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, Vol. 38 ›› Issue (5): 924-940.

带可加噪声的非自治随机Boussinesq格点方程的随机吸引子

赵敏^1,*(),周盛凡²

¹ 温州大学数理与电子信息工程学院浙江温州 325035
² 浙江师范大学数理与信息工程学院浙江金华 321004

收稿日期:2017-02-13 出版日期:2018-10-26 发布日期:2018-11-09
通讯作者: 赵敏 E-mail:zhaomin1223@126.com
基金资助:
国家自然科学基金(11471290);温州大学基金(135010121413)

Random Attractor for Non-Autonomous Stochastic Boussinesq Lattice Equations with Additive White Noises

Min Zhao^1,*(),Shengfan Zhou²

¹ College of Mathematics, Physics and Electronic Information Engineering, Wenzhou University, Zhejiang Wenzhou 325035
² College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Zhejiang Jinhua 321004

Received:2017-02-13 Online:2018-10-26 Published:2018-11-09
Contact: Min Zhao E-mail:zhaomin1223@126.com
Supported by:
the NSFC(11471290);the Foundation of Wenzhou University(135010121413)

摘要/Abstract

摘要：

该文研究具时变耦合系数的非自治随机Boussinesq格点系统同时受依时间确定性外力和可加白噪声影响时的渐近行为.首先证明非自治随机Boussinesq格点方程的解生成的连续余圈的随机吸引子的存在性.然后证明此系统的随机吸引子在噪声项系数趋于零时的上半连续性.

关键词: 随机Boussinesq格点方程, 连续余圈, 随机吸引子, 上半连续性, 可加白噪声

Abstract:

In this paper, we study the asymptomatic behavior of non-autonomous stochastic Boussinesq lattice equations with time-dependent coupled coefficients, time-dependent deterministic forces and additive white noises. Firstly, we prove the existence of a random attractor for the continuous cocycle generated by the solutions of the non-autonomous stochastic Boussinesq lattice equations. Lastly we establish the upper semi-continuity of random attractors for the random systems as the intensity of the noises tends to zero.

Key words: Stochastic Boussinesq lattice equations, Continuous cocycle, Random attractor, Upper semi-continuity, Additive white noise

中图分类号:

O193

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[J]. 数学物理学报, 2018, 38(5): 924-940.

Min Zhao,Shengfan Zhou. Random Attractor for Non-Autonomous Stochastic Boussinesq Lattice Equations with Additive White Noises[J]. Acta mathematica scientia,Series A, 2018, 38(5): 924-940.

参考文献

1	Jia Y , Zhang X , Dong B . Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Commun Pure Appl Anal, 2013, 12 (2): 923- 937
2	Wu G , Zheng X . Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. J Differential Equations, 2013, 255 (9): 2891- 2926
3	Yang X , Fan Z , Li K . Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity. Math Method Appl Sci, 2016, 39 (11): 3075- 3087
4	Yang Z . Longtime dynamics of the damped Boussinesq equation. J Math Anal Appl, 2013, 399 (1): 180- 190
5	Zhao L , Yang M , Zhang C . Dynamics for the damped Boussinesq equation with delay. Stoch Dynam, 2016, 16 (6): 1650021
6	耿范, 李瑞斋, 葛翔宇. 具阻尼项的Boussinesq型方程的长时间行为. 数学物理学报, 2016, 36A (6): 1196- 1210
6	Geng F , Li R , Ge X . Long time behavior of Boussinesq type equation with damping term. Acta Math Sci, 2016, 36A (6): 1196- 1210
7	Abdallah A . Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation. Abstr Appl Anal, 2005, 6: 655- 671
8	Zhao M , Zhou S . Exponential attractor for lattice system of nonlinear Boussinesq equation. Discrete Dyn Nat Soc, 2013, 2013: Art ID:869621
9	Zhao M , Zhou S . Pullback and uniform exponential attractors for nonautonomous Boussinesq lattice system. Internat J Bifur Chaos, 2015, 25 (8): 1550100
10	Zhao M , Zhou S . Random attractor of non-autonomous stochastic Boussinesq lattice system. J Math Phys, 2015, 56 (9): 092702
11	Crauel H , Debussche A , Flandoli F . Random attractors. J Dynam Differential Equations, 1997, 9 (2): 307- 341
12	Crauel H , Flandoli F . Attractors for random dynamical systems. Probab Theory Related Fields, 1994, 100 (3): 365- 393
13	Flandoli F , Schmalfuss B . Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch Stoch Rep, 1996, 59 (1/2): 21- 45
14	Imkeller P , Schmalfuss B . The conjugacy of stochastic and random differential equations and the existence of global attractors. J Dynam Differential Equations, 2001, 13 (2): 215- 249
15	Arnold L . Random Dynamical Systems. Berlin: Springer-Verlag, 1998
16	Zhao M , Zhou S . Random attractor for stochastic boissonade system with time-dependent deterministic forces and white noises. Discrete Cont Dyn-B, 2017, 22 (4): 1683- 1717
17	Caraballo T , Lu K . Attractors for stochastic lattice dynamical systems with a multiplication noise. Front Math China, 2008, 3 (3): 317- 335
18	Chueshov I . Monotone Random Systems Theory and Applications. Berlin: Springer-Verlag, 2002
19	Zhou S , Zhao M . Random attractors for damped non-autonomous wave equations with memory and white noise. Nonlinear Anal, 2015, 120: 202- 226
20	Shen Z , Zhou S , Shen W . One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J Differential Equations, 2010, 248 (6): 1432- 1457
21	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, 37 (1): 158- 172
21	Li D , Fan Y . Random attractors for the stochastic nonclassical diffusion equation on unbounded domains. Acta Math Sci, 2017, 37 (1): 158- 172
22	Wang B . Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differential Equations, 2012, 253 (5): 1544- 1583
23	Zhou S , Wei L . A random attractor for a stochastic second order lattice system with random coupled coefficients. J Math Anal Appl, 2012, 395 (1): 42- 55
24	Caraballo T , Langa J , Robinson J . Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commu Partial Differential Equations, 1998, 23 (9/10): 1557- 1581
25	Caraballo T , Langa J . On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dynam Cont Dis Ser A, 2003, 10 (4): 491- 513
26	Robinson J . Infinite-Dimensional Dynamical Systems. Cambridge: Cambridge University Press, 2001
27	Zhou S . Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises. Nonlinear Anal, 2012, 75 (5): 2793- 2805
28	Zhou S , Zhao C , Wang Y . Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete Contin Dyn Syst, 2008, 21 (4): 1259- 1277
29	Wang B . Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch Dyn, 14 (4): 1450009
30	Bates P , Lu K , Wang B . Attractors of non-autonomous stochastic lattice systems in weighted spaces. Physica D Nonlinear Phenomena, 2014, 289: 32- 50
31	Xiang X , Zhou S . Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term. J Difference Equ Appl, 2016, 22 (2): 235- 252
32	Pazy A . Semigroups of Linear Operator and Applications to Partial Differential Equations. New York: Springer-Verlag, 2007

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