离散的三分量可逆Gray-Scott模型的不变Borel概率测度

数学物理学报 ›› 2021, Vol. 41 ›› Issue (2): 523-537.

离散的三分量可逆Gray-Scott模型的不变Borel概率测度

肖巧懿(),李春秋*()

^{温州大学数学系浙江温州 325035}

收稿日期:2019-11-03 出版日期:2021-04-26 发布日期:2021-04-29
通讯作者: 李春秋 E-mail:xiaoqiaoyi1127@139.com;licqmath@tju.edu.cn
作者简介:肖巧懿, E-mail: xiaoqiaoyi1127@139.com
基金资助:
国家自然科学基金(11871368)

Invariant Borel Probability Measures for the Discrete Three Component Reversible Gray-Scott Model

Qiaoyi Xiao(),Chunqiu Li*()

^{Department of Mathematics, Wenzhou University, Zhejiang Wenzhou 325035}

Received:2019-11-03 Online:2021-04-26 Published:2021-04-29
Contact: Chunqiu Li E-mail:xiaoqiaoyi1127@139.com;licqmath@tju.edu.cn
Supported by:
the NSFC(11871368)

摘要/Abstract

摘要：

研究由非自治三分量可逆Gray-Scott模型在无穷格点上所生成过程的相关时均观测的Borel概率测度.首先，证明该过程存在拉回-$ {\cal D} $吸引子.进一步，建立由该拉回吸引子所支撑的唯一一族不变Borel概率测度的存在性.

关键词: 不变测度, 格点动力系统, 拉回吸引子, 非自治Gray-Scott模型

Abstract:

In this paper, we study the Borel probability measures that can be associated to the time averaged observation of the process generated by the nonautonomous three component reversible Gray-Scott model on infinite lattices. We first show that there exists a pullback-$ {\cal D} $ attractor for the process. Furthermore, we establish the existence of a unique family of invariant Borel probability measures carried by this pullback attractor.

Key words: Invariant measures, Lattice dynamical system, Pullback attractor, Nonautonomous Gray-Scott model

中图分类号:

O192

肖巧懿,李春秋. 离散的三分量可逆Gray-Scott模型的不变Borel概率测度[J]. 数学物理学报, 2021, 41(2): 523-537.

Qiaoyi Xiao,Chunqiu Li. Invariant Borel Probability Measures for the Discrete Three Component Reversible Gray-Scott Model[J]. Acta mathematica scientia,Series A, 2021, 41(2): 523-537.

参考文献 36

1	Bates P W , Lu K , Wang B . Attractors for lattice dynamical systems. Inter J Bifur Chaos, 2001, 11, 143- 153 doi: 10.1142/S0218127401002031
2	Beyn W J , Pilyugin S Yu . Attractors of reaction diffusion systems on infinite lattices. J Dyn Differential Equations, 2003, 15, 485- 515 doi: 10.1023/B:JODY.0000009745.41889.30
3	Babin A V , Vishik M I . Attractors of Evolution Equations. Amsterdam: North-Holland, 1992
4	Chow S N . Lattice dynamical systems. Lecture Notes in Math, 2003, 1822, 1- 102
5	Caraballo T , Łukaszewicz G , Real J . Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal, 2006, 64, 484- 498 doi: 10.1016/j.na.2005.03.111
6	Caraballo T , Łukaszewicz G , Real J . Pullback attractors for nonautonomous 2D-Navier-Stokes equations in some unbounded domains. C R Acad Sci Paris, Ser I, 2006, 342, 263- 268 doi: 10.1016/j.crma.2005.12.015
7	Chueshov I , Lasiecka I . Attractors for second-order evolution equations with a nonlinear damping. J Dyn Differential Equations, 2004, 16, 469- 512 doi: 10.1007/s10884-004-4289-x
8	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
9	Chepyzhov V V , Vishik M I . Attractors for Equations of Mathematical Physics. Providence, RI: Amer Math Soc, 2002
10	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
11	Foias C , Manley O , Rosa R , Temam R . Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press, 2001
12	García-Luengo J , Marín-Rubio P , Real J . Pullback attractors in V for nonautonomous 2D-Navier-Stokes equations and their tempered behavior. J Differential Equations, 2012, 252, 4333- 4356 doi: 10.1016/j.jde.2012.01.010
13	García-Luengo J , Marín-Rubio P , Real J . Pullback attractors for three-dimensional nonautonomous Navier-Stokes-Voigt equations. Nonlinearity, 2012, 25, 905- 930 doi: 10.1088/0951-7715/25/4/905
14	Gu A , Zhou S , Wang Z . Uniform attractor of nonautonomous three component reversible Gray-Scott system. Appl Math Comp, 2013, 219, 8718- 8729 doi: 10.1016/j.amc.2013.02.056
15	Jia X , Zhao C , Yang X . Global attractor and Kolmogorov entropy of three component reversible Gray-Scott model on infinite lattices. Appl Math Comp, 2012, 218, 9781- 9789 doi: 10.1016/j.amc.2012.03.036
16	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
17	Kloeden P E , Marín-Rubio P , Real J . Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Comm Pure Appl Anal, 2009, 8, 785- 802 doi: 10.3934/cpaa.2009.8.785
18	Ladyzhenskaya O . Attractors for Semigroups and Evolution Equations. Cambridge: Cambridge University Press, 1991
19	Łukaszewicz G . Pullback attractors and statistical solutions for 2D Navier-Stokes equations. Discrete Cont Dyn Syst B, 2008, 9, 643- 659 doi: 10.3934/dcdsb.2008.9.643
20	Li C , Li D , Ju X . On the forward dynamical behavior of nonautonomous systems. Discrete Cont Dyn Syst B, 2020, 25 (1): 473- 487
21	Łukaszewicz G , Robinson J C . Invariant measures for nonautonomous dissipative dynamical systems. Discrete Cont Dyn Syst, 2014, 34 (10): 4211- 4222 doi: 10.3934/dcds.2014.34.4211
22	Łukaszewicz G , Real J , 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
23	Mahara H , et al. Three-variable reversible Gray-Scott model. J Chem Phys, 2004, 121, 8968- 8972 doi: 10.1063/1.1803531
24	Pecora L M , Carroll T L . Synchronization in chaotic systems. Phys Rev Lett, 1990, 64, 821- 824 doi: 10.1103/PhysRevLett.64.821
25	Temam R . Infinite Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag, 1988
26	Wang X . Upper-semicontinuity of stationary statistical properties of dissipative systems. Discrete Cont Dyn Syst, 2009, 23, 521- 540
27	You Y . Dynamics of two-compartment Gray-Scott equations. Nonlinear Anal, 2011, 74, 1969- 1986 doi: 10.1016/j.na.2010.11.004
28	You Y . Dynamics of three-component reversible Gray-Scott model. Discrete Cont Dyn Syst B, 2010, 14 (4): 1671- 1688
29	You Y . Global attractor of the Gray-Scott equations. Comm Pure Appl Anal, 2008, 7, 947- 970 doi: 10.3934/cpaa.2008.7.947
30	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, 266, 7205- 7229 doi: 10.1016/j.jde.2018.11.032
31	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
32	Zhao C , Xue G , Łukaszewicz G . Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete Cont Dyn Syst B, 2018, 23 (9): 4021- 4044
33	Zhao C , Zhou S . Compact uniform attractors for dissipative lattice dynamical systems with delays. Discrete Cont Dyn Syst, 2008, 21 (2): 643- 663 doi: 10.3934/dcds.2008.21.643
34	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
35	Zhou S . Attractors and approximations for lattice dynamical systems. J Differential Equations, 2004, 200, 342- 368 doi: 10.1016/j.jde.2004.02.005
36	Zhou S . Attractors for first order dissipative lattice dynamical systems. Physica D, 2003, 178, 51- 61 doi: 10.1016/S0167-2789(02)00807-2

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