2019 Volume 9 Issue 6
Article Contents

Zeqi Zhu, Yanmiao Sang, Caidi Zhao. PULLBACK ATTRACTORS AND INVARIANT MEASURES FOR THE DISCRETE ZAKHAROV EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2333-2357. doi: 10.11948/20190091
Citation: Zeqi Zhu, Yanmiao Sang, Caidi Zhao. PULLBACK ATTRACTORS AND INVARIANT MEASURES FOR THE DISCRETE ZAKHAROV EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2333-2357. doi: 10.11948/20190091

PULLBACK ATTRACTORS AND INVARIANT MEASURES FOR THE DISCRETE ZAKHAROV EQUATIONS

  • Corresponding author: Email address: zhaocaidi2013@163.com(C. Zhao)
  • Fund Project: The authors are supported by NSF of China(Nos. 51279202, 11271290) and by NSF of Zhejiang Province(No. LY17A010011)
  • This article studies the probability distributions of solutions in the phase space for the discrete Zakharov equations. The authors first prove that the generated process of the solutions operators possesses a pullback-${\mathcal D}$ attractor, and then they establish that there exists a unique family of invariant Borel probability measures supported by the pullback attractor.
    MSC: 35B41, 37Lxx, 35Qxx
  • 加载中
  • [1] Ahmed Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems, J. Differential Equations, 2011, 251, 1489-1504. doi: 10.1016/j.jde.2011.05.030

    CrossRef Google Scholar

    [2] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 2011, 11, 143-153.

    Google Scholar

    [3] W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 2003, 15, 485-515. doi: 10.1023/B:JODY.0000009745.41889.30

    CrossRef Google Scholar

    [4] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 2012, 253, 667-693. doi: 10.1016/j.jde.2012.03.020

    CrossRef Google Scholar

    [5] T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Syst.-B, 2014, 34, 51-77.

    Google Scholar

    [6] M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 2012, 316, 723-761. doi: 10.1007/s00220-012-1515-y

    CrossRef Google Scholar

    [7] S. N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 1995, 42, 746-751. doi: 10.1109/81.473583

    CrossRef Google Scholar

    [8] S. N. Chow, J. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyn., 1996, 4, 109-178.

    Google Scholar

    [9] S. N. Chow, Lattice dynamical systems, Lecture Notes in Math., 2003, 1822, 1-102.

    Google Scholar

    [10] L. Fabiny, P. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 1993, 47, 4287-4296. doi: 10.1103/PhysRevA.47.4287

    CrossRef Google Scholar

    [11] S. Fang, L. Jin and B. Guo, Existence of weak solution for quantum Zakharov equations for plasmas mode, Appl. Math. Mech., 2011, 32, 1339-1344. doi: 10.1007/s10483-011-1504-7

    CrossRef Google Scholar

    [12] S. Fang, L. Jin and B. Guo, Exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction, Acta Math. Sci., 2012, 32, 1073-1082. doi: 10.1016/S0252-9602(12)60080-0

    CrossRef Google Scholar

    [13] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.

    Google Scholar

    [14] Y. Guo, B. Guo and D. Li, The asymptotic behavior of solutions of the modified Zakharov equations with a quantum correction, Appl. Math. Mech., 2012, 33, 486-499.

    Google Scholar

    [15] C. Guo, S. Fang and B. Guo, Long time behavior of the solutions for the dissipative modified Zakharov equations for plasmas with a quantum correction, J. Math. Anal. Appl., 2013, 403, 183-192. doi: 10.1016/j.jmaa.2013.01.058

    CrossRef Google Scholar

    [16] X. Han, W. Shen and S. Zhou, 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

    CrossRef Google Scholar

    [17] X. Han and P. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 2016, 261, 2986-3009. doi: 10.1016/j.jde.2016.05.015

    CrossRef Google Scholar

    [18] X. Jia, C. Zhao and X. Yang, 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

    CrossRef Google Scholar

    [19] X. Jia, C. Zhao and X. Yang, Uniform attractor for discrete Selkov equations, Discrete Cont. Dyn. Syst.-A, 2014, 34), 229-248.

    Google Scholar

    [20] R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 1991, 6, 113-163. doi: 10.1007/BF01192578

    CrossRef Google Scholar

    [21] J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 1987, 47, 556-572. doi: 10.1137/0147038

    CrossRef Google Scholar

    [22] Y. Li, S. Wang and T. Zhao, The existence of pullback exponential attractors for nonautonomous dynamical system and applications to non-autonomous reaction diffusion equations, J. Appl. Anal. Comp., 2015, 5, 388-405.

    Google Scholar

    [23] Y. Li, S. Wang and T. Zhao, Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity, J. Appl. Anal. Comp., 2016, 6, 242-253.

    Google Scholar

    [24] Y. Liang, C. Li and C. Zhao, Compact kernel sections of the dissipative modified Zakharov equations for plasmas with a quantum correction on infinite lattices (in Chinese), Acta Math. Sci., 2014, 34, 1203-1218.

    Google Scholar

    [25] Y. Liang, Z. Guo, Y. Ying and C. Zhao, Finite dimensionality and upper semicontinuity of kernel sections for the discrete Zakharov equations, Bull. Malays. Math. Sci. soc., 2017, 40, 135-161. doi: 10.1007/s40840-016-0314-6

    CrossRef Google Scholar

    [26] G. Łukaszewicz, J. Real and J. C. Robinson, 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

    CrossRef Google Scholar

    [27] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyn. Syst.-A, 2014, 34, 4211-4222. doi: 10.3934/dcds.2014.34.4211

    CrossRef Google Scholar

    [28] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor equations, Springer, Vienna, 2002.

    Google Scholar

    [29] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 2006, 221, 224-245. doi: 10.1016/j.jde.2005.01.003

    CrossRef Google Scholar

    [30] X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst.-A, 2009, 23, 521-540.

    Google Scholar

    [31] Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Cont. Dyn. Syst.-A, 2015, 20, 1213-1230. doi: 10.3934/dcdsb.2015.20.1213

    CrossRef Google Scholar

    [32] C. Wang, G. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 2018, 339, 853-865. doi: 10.1016/j.amc.2018.06.059

    CrossRef Google Scholar

    [33] X. Yang, C. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schroinger-Boussinesq equations, Appl. Math. Comp., 2013, 219, 8508-8524. doi: 10.1016/j.amc.2013.01.053

    CrossRef Google Scholar

    [34] F. Yin, et al. Attractor for lattice system of dissipative Zakharov equation, Acta Math. Sinica., 2009, 25, 321-342. doi: 10.1007/s10114-008-5595-8

    CrossRef Google Scholar

    [35] C. Zhao and L. Yang, Pullback attractor and invariant measures for the non-autonomous globally modified Navier-Stokes equations, Comm. Math. Sci., 2017, 15, 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4

    CrossRef Google Scholar

    [36] C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractors and trajectory statistical solutions for three-dimensional globally modified Navier -Stokes equations, J. Differential Equations, 2019, 266, 7205-7229. doi: 10.1016/j.jde.2018.11.032

    CrossRef Google Scholar

    [37] C. Zhao, G. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B, 2018, 23, 4021-4044.

    Google Scholar

    [38] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Cont. Dyn. Syst.-B, 2008, 9, 763-785. doi: 10.3934/dcdsb.2008.9.763

    CrossRef Google Scholar

    [39] S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 2002, 179, 605-624. doi: 10.1006/jdeq.2001.4032

    CrossRef Google Scholar

    [40] S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 2003, 178, 51-61. doi: 10.1016/S0167-2789(02)00807-2

    CrossRef Google Scholar

    [41] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 2004, 200, 342-368. doi: 10.1016/j.jde.2004.02.005

    CrossRef Google Scholar

    [42] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 2006, 224, 172-204. doi: 10.1016/j.jde.2005.06.024

    CrossRef Google Scholar

    [43] S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 17, 263, 2247-2279.

    Google Scholar

    [44] Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete Cont. Dyn. Syst.-A, 2018, 38, 1461-1477. doi: 10.3934/dcds.2018060

    CrossRef Google Scholar

Article Metrics

Article views(2059) PDF downloads(378) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint