-Navier-Stokes Equations, J. Differ. Equ., 211 (2005), 452-484.  doi: 10.1016/j.jde.2004.08.016.  Google Scholar

[23]

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

[24]

M. Wang and Y. Tang, Attractors in $H^2$ and $L^{2p-2}$ for reaction-diffusion equations on unbounded domains, Commun. Pure Appl. Anal., 12 (2013), 1111-1121.  doi: 10.3934/cpaa.2013.12.1111.  Google Scholar

[25]

J. YinY. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.  Google Scholar

[26]

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

[27]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[28]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

show all references

References:
[1]

C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous $g$-Navier-Stokes equations, Ann. Polon. Math., 103 (2012), 277-302.  doi: 10.4064/ap103-3-5.  Google Scholar

[2]

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

[3]

H. O. Bae and J. Roh, Existence of solutions of the $g$-Navier-Stokes equations, Taiwan. J. Math., 8 (2004), 85-102.  doi: 10.11650/twjm/1500558459.  Google Scholar

[4]

I. Chueshov, Monotone Random Systems Theory and Applications, Vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.  Google Scholar

[5]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 38 (2018), 187-208.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[6]

H. CuiJ. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.  Google Scholar

[7]

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

[8]

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

[9]

D. Iftimie and G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Differ. Equ., 169 (2001), 281-331.  doi: 10.1006/jdeq.2000.3900.  Google Scholar

[10]

J. Jiang and Y. Hou, The global attractor of $g$-Navier-Stokes equations with linear dampness on $\mathbb R^2$, Appl. Math. Comput., 215 (2009), 1068-1076.  doi: 10.1016/j.amc.2009.06.035.  Google Scholar

[11]

J. Jiang and Y. Hou, Pullback attractor of 2D non-autonomous $g$-Navier-Stokes equations on some bounded domains, Appl. Math. Mech. (English Ed.), 31 (2010), 697-708.  doi: 10.1007/s10483-010-1304-x.  Google Scholar

[12]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[13]

M. KwakH. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier-Stokes equations, J. Math. Anal. Appl., 315 (2006), 436-461.  doi: 10.1016/j.jmaa.2005.04.050.  Google Scholar

[14]

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.  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. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar

[17]

Y. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous Sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[18]

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

[19]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. Ⅰ. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.1090/s0894-0347-1993-1179539-4.  Google Scholar

[20]

G. Raugel and G. R. Sell, Navier-Stokes Equations in Thin 3D Domains Ⅲ: Existence of a Global Attractor, Turbulence in Fluid Flows, 55 (1993), 137-163.  doi: 10.1007/978-1-4612-4346-5_9.  Google Scholar

[21]

J. Roh, $g$-Navier-Stokes Equations, Thesis, University of Minnesota, 2001. Google Scholar

[22]

J. Roh, Dynamics of the $g$-Navier-Stokes Equations, J. Differ. Equ., 211 (2005), 452-484.  doi: 10.1016/j.jde.2004.08.016.  Google Scholar

[23]

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

[24]

M. Wang and Y. Tang, Attractors in $H^2$ and $L^{2p-2}$ for reaction-diffusion equations on unbounded domains, Commun. Pure Appl. Anal., 12 (2013), 1111-1121.  doi: 10.3934/cpaa.2013.12.1111.  Google Scholar

[25]

J. YinY. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.  Google Scholar

[26]

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

[27]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[28]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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