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

Fuzhi Li; Dongmei Xu; Jiali Yu

doi:10.3934/cpaa.2020136

Article Contents

2020, Volume 19, Issue 6: 3137-3157. Doi: 10.3934/cpaa.2020136

This issue Previous Article Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off Next Article Radial solutions for a class of Hénon type systems with partial interference with the spectrum

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
^* Corresponding author

Received: June 2019

Revised: December 2019

Published: March 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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. Cui, Y. 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. Cui, J. 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. 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.
[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. Kwak, H. 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. Li, H. 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. 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, R. 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. Yin, Y. 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. 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.
[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. Zhong, M. 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.