半耗散格点Schrödinger方程组的统计解与Liouville型定理
Statistical Solutions of the Semi-Dissipative Lattice Schrödinger System of Equations and Liouville-Type Theorem
摘要: 本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型，模型中的Bose波函数具有耗散性，杂质波函数的能量守恒。作者首先证明该问题的整体适定性，然后研究Bose波函数在适当意义下拉回吸引子的存在性，接着应用该拉回吸引子和广义Banach极限构造统计解，并证明统计解满足Liouville型定理。
Abstract: In this paper, the pullback asymptotic behavior of solutions to the nonlinear Schrödinger system of equations with semi-dissipative lattices and their probability distributions are studied. The equations describe the Bose-Einstein condensation model with impurities, in which the Bose wave function is dissipative, and the energy of the impurity wave function is conserved. The authors first prove the global well-posed of the problem and then investigate the existence of a pullback attractor for the Bose wave function in a suitable sense. The authors then apply the pullback attractor and the generalized Banach limit to construct a statistical solution and show that the statistical solution satisfies the Liouville-type theorem.
文章引用：何乐乐, 赵才地. 半耗散格点Schrödinger方程组的统计解与Liouville型定理[J]. 应用数学进展, 2025, 14(5): 450-464. https://doi.org/10.12677/aam.2025.145274

参考文献

[1] Bourgain, J. (1999) Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society. [Google Scholar] [CrossRef] 
[2] Bourgain, J. and Bulut, A. (2014) Almost Sure Global Well-Posedness for the Radial Nonlinear Schrödinger Equation on the Unit Ball II: The 3d Case. Journal of the European Mathematical Society, 16, 1289-1325. [Google Scholar] [CrossRef] 
[3] Deng, Y., Nahmod, A.R. and Yue, H. (2020) Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrödinger Equation. Communications in Mathematical Physics, 384, 1061-1107. [Google Scholar] [CrossRef] 
[4] Deng, Y., Nahmod, A. and Yue, H. (2024) Invariant Gibbs Measures and Global Strong Solutions for Nonlinear Schrödinger Equations in Dimension Two. Annals of Mathematics, 200, 399-486. [Google Scholar] [CrossRef] 
[5] Carvalho, A.N., Langa, J.A. and Robinson, J.C. (2013) Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. Springer.
[6] Goubet, O. and Kechiche, W. (2010) Uniform Attractor for Non-Autonomous Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 10, 639-651. [Google Scholar] [CrossRef] 
[7] 杨新波, 赵才地, 贾晓琳. 自治耦合格点非线性 Schrödinger方程组的一致吸引子及熵的估计[J]. 数学物理学报, 2013, 33(4): 636-645.
[8] Carroll, T.L. and Pecora, L.M. (1993) Cascading Synchronized Chaotic Systems. Physica D: Nonlinear Phenomena, 67, 126-140. [Google Scholar] [CrossRef] 
[9] Firth, W.J. (1988) Optical Memory and Spatial Chaos. Physical Review Letters, 61, 329-332. [Google Scholar] [CrossRef] [PubMed]
[10] Keener, J.P. (1987) Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells. SIAM Journal on Applied Mathematics, 47, 556-572. [Google Scholar] [CrossRef] 
[11] Abdallah, A.Y. (2011) Uniform Exponential Attractors for First Order Non-Autonomous Lattice Dynamical Systems. Journal of Differential Equations, 251, 1489-1504. [Google Scholar] [CrossRef] 
[12] Bates, P.W., Lu, K. and Wang, B. (2014) Attractors of Non-Autonomous Stochastic Lattice Systems in Weighted Spaces. Physica D: Nonlinear Phenomena, 289, 32-50. [Google Scholar] [CrossRef] 
[13] Caraballo, T., Morillas, F. and Valero, J. (2012) Attractors of Stochastic Lattice Dynamical Systems with a Multiplicative Noise and Non-Lipschitz Nonlinearities. Journal of Differential Equations, 253, 667-693. [Google Scholar] [CrossRef] 
[14] Caraballo, T., Morillas, F. and Valero, J. (2014) On Differential Equations with Delay in Banach Spaces and Attractors for Retarded Lattice Dynamical Systems. Discrete & Continuous Dynamical Systems A, 34, 51-77. [Google Scholar] [CrossRef] 
[15] Zhou, S. and Han, X. (2012) Pullback Exponential Attractors for Non-Autonomous Lattice Systems. Journal of Dynamics and Differential Equations, 24, 601-631. [Google Scholar] [CrossRef] 
[16] Wang, X., Lu, K. and Wang, B. (2015) Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise. Journal of Dynamics and Differential Equations, 28, 1309-1335. [Google Scholar] [CrossRef] 
[17] Zhou, S. (2017) Random Exponential Attractor for Cocycle and Application to Non-Autonomous Stochastic Lattice Systems with Multiplicative White Noise. Journal of Differential Equations, 263, 2247-2279. [Google Scholar] [CrossRef] 
[18] Foias, C., Manley, O., Rosa, R. and Temam, R. (2001) Navier-Stokes Equations and Turbulence. Cambridge University Press. [Google Scholar] [CrossRef] 
[19] Foias, C. and Prodi, G. (1976) Sur les solutions statistiques des équations de Navier-Stokes. Annali di Matematica Pura ed Applicata, 111, 307-330. [Google Scholar] [CrossRef] 
[20] Vishik, M.I. and Fursikov, A.V. (1979) Translationally Homogeneous Statistical Solutions and Individual Solutions with Infinite Energy of a System of Navier-Stokes Equations. Siberian Mathematical Journal, 19, 710-729. [Google Scholar] [CrossRef] 
[21] Bronzi, A.C., Mondaini, C.F. and Rosa, R.M.S. (2014) Trajectory Statistical Solutions for Three-Dimensional Navier-Stokes-Like Systems. SIAM Journal on Mathematical Analysis, 46, 1893-1921. [Google Scholar] [CrossRef] 
[22] Bronzi, A.C., Mondaini, C.F. and Rosa, R.M.S. (2016) Abstract Framework for the Theory of Statistical Solutions. Journal of Differential Equations, 260, 8428-8484. [Google Scholar] [CrossRef] 
[23] Zhao, C., Li, Y. and Song, Z. (2020) Trajectory Statistical Solutions for the 3D Navier-Stokes Equations: The Trajectory Attractor Approach. Nonlinear Analysis: Real World Applications, 53, Article 103077. [Google Scholar] [CrossRef] 
[24] Zhao, C., Li, Y. and Caraballo, T. (2020) Trajectory Statistical Solutions and Liouville Type Equations for Evolution Equations: Abstract Results and Applications. Journal of Differential Equations, 269, 467-494. [Google Scholar] [CrossRef] 
[25] Zhao, C., Caraballo, T. and Łukaszewicz, G. (2021) Statistical Solution and Liouville Type Theorem for the Klein-Gordon-Schrödinger Equations. Journal of Differential Equations, 281, 1-32. [Google Scholar] [CrossRef] 
[26] Jiang, H. and Zhao, C. (2021) Trajectory Statistical Solutions and Liouville Type Theorem for Nonlinear Wave Equations with Polynomial Growth. Advances in Differential Equations, 26, 107-132. [Google Scholar] [CrossRef] 
[27] Zhao, C. and Zhuang, R. (2023) Statistical Solutions and Liouville Theorem for the Second Order Lattice Systems with Varying Coefficients. Journal of Differential Equations, 372, 194-234. [Google Scholar] [CrossRef] 
[28] Zhao, C. (2024) Absorbing Estimate Implies Trajectory Statistical Solutions for Nonlinear Elliptic Equations in Half-Cylindrical Domains. Mathematische Annalen, 391, 1711-1730. [Google Scholar] [CrossRef] 
[29] Chekroun, M.D. and Glatt-Holtz, N.E. (2012) Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications. Communications in Mathematical Physics, 316, 723-761. [Google Scholar] [CrossRef] 
[30] Łukaszewicz, G., Real, J. and Robinson, J.C. (2011) Invariant Measures for Dissipative Systems and Generalised Banach Limits. Journal of Dynamics and Differential Equations, 23, 225-250. [Google Scholar] [CrossRef] 
[31] Robinson, J.C. and Łukaszewicz, G. (2014) Invariant Measures for Non-Autonomous Dissipative Dynamical Systems. Discrete and Continuous Dynamical Systems, 34, 4211-4222. [Google Scholar] [CrossRef] 
[32] 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.
[33] Zhao, C., Wang, J. and Caraballo, T. (2022) Invariant Sample Measures and Random Liouville Type Theorem for the Two-Dimensional Stochastic Navier-Stokes Equations. Journal of Differential Equations, 317, 474-494. [Google Scholar] [CrossRef] 
[34] Zhao, C., Xue, G. and Łukaszewicz, G. (2018) Pullback Attractors and Invariant Measures for Discrete Klein-Gordon-Schrödinger Equations. Discrete & Continuous Dynamical Systems B, 23, 4021-4044. [Google Scholar] [CrossRef] 
[35] 邹天芳, 赵才地. 加权空间中一阶格点系统的统计解及其Kolmogorov熵[J]. 数学物理学报, 2023, 43(4): 1-17.
[36] 郑志刚, 胡岗. 从动力学到统计物理学[M]. 北京: 北京大学出版社, 2016.

为你推荐