带非线性噪音的随机${g}$-Navier-Stokes方程的后向弱紧均值动力学

Abstract

Abstract:

We study the mean dynamics for the stochastic 2D $g$-Navier-Stokes matrix driven by infinitely dimensional cylindrical noise with a nonlinear diffusion term and a time-dependent external force. We first obtain a mean random dynamical system if the nonlinear diffusion term is Lipschtz continuous and the force is locally integrable. We then show that the mean RDS possesses a unique mean pullback weak attractor in the Bochner space of even power if the force is also tempered. Moreover, by using the monotonicity of Bochner spaces with respect to the time, we show that the backward union of the mean pullback w-attractor is well-defined and weakly compact in progressive Bochner spaces if the force is backward tempered. We finally provide three examples of backward $w$-compact $w$-attractors when the force is null, periodic or increasing, respectively.

Key words: Mean pullback $w$-attractor, $g$-Navier-Stokes matrix, Nonlinear noise, Bochner spaces, Backward $w$-compactness

CLC Number:

O211.4

Li Yangrong, Wang Fengling, Yang Shuang. Backward $W$-compact Mean Dynamics for Stochastic ${g}$-Navier-Stokes Equations with Nonlinear Noise[J].Acta mathematica scientia,Series A, 2023, 43(2): 531-548.

Trendmd

References 42

[1]	Kloeden P E, Lorenz T. Mean-square random dynamical systems. J Differential Equations, 2012, 253: 1422-1438 doi: 10.1016/j.jde.2012.05.016
[2]	Wang B X. Weak pullback attractors for mean random dynamical systems in Bochner spaces. J Dynam Differential Equations, 2019, 31: 2177-2204 doi: 10.1007/s10884-018-9696-5
[3]	Wang R H, Wang B X. Random dynamics of $p$-Laplacian lattice systems driven by infinite-dimensional nonlinear noise. Stoch Proc Appl, 2020, 130: 7431-7462 doi: 10.1016/j.spa.2020.08.002
[4]	Wang R H, Wang B X. Random dynamics of lattice wave matrixs driven by infinite-dimensional nonlinear noise. Discrete Cont Dyn Syst Ser B, 2020, 25: 2461-2493
[5]	Wang X H, Kloeden P E, Han X Y. Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise. Nonl Differ Equ Appl, 2021, 28: 43
[6]	Yang S, Li Y R. Dynamics and invariant measures of multi-stochastic sine-Gordon lattices with random viscosity and nonlinear noise. J Math Phys, 2021, 25: 051510
[7]	Wang B X. Dynamics of fractional stochastic reaction-diffusion matrixs on unbounded domains driven by nonlinear noise. J Differential Equations, 2019, 268: 1-59 doi: 10.1016/j.jde.2019.08.007
[8]	Wang B X. Dynamics of stochastic reaction diffusion lattice systems driven by nonlinear noise. J Math Anal Appl, 2019, 477: 104-132 doi: 10.1016/j.jmaa.2019.04.015
[9]	Wang B X. Weak pullback attractors for stochastic Navier-Stokes matrixs with nonlinear diffusion terms. Proc Amer Math Soc, 2019, 147: 1627-1638 doi: 10.1090/proc/2019-147-04
[10]	Prévôt C, Röckner M. A concise Course on Stochastic Partial Differential Equations. Berlin: Springer, 2007
[11]	Kloeden P E, Simsen J. Attractors of asymptotically autonomous quasi-linear parabolic matrix with spatially variable exponents. J Math Anal Appl, 2015, 425: 911-918 doi: 10.1016/j.jmaa.2014.12.069
[12]	Kloeden P E, Simsen J, Simsen M S. Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents. J Math Anal Appl, 2017, 445: 513-531 doi: 10.1016/j.jmaa.2016.08.004
[13]	Li Y R, She L B, Yin J Y. Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau matrixs. Acta Math Scientia, 2018, 40B: 591-609
[14]	Li Y R, She L B, Wang R H. Asymptotically autonomous dynamics for parabolic matrix. J Math Anal Appl, 2018, 459: 1106-1123 doi: 10.1016/j.jmaa.2017.11.033
[15]	She L B, Wang R H. Regularity, forward-compactness and measurability of attractors for non-autonomous stochastic lattice systems. J Math Anal Appl, 2019, 479: 2007-2031 doi: 10.1016/j.jmaa.2019.07.038
[16]	Wang R H, Li Y R. Asymptotic autonomy of kernel sections for Newton-Boussinesq matrixs on unbounded zonary domains. Dynmics of PDE, 2019, 16: 295-316
[17]	Cui H Y, Kloeden P E, Wu F K. Pathwise upper semi-continuity of random pullback attractors along the time axis. Physica D, 2018, 16: 21-34
[18]	Li Y R, Yang S. Backward compact and periodic random attractors for non-autonomous Sine-Gordon matrixs with multiplicative noise. Commun Pure Appl Anal, 2019, 18: 1155-1175
[19]	Wang S L, Li Y R. Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics matrixs. Physica D, 2018, 382: 46-57
[20]	Roh J. Dynamics of the g-Navier-Stokes matrixs. J Differential Equations, 2005, 211: 452-484 doi: 10.1016/j.jde.2004.08.016
[21]	Gao X C, Gao H J. Existence and uniqueness of weak solutions to stochastic 3D Navier-Stokes matrixs with delays. Appl Math Letters, 2019, 95: 158-164 doi: 10.1016/j.aml.2019.03.037
[22]	Raugel G, Sell G R. Navier-stokes matrixs on thin 3D domains. I. Global attractors and global regularity of solutions. J Amer Math Soc, 1993, 6: 503-568
[23]	徐明月, 赵才地, Caraballo T. 三维不可压Navier-Stokes方程组轨道统计解的退化正则性. 数学物理学报, 2021, 41A(2): 336-344
	Xu M Y, Zhou C D, Caraballo T. Degenerate regularity of trajectory statistical solutions for the 3D incompressible Navier-Stokes matrixs. Acta Math Sci, 2021, 41A(2):336-344
[24]	Feng X L, You B. Random attractors for the two-dimensional stochastic g-Navier-Stokes matrixs. Stochastics, 2020, 92: 613-626 doi: 10.1080/17442508.2019.1642340
[25]	Li F Z, Li Y R. Asymptotic behavior of stochastic $g$-Navier-Stokes matrixs on a sequence of expanding domains. J Math Phys, 2019, 60: 061505
[26]	Li Y R, Yang S. Hausdorff sub-norm spaces and continuity of random attractors for bi-stochastic $g$-Navier-Stokes matrixs with respect to tempered forces. J Dynam Differential Equations, 2023, 35(1): 543-574 doi: 10.1007/s10884-021-10026-0
[27]	Brzeźniak Z, Li Y H. Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes matrixs on some unbounded domains. Trans Amer Math Soc, 2006, 358: 5587-5629 doi: 10.1090/S0002-9947-06-03923-7
[28]	Brzeźniak Z, Goldys B, Le Gia Q T. Random attractors for the stochastic Navier-Stokes matrixs on the 2D Unit Sphere. J Math Fluid Mech, 2018, 20: 227-253 doi: 10.1007/s00021-017-0351-4
[29]	Caraballo T, Łukaszewiczd G, Real J. Pullback attractors for non-autonomous 2D-Navier-Stokes matrixs in some unbounded domains. C R Math Acad Sci Paris, 2006, 342: 263-268 doi: 10.1016/j.crma.2005.12.015
[30]	Chueshov I, Millet A. Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl Math Optim, 2010, 61: 379-420 doi: 10.1007/s00245-009-9091-z
[31]	Guo B L, Guo C X, Pu X K. Random attractors for a stochastic hydrodynamical matrix in heisenberg paramagent. Acta Math Scientia, 2011, 31B: 529-540
[32]	Zhao C D, Li Y S, Zhou S F. Random attractor for a two-dimensional incompressible non-Newtonian fluid with multiplicative noise. Acta Math Scientia, 2011, 31B: 567-575
[33]	Caraballo T, Han X Y, Sehmalfuss B, Valero J. Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise. Nonlinear Anal, 2016, 130: 255-278 doi: 10.1016/j.na.2015.09.025
[34]	Gess B. Random attractors for stochastic porous media matrixs perturbed by space-time linear multiplicative noise. Annals Probab, 2014, 42: 818-864
[35]	黎定仕, 范英飞. 随机非局部扩散方程的随机吸引子的存在性. 数学物理学报, 2017, 37A(1): 158-172
	Li D S, Fan Y F. Random attractors for the stochastic nonclassical diffusion matrix on unbounded domains. Acta Math Sci, 2017, 37A(1): 158-172
[36]	Li Y R, Gu A H, Li J. Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian matrixs. J Differential Equations, 2015, 258: 504-534 doi: 10.1016/j.jde.2014.09.021
[37]	Li Y R, Yin J Y. A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo matrixs. Discrete Contin Dyn Syst Ser B, 2016, 21: 1203-1223 doi: 10.3934/dcdsb
[38]	Lu K N, Wang B X. Wong-Zakai approximations and long term behavior of stochastic partial differential matrixs. J Dynam Differential Equations, 2019, 31: 1341-1371 doi: 10.1007/s10884-017-9626-y
[39]	Zhao W Q. Regularity of random attractors for a stochastic degenerate parabolic matrixs driven by multiplicative noise. Acta Math Scientia, 2016, 36B: 409-427
[40]	Zhao W Q, Zhang Y J. Upper semi-continuity of random attractors for a non-autonomous dynamical system with a weak convergence condition. Acta Math Scientia, 2020, 40B: 921-933
[41]	Zhou S F, Wang Z J. Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise. J Math Anal Appl, 2016, 441: 648-667 doi: 10.1016/j.jmaa.2016.04.038
[42]	赵敏, 周盛凡. 带可加噪声的非自治随机Boussinesq格点方程的随机吸引子. 数学物理学报, 2018, 38A(5): 924-940
	Zhao M, Zhou S F. Random attractor for non-autonomous stochastic Boussinesq lattice matrixs with additive white noises. Acta Math Sci, 2018, 38A(5): 924-940

Backward $W$-compact Mean Dynamics for Stochastic ${g}$-Navier-Stokes Equations with Nonlinear Noise

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 42

Related Articles 2

Metrics

Comments

Recommended 10

[1]	Cai Gang. The Well-Posedness of Degenerate Differential Equations with Finite Delay in Banach Spaces [J]. Acta mathematica scientia,Series A, 2018, 38(2): 264-275.
[2]	DONG Ge, WEI Wen-Zhan. Noncreasy Orlicz-Bochner Function Spaces with Luxemburg Norm [J]. Acta mathematica scientia,Series A, 2009, 29(2): 416-422.