# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021311
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## Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

 1 Department of Data science and Big data technology, Wenzhou University, Wenzhou, Zhejiang 325035, China 2 Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang 325035, China

* Corresponding author: Chunqiu Li

Received  August 2021 Revised  November 2021 Early access January 2022

In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-${\mathcal{D}}$ attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-${\mathcal{D}}$ attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.

Citation: Congcong Li, Chunqiu Li, Jintao Wang. Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021311
##### References:
 [1] A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030. [2] A. C. Bronzi, C. F. Mondaini and R. M. S. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), 1893-1921.  doi: 10.1137/130931631. [3] A. C. Bronzi, C. F. Mondaini and R. M. S. Rosa, Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), 8428-8484.  doi: 10.1016/j.jde.2016.02.027. [4] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. -B, 10 (2008), 761-781.  doi: 10.3934/dcdsb.2008.10.761. [5] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [6] M. D. Chekroun and N. E. 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. [7] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations, Evolution Equations and Control Theory, 1 (2012), 57-80.  doi: 10.3934/eect.2012.1.57. [8] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Phys. D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [9] L. G. Farah and A. Pastor, On the periodic Schrödinger-Boussinesq system, J. Math. Anal. Appl., 368 (2010), 330-349.  doi: 10.1016/j.jmaa.2010.03.007. [10] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754. [11] B. Guo and F. Chen, Finite dimensional behavior of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equation, Phys. D, 93 (1996), 101-118.  doi: 10.1016/0167-2789(95)00277-4. [12] B. Guo and X. Duo, The behavior of attractors for damped Schrödinger-Boussinesq equation, Comm. Nonlinear Sci. Numer. Simul., 6 (2001), 54-60.  doi: 10.1016/S1007-5704(01)90030-9. [13] L. Han, J. Zhang and B. Guo, Global well-posedness for the fractional Schrödinger-Boussinesq system, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 2644-2652.  doi: 10.1016/j.cnsns.2013.12.032. [14] X. Han and P. E. Kloden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [15] X. Han, P. E. Kloden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813. [16] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018. [17] Y. He, C. Li and J. Wang, Invariant measures and statistical solutions for the nonautonomous discrete modified Swift-Hohenberg equation, Bull. Malays. Math. Sci. Soc., 44 (2021), 3819-3837.  doi: 10.1007/s40840-021-01143-6. [18] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [19] 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. [20] C. Li and J. Wang, On the forward dynamical behaviour of nonautonomous lattice dynamical systems, J. Differ. Equ. Appl., 27 (2021), 1052-1080.  doi: 10.1080/10236198.2021.1962850. [21] Y. Li, On the initial boundary value problems for two dimensional systems of Zakharov equations and of complex-Schrödinger-real-Boussinesq equations, J. Partial Differential Equations, 5 (1992), 81-93. [22] Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations, J. Math. Anal. Appl., 205 (1997), 107-132.  doi: 10.1006/jmaa.1996.5148. [23] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Contin. Dyn. Syst. -B, 9 (2008), 643-659.  doi: 10.3934/dcdsb.2008.9.643. [24] G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.  doi: 10.1007/s10884-011-9213-6. [25] G. Łukaszewicz and J. C. Robinson, Invariant measures for nonautonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [26] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821. [27] 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. [28] J. Wang, X. Zhang and C. Zhao, Statistical solutions for a nonautonomous modified Swift-Hohenberg equation, Math. Methods Appl. Sci., 44 (2021), 14502-14516.  doi: 10.1002/mma.7719. [29] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Comm. Nonlinear Sci. Numer. Simul., 91 (2020), 105459, 14 pp.  doi: 10.1016/j.cnsns.2020.105459. [30] X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521. [31] Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete. Contin. Dyn. Syst. -B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213. [32] Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal. TMA, 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020. [33] S. Wu and J. Huang, Invariant measure and statistical solutions for nonautonomous discrete Klein-Gordon-Schrödinger type equations, J. Appl. Anal. Comput., 10 (2020), 1516-1533.  doi: 10.11948/20190243. [34] Q. Xiao and C. Q. Li, Invariant Borel probability measures for the discrete three component reversible Gray-Scott model, Acta Mathematica Scientia-Series A, 2 (2021), 523-537. [35] X. Yang, C. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger-Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524.  doi: 10.1016/j.amc.2013.01.053. [36] 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. [37] 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. [38] 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. [39] C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp.  doi: 10.1016/j.aml.2019.07.012. [40] C. Zhao, G. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst. -B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122. [41] C. Zhao and L. Yang, Pullback attractors and invariant measures for the globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4. [42] 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. [43] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643. [44] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. -B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763. [45] S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2. [46] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.

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##### References:
 [1] A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030. [2] A. C. Bronzi, C. F. Mondaini and R. M. S. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), 1893-1921.  doi: 10.1137/130931631. [3] A. C. Bronzi, C. F. Mondaini and R. M. S. Rosa, Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), 8428-8484.  doi: 10.1016/j.jde.2016.02.027. [4] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. -B, 10 (2008), 761-781.  doi: 10.3934/dcdsb.2008.10.761. [5] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [6] M. D. Chekroun and N. E. 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. [7] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations, Evolution Equations and Control Theory, 1 (2012), 57-80.  doi: 10.3934/eect.2012.1.57. [8] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Phys. D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [9] L. G. Farah and A. Pastor, On the periodic Schrödinger-Boussinesq system, J. Math. Anal. Appl., 368 (2010), 330-349.  doi: 10.1016/j.jmaa.2010.03.007. [10] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754. [11] B. Guo and F. Chen, Finite dimensional behavior of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equation, Phys. D, 93 (1996), 101-118.  doi: 10.1016/0167-2789(95)00277-4. [12] B. Guo and X. Duo, The behavior of attractors for damped Schrödinger-Boussinesq equation, Comm. Nonlinear Sci. Numer. Simul., 6 (2001), 54-60.  doi: 10.1016/S1007-5704(01)90030-9. [13] L. Han, J. Zhang and B. Guo, Global well-posedness for the fractional Schrödinger-Boussinesq system, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 2644-2652.  doi: 10.1016/j.cnsns.2013.12.032. [14] X. Han and P. E. Kloden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [15] X. Han, P. E. Kloden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813. [16] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018. [17] Y. He, C. Li and J. Wang, Invariant measures and statistical solutions for the nonautonomous discrete modified Swift-Hohenberg equation, Bull. Malays. Math. Sci. Soc., 44 (2021), 3819-3837.  doi: 10.1007/s40840-021-01143-6. [18] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [19] 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. [20] C. Li and J. Wang, On the forward dynamical behaviour of nonautonomous lattice dynamical systems, J. Differ. Equ. Appl., 27 (2021), 1052-1080.  doi: 10.1080/10236198.2021.1962850. [21] Y. Li, On the initial boundary value problems for two dimensional systems of Zakharov equations and of complex-Schrödinger-real-Boussinesq equations, J. Partial Differential Equations, 5 (1992), 81-93. [22] Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations, J. Math. Anal. Appl., 205 (1997), 107-132.  doi: 10.1006/jmaa.1996.5148. [23] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Contin. Dyn. Syst. -B, 9 (2008), 643-659.  doi: 10.3934/dcdsb.2008.9.643. [24] G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.  doi: 10.1007/s10884-011-9213-6. [25] G. Łukaszewicz and J. C. Robinson, Invariant measures for nonautonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [26] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821. [27] 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. [28] J. Wang, X. Zhang and C. Zhao, Statistical solutions for a nonautonomous modified Swift-Hohenberg equation, Math. Methods Appl. Sci., 44 (2021), 14502-14516.  doi: 10.1002/mma.7719. [29] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Comm. Nonlinear Sci. Numer. Simul., 91 (2020), 105459, 14 pp.  doi: 10.1016/j.cnsns.2020.105459. [30] X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521. [31] Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete. Contin. Dyn. Syst. -B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213. [32] Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal. TMA, 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020. [33] S. Wu and J. Huang, Invariant measure and statistical solutions for nonautonomous discrete Klein-Gordon-Schrödinger type equations, J. Appl. Anal. Comput., 10 (2020), 1516-1533.  doi: 10.11948/20190243. [34] Q. Xiao and C. Q. Li, Invariant Borel probability measures for the discrete three component reversible Gray-Scott model, Acta Mathematica Scientia-Series A, 2 (2021), 523-537. [35] X. Yang, C. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger-Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524.  doi: 10.1016/j.amc.2013.01.053. [36] 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. [37] 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. [38] 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. [39] C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp.  doi: 10.1016/j.aml.2019.07.012. [40] C. Zhao, G. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst. -B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122. [41] C. Zhao and L. Yang, Pullback attractors and invariant measures for the globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4. [42] 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. [43] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643. [44] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. -B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763. [45] S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2. [46] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.
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