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

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

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  • 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.

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


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  • [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.
    [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
    [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.
    [4] I. Chueshov, Monotone Random Systems Theory and Applications, Vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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. 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.
    [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.
    [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.
    [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.
    [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.
    [21] J. Roh, $g$-Navier-Stokes Equations, Thesis, University of Minnesota, 2001.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
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