# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022065
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## Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations

 1 Department of Mathematics, Wenzhou University, Wenzhou, 325035, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei Province, 430071, China

* Corresponding author: Caidi Zhao

Received  December 2021 Early access March 2022

This paper studies the impulsive discrete Klein-Gordon-Schrödinger-type equations. We first prove that the problem of the discrete Klein-Gordon-Schrödinger-type equations with initial and impulsive conditions is global well-posedness. Then we establish that the solution operators form a continuous process and that this process possesses a pullback attractor and a family of invariant Borel probability measures. Further, we prove that this family of Borel probability measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Klein-Gordon-Schrödinger-type equations. Finally, we formulate the concept of Kolmogorov entropy for the statistical solution and estimate its upper bound.

Citation: Zehan Lin, Chongbin Xu, Caidi Zhao, Chujin Li. Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022065
##### References:
 [1] M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Differential Integral Equation, 16 (2003), 573-581. [2] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Application, John Wiley, New York, 1993. [3] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B., 10 (2008), 761-781.  doi: 10.3934/dcdsb.2008.10.761. [4] M. Chekroun and N. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761.  doi: 10.1007/s00220-012-1515-y. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. [6] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754. [7] B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differential Equations, 136 (1997), 356-377.  doi: 10.1006/jdeq.1996.3242. [8] H. Jiang and C. Zhao, Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth, Adv. Differential Equations, 26 (2021), 107-132. [9] P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802.  doi: 10.3934/cpaa.2009.8.785. [10] C. Li, C. Hsu, J. Lin and C. Zhao, Global attractors for the discrete Klein-Gordon-Schrödinger type equations, J. Differential Equ. Appl., 20 (2014), 1404-1426.  doi: 10.1080/10236198.2014.933821. [11] G. Lorentz, M. Golistschek and Y. Makovoz, Constructive Approximation, Advanced Problem, [Fundamental Principles of Mathematical Sciences], 304. Springer-Verlag, Berlin, 1996. [12] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827. [13] K. Lu and B. Wang, Upper semicontinuity of attractors for Klein-Gordon-Schrödinger equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 157-168.  doi: 10.1142/S0218127405012077. [14] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B., 9 (2008), 643-659.  doi: 10.3934/dcdsb.2008.9.643. [15] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [16] M. N. Poulou and N. M. Stavrakakis, Global attractor for a system of Klein-Gordon-Schrödinger type in all $\mathbb{R}$, Nonlinear Anal., 74 (2011), 2548-2562.  doi: 10.1016/j.na.2010.12.009. [17] C. Wang, G. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865.  doi: 10.1016/j.amc.2018.06.059. [18] S. Wu and J. Huang, Invariant measure and statistical solutions for non-autonomous discrete Klein-Gordon-Schrödinger-type equations, J. Appl. Anal. Comp., 10 (2020), 1516-1533.  doi: 10.11948/20190243. [19] X. Yan, Y. Wu and C. Zhong, Uniform attractors for impulsive reaction-diffusion equations, Appl. Math. Comp., 216 (2010), 2534-2543.  doi: 10.1016/j.amc.2010.03.095. [20] C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032. [21] C. Zhao, T. Caraballo and G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations, J. Differential Equations, 281 (2021), 1-32.  doi: 10.1016/j.jde.2021.01.039. [22] C. Zhao, H. Jiang and T. Caraballo, Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices,, Appl. Math. Comp., 404 (2021), Paper No. 126103, 14 pp. doi: 10.1016/j.amc.2021.126103. [23] C. Zhao, Y. Li and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011. [24] C. Zhao, Y. Li and G. Łukaszewicz, Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys., 71 (2020), Paper No. 141, 24 pp. doi: 10.1007/s00033-020-01368-8. [25] C. Zhao, Y. Li and Y. Sang, Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows, Z. Angew. Math. Mech., 100 (2020), e201800197, 15 pp. doi: 10.1002/zamm.201800197. [26] C. Zhao, Y. Li and Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Anal.-RWA, 53 (2020), 103077, 10 pp. doi: 10.1016/j.nonrwa.2019.103077. [27] C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equations for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp. doi: 10.1016/j.aml.2019.07.012. [28] C. Zhao, J. Wang and T. Caraballo, Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations, J. Differential Equations, 317 (2022), 474-494.  doi: 10.1016/j.jde.2022.02.007. [29] C. Zhao, G. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B., 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122. [30] C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.  doi: 10.1016/j.jmaa.2006.10.002. [31] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001. [32] S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277.  doi: 10.3934/dcds.2008.21.1259. [33] Z. Zhu, Y. Sang and C. Zhao, Pullback attractor and invariant measures for the discrete Zakharov equations, J. Appl. Anal. Comp., 9 (2019), 2333-2357.  doi: 10.11948/20190091. [34] Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete Cont. Dyn. Syst., 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060.

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##### References:
 [1] M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Differential Integral Equation, 16 (2003), 573-581. [2] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Application, John Wiley, New York, 1993. [3] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B., 10 (2008), 761-781.  doi: 10.3934/dcdsb.2008.10.761. [4] M. Chekroun and N. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761.  doi: 10.1007/s00220-012-1515-y. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. [6] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754. [7] B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differential Equations, 136 (1997), 356-377.  doi: 10.1006/jdeq.1996.3242. [8] H. Jiang and C. Zhao, Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth, Adv. Differential Equations, 26 (2021), 107-132. [9] P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802.  doi: 10.3934/cpaa.2009.8.785. [10] C. Li, C. Hsu, J. Lin and C. Zhao, Global attractors for the discrete Klein-Gordon-Schrödinger type equations, J. Differential Equ. Appl., 20 (2014), 1404-1426.  doi: 10.1080/10236198.2014.933821. [11] G. Lorentz, M. Golistschek and Y. Makovoz, Constructive Approximation, Advanced Problem, [Fundamental Principles of Mathematical Sciences], 304. Springer-Verlag, Berlin, 1996. [12] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827. [13] K. Lu and B. Wang, Upper semicontinuity of attractors for Klein-Gordon-Schrödinger equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 157-168.  doi: 10.1142/S0218127405012077. [14] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B., 9 (2008), 643-659.  doi: 10.3934/dcdsb.2008.9.643. [15] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [16] M. N. Poulou and N. M. Stavrakakis, Global attractor for a system of Klein-Gordon-Schrödinger type in all $\mathbb{R}$, Nonlinear Anal., 74 (2011), 2548-2562.  doi: 10.1016/j.na.2010.12.009. [17] C. Wang, G. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865.  doi: 10.1016/j.amc.2018.06.059. [18] S. Wu and J. Huang, Invariant measure and statistical solutions for non-autonomous discrete Klein-Gordon-Schrödinger-type equations, J. Appl. Anal. Comp., 10 (2020), 1516-1533.  doi: 10.11948/20190243. [19] X. Yan, Y. Wu and C. Zhong, Uniform attractors for impulsive reaction-diffusion equations, Appl. Math. Comp., 216 (2010), 2534-2543.  doi: 10.1016/j.amc.2010.03.095. [20] C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032. [21] C. Zhao, T. Caraballo and G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations, J. Differential Equations, 281 (2021), 1-32.  doi: 10.1016/j.jde.2021.01.039. [22] C. Zhao, H. Jiang and T. Caraballo, Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices,, Appl. Math. Comp., 404 (2021), Paper No. 126103, 14 pp. doi: 10.1016/j.amc.2021.126103. [23] C. Zhao, Y. Li and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011. [24] C. Zhao, Y. Li and G. Łukaszewicz, Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys., 71 (2020), Paper No. 141, 24 pp. doi: 10.1007/s00033-020-01368-8. [25] C. Zhao, Y. Li and Y. Sang, Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows, Z. Angew. Math. Mech., 100 (2020), e201800197, 15 pp. doi: 10.1002/zamm.201800197. [26] C. Zhao, Y. Li and Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Anal.-RWA, 53 (2020), 103077, 10 pp. doi: 10.1016/j.nonrwa.2019.103077. [27] C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equations for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp. doi: 10.1016/j.aml.2019.07.012. [28] C. Zhao, J. Wang and T. Caraballo, Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations, J. Differential Equations, 317 (2022), 474-494.  doi: 10.1016/j.jde.2022.02.007. [29] C. Zhao, G. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B., 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122. [30] C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.  doi: 10.1016/j.jmaa.2006.10.002. [31] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001. [32] S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277.  doi: 10.3934/dcds.2008.21.1259. [33] Z. Zhu, Y. Sang and C. Zhao, Pullback attractor and invariant measures for the discrete Zakharov equations, J. Appl. Anal. Comp., 9 (2019), 2333-2357.  doi: 10.11948/20190091. [34] Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete Cont. Dyn. Syst., 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060.
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