一阶格点系统的不变测度与Liouville型方程

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 328-339.

一阶格点系统的不变测度与Liouville型方程

李永军¹,桑燕苗²,赵才地^2,*()

¹ 兰州城市学院数学学院兰州 730070
² 温州大学数学系浙江温州 325035

收稿日期:2019-03-07 出版日期:2020-04-26 发布日期:2020-05-21
通讯作者: 赵才地 E-mail:zhaocaidi2013@163.com; zhaocaidi@wzu.edu.cn
基金资助:
国家自然科学基金(11971356);国家自然科学基金(11761044);浙江省自然科学基金(LY17A010011);兰州城市学院重点建设项目(LZCUZDJSXK-201706)

Invariant Measures and Liouville Type Equation for First-Order Lattice System

Yongjun Li¹,Yanmiao Sang²,Caidi Zhao^2,*()

¹ School of Mathematics, Lanzhou City University, Lanzhou 730070
² Department of Mathematics, Wenzhou University, Zhejiang Wenzhou 325035

Received:2019-03-07 Online:2020-04-26 Published:2020-05-21
Contact: Caidi Zhao E-mail:zhaocaidi2013@163.com; zhaocaidi@wzu.edu.cn
Supported by:
the NSFC(11971356);the NSFC(11761044);the NSF of Zhejiang Province(LY17A010011);the Key Constructive Discipline of Lanzhou City University(LZCUZDJSXK-201706)

摘要/Abstract

摘要：

该文讨论一阶格点系统的解在相空间中的概率分布问题.作者先证明该格点系统的解算子生成的过程存在拉回吸引子，然后证明拉回吸引子上存在唯一的Borel不变概率测度，且该不变测度满足Liouville型方程.

关键词: 格点系统, 拉回吸引子, 不变测度, Liouville型方程

Abstract:

This article discusses the probability distribution of solutions in the phase space for the first-order lattice system. The authors first prove the the process generated by the solutions operator of the lattice system possesses a pullback attractor. Then they prove that there is a unique family of invariant probability measures contained in pullback attractor and the measures satisfy a Liouville type equation.

Key words: Lattice system, Pullback attractor, Invariant measure, Liouville type equation

中图分类号:

O175

李永军,桑燕苗,赵才地. 一阶格点系统的不变测度与Liouville型方程[J]. 数学物理学报, 2020, 40(2): 328-339.

Yongjun Li,Yanmiao Sang,Caidi Zhao. Invariant Measures and Liouville Type Equation for First-Order Lattice System[J]. Acta mathematica scientia,Series A, 2020, 40(2): 328-339.

参考文献 36

1	Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. Berlin: Springer, 1997
2	Feireisl E , Laurencot P , Simondon F . Global attarctor for degenetate parabolic equations unbounded domains. J Differential Equations, 1996, 129, 239- 261 doi: 10.1006/jdeq.1996.0117
3	Li Y , Wei J . Asymptotic dynamics for reaction diffusion equations in unbounded domain. J Appl Anal Comp, 2018, 8, 1186- 1193
4	Wang B . Attractors for reaction diffusion equations in unbounded domains. Physica D, 1999, 128, 41- 52 doi: 10.1016/S0167-2789(98)00304-2
5	Li C , Li D , Zhang Z . Dynamic bifurcation from infinity of nonlinear evolution equations. SIAM J Appl Dyn Syst, 2017, 16, 1831- 1868 doi: 10.1137/16M1107358
6	Bates P W , Lu K , Wang B . Attractors for lattice dynamical systems. Inter J Bifur Choas, 2001, 11 (1): 143- 153 doi: 10.1142/S0218127401002031
7	Chow S N. Lattice Dynamical Systems//Macki J W, Zecca P, et al. Dynamical Systems. Berlin: Springer, 2003: 1-102
8	Keener J P . Propagation and its failure in coupled systems of discrete excitable cells. SIAM J Appl Math, 1987, 47, 556- 572 doi: 10.1137/0147038
9	Erneux T , Nicolis G . Propagating waves in discrete bistable reaction diffusion systems. Physica D, 1993, 67, 237- 244 doi: 10.1016/0167-2789(93)90208-I
10	Fabiny L , Colet P , Roy R . Coherence and phase dynamics of spatially coupled solid-state lasers. Phys Rev A, 1993, 47, 4287- 4296 doi: 10.1103/PhysRevA.47.4287
11	Hillert M . A solid-solution model for inhomogeneous systems. Acta Metall, 1961, 9, 525- 535 doi: 10.1016/0001-6160(61)90155-9
12	Chow S N , Mallet Paret J , Van Vleck E S . Pattern formation and spatial chaos in spatially discrete evolution equations. Rand Comp Dyn, 1996, 4, 109- 178
13	Wang B . Dynamics of systems on infinite lattices. J Differential Equations, 2006, 221, 224- 245 doi: 10.1016/j.jde.2005.01.003
14	Zhou S , Shi W . Attractors and dimension of dissipative lattice systems. J Differential Equations, 2006, 224, 172- 204 doi: 10.1016/j.jde.2005.06.024
15	Wang B . Asymptotic behavior of non-autonomous lattice systems. J Math Anal Appl, 2007, 331, 121- 136 doi: 10.1016/j.jmaa.2006.08.070
16	Zhao C , Zhou S . Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices. J Math Anal Appl, 2007, 332, 32- 56 doi: 10.1016/j.jmaa.2006.10.002
17	Zhao X , Zhou S . Kernel sections for processes and nonautonomous lattice systems. Discrete Cont Dyn Syst-B, 2008, 9, 763- 785 doi: 10.3934/dcdsb.2008.9.763
18	Zhou S , Zhao C . Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Comm Pure Appl Anal, 2007, 21, 1087- 1111
19	Zhao C , Zhou S . Attractors of retarded first order lattice systems. Nonlinearity, 2007, 20, 1987- 2006 doi: 10.1088/0951-7715/20/8/010
20	Han X , Shen W , Zhou S . Random attractors for stochastic lattice dynamical systems in weighted spaces. J Differential Equations, 2011, 250, 1235- 1266 doi: 10.1016/j.jde.2010.10.018
21	Zhao C , Zhou S . Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications. J Math Anal Appl, 2009, 354, 78- 95 doi: 10.1016/j.jmaa.2008.12.036
22	Zhou S . Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J Differential Equations, 2017, 263, 2247- 2279 doi: 10.1016/j.jde.2017.03.044
23	Abdallah Ahmed Y . Exponential attractors for first-order lattice dynamical systems. J Math Anal Appl, 2008, 339, 217- 224 doi: 10.1016/j.jmaa.2007.06.054
24	Abdallah Ahmed Y . Uniform exponential attractors for first order non-autonomous lattice dynamical systems. J Differential Equations, 2011, 251, 1489- 1504 doi: 10.1016/j.jde.2011.05.030
25	赵才地, 周盛凡. 格点系统存在指数吸引子的充分条件及应用. 数学学报, 2010, 53, 233- 242 doi: 10.3969/j.issn.1005-3085.2010.02.006
	Zhao C , Zhou S . Sufficient conditions for the existence of exponential attractor for lattice system. Acta Math Sinica, 2010, 53, 233- 242 doi: 10.3969/j.issn.1005-3085.2010.02.006
26	Zhao C , Xue G , Łukaszewicz G . Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete Cont Dyn Syst-B, 2018, 23, 4021- 4044
27	Foias C , Manley O , Rosa R , Temam R . Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press, 2001
28	Wang X . Upper-semicontinuity of stationary statistical properties of dissipative systems. Discrete Cont Dyn Syst, 2009, 23, 521- 540
29	Lukaszewicz G , Real R , Robinson J C . Invariant measures for dissipative dynamical systems and generalised Banach limits. J Dyn Differential Equations, 2011, 23, 225- 250 doi: 10.1007/s10884-011-9213-6
30	Chekroun M , Glatt-Holtz N E . Invariant measures for dissipative dynamical systems:Abstract results and applications. Comm Math Phys, 2012, 316, 723- 761 doi: 10.1007/s00220-012-1515-y
31	Lukaszewicz G , Robinson J C . Invariant measures for non-autonomous dissipative dynamical systems. Discrete Cont Dyn Syst, 2014, 34, 4211- 4222 doi: 10.3934/dcds.2014.34.4211
32	Zhao C , Yang L . Pullback attractor and invariant measure for the globally modified Navier-Stokes equations. Comm Math Sci, 2017, 15, 1565- 1580 doi: 10.4310/CMS.2017.v15.n6.a4
33	赵才地, 李艳娇, 阳玲, 张明书. Ladyzhenskaya流体力学方程组的拉回吸引子与不变测度. 数学学报, 2018, 61, 823- 834 doi: 10.3969/j.issn.0583-1431.2018.05.012
	Zhao C , Li Y , Yang L , Zhang M . Pullback attractor and invariant measures for Ladyzhenskaya model. Acta Math Sinica, 2018, 61, 823- 834 doi: 10.3969/j.issn.0583-1431.2018.05.012
34	Zhu Z , Zhao C . Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete Cont Dyn Syst, 2018, 38, 1461- 1477 doi: 10.3934/dcds.2018060
35	Zhao C , Caraballo T . Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations. J Differential Equations, 2019, 66, 7205- 7229
36	郑志刚, 胡岗. 从动力学到统计物理学. 北京: 北京大学出版社, 2016
	Zheng Z , Hu G . From Dynamics to Statistical Mechanics. Beijing: Beijing University Press, 2016

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