CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS

doi:10.1007/s10473-024-0611-3

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2249-2273.doi: 10.1007/s10473-024-0611-3

CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS

Ningning LI, Jing ZHANG, Yongqian ZHANG^†

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

收稿日期:2023-08-09 修回日期:2024-04-29 发布日期:2024-12-06
通讯作者: † Yongqian ZHANG, E-mail: yongqianz@fudan.edu.cn
作者简介:Ningning LI, E-mail: nnli20@fudan.edu.cn; Jing ZHANG, E-mail: jing zhang@fudan.edu.cn
基金资助:
NSFC (11421061, 12271507) and the Natural Science Foundation of Shanghai (15ZR1403900).

CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS

Ningning LI, Jing ZHANG, Yongqian ZHANG^†

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received:2023-08-09 Revised:2024-04-29 Published:2024-12-06
Contact: † Yongqian ZHANG, E-mail: yongqianz@fudan.edu.cn
About author:Ningning LI, E-mail: nnli20@fudan.edu.cn; Jing ZHANG, E-mail: jing zhang@fudan.edu.cn
Supported by:
NSFC (11421061, 12271507) and the Natural Science Foundation of Shanghai (15ZR1403900).

摘要/Abstract

摘要： This paper studies the strong convergence of the quantum lattice Boltzmann (QLB) scheme for the nonlinear Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in $C([0,T];L^2(\R^1))$ to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero. In the proof, at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding quantum lattice Boltzmann scheme, which leads to the compactness of the set of the solutions for the quantum lattice Boltzmann scheme. Finally the limit of any convergent subsequence of the solutions for the quantum lattice Boltzmann scheme is shown to coincide with the strong solution to a Cauchy problem for the nonlinear Dirac equations.

关键词: lattice Boltzmann scheme, nonlinear Dirac equation, Gross-Neveu model, global strong solution, Glimm type functional

Abstract: This paper studies the strong convergence of the quantum lattice Boltzmann (QLB) scheme for the nonlinear Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in $C([0,T];L^2(\R^1))$ to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero. In the proof, at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding quantum lattice Boltzmann scheme, which leads to the compactness of the set of the solutions for the quantum lattice Boltzmann scheme. Finally the limit of any convergent subsequence of the solutions for the quantum lattice Boltzmann scheme is shown to coincide with the strong solution to a Cauchy problem for the nonlinear Dirac equations.

Key words: lattice Boltzmann scheme, nonlinear Dirac equation, Gross-Neveu model, global strong solution, Glimm type functional

中图分类号:

35L60

Ningning LI, Jing ZHANG, Yongqian ZHANG. CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS[J]. 数学物理学报(英文版), 2024, 44(6): 2249-2273.

Ningning LI, Jing ZHANG, Yongqian ZHANG. CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS[J]. Acta mathematica scientia,Series B, 2024, 44(6): 2249-2273.

参考文献

[1] Alvarez A.Linearized Crank-Nicholson schems for nonlinear Dirac equations. J Comput Phys, 1992, 99(2): 348-350
[2] Alvarez A, Kuo P Y, Vazquez L.The numerical study of a nonlinear one-dimensional Dirac model. Appl Math Comput, 1983, 13(1/2): 1-15
[3] Arrighi P, Nesme V, Forets M. The Dirac equation as a quantum walk: higher dimensions, observational convergence. J Phys A, 2014, 47(46): Art 465302
[4] Bony J M. Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension $1$ d'espace // Journées "Équations aux derivées partielles'': Held in Saint Jean de Monts. Palaiseau: École Polytechnique, 1987: Exp No XVI
[5] Bournaveas N, Zouraris G E.Theory and numerical approximations for a nonlinear $1+1$ Dirac system. ESAIM Math Model Numer Anal, 2012, 46(4): 841-874
[6] Bressan A.Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem. Oxford: Oxford University Press, 2000
[7] Cacciafesta F.Global small solutions to the critical radial Dirac equation with potential. Nonlinear Anal, 2011, 74(17): 6060-6073
[8] Candy T.Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension. Adv Differential Equations, 2011, 16(7/8): 643-666
[9] Contreras A, Pelinovsky D E, Shimabukuro Y.$L^2$ orbital stability of Dirac solitons in the massive Thirring model. Comm Partial Differential Equations, 2016, 41(2): 227-255
[10] Dafermos C M.Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer-Verlag, 2010
[11] Delgado V.Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension. Proc Amer Math Soc, 1978, 69(2): 289-296
[12] Dellar P J.Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms. ESAIM Proc Surveys, 2015, 52: 76-104
[13] Escobedo M, Vega L.A semilinear Dirac equation in $H^s(R^3)$ for $s>1$. SIAM J Math Anal, 1997, 28(2): 338-362
[14] Glimm J.Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697-715
[15] Gosse L.A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation. BIT, 2015, 55(2): 433-458
[16] Gross D J, Neveu A.Dynamical symmetry breaking in asymptotically free field theories. Phys Rev D, 1974, 10(10): 3235-3253
[17] Ha S Y, Tzavaras A E.Lyapunov functionals and $L^1$-stability for discrete velocity Boltzmann equations. Comm Math Phys, 2003, 239(1/2): 65-92
[18] Huh H.Global strong solution to the Thirring model in critical space. J Math Anal Appl, 2011, 381(2): 513-520
[19] Huh H.Global solutions to Gross-Neveu equation. Lett Math Phys, 2013, 103(8): 927-931
[20] Huh H, Moon B.Low regularity well-posedness for Gross-Neveu equations. Commun Pure Appl Anal, 2015, 14(5): 1903-1913
[21] Junk M, Yang Z.$L_2$ convergence of the lattice Boltzmann method for one dimensional convection-diffusion-reaction equations. Commun Comput Phys, 2015, 17(5): 1225-1245
[22] Junk M, Yong W.Weighted $L^2$ stability of the lattice Boltzmann method. SIAM J Numer Anal, 2009, 47(3): 1651-1665
[23] Lapitski D, Dellar P J.Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation. Phil Trans R Soc A, 2011, 369(1944): 2155-2163
[24] Maeda M. Asymptotic stability of small bound state of nonlinear quantum walks. Phys D, 2022, 439: Art 133408
[25] Maeda M, Suzuki A. Continuous limits of linear and nonlinear quantum walks. Rev Math Phys, 2020, 32(4): Art 2050008
[26] Palpacelli S, Romatschke P, Succi S. One-dimensional quantum lattice {B}oltzmann scheme for the nonlinear {D}irac equation. Internat J Modern Phys C, 2013, 24(12): Art 1340001
[27] Pelinovsky D.Survey on global existence in the nonlinear Dirac equations in one dimension// Ozawa T, Sugimoto M. Harmonic Analysis and Nonlinear Partial Differential
[28] Equations: Papers from the RIMS Symposium held at Kyoto University. Kyoto: RIMS, 2011: 37-50Selberg S, Tesfahun A. Low regularity well-posedness for some nonlinear Dirac equations in one space dimension. Differential Integral Equations, 2010, 23(3/4): 265-278
[29] Succi S.The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. New York: Clarendon Press, 2001
[30] Succi S, Benzi R.Lattice Boltzmann equation for quantum mechanics. Phys D, 1993, 69(3/4): 327-332
[31] Thirring W E.A soluble relativistic field theory. Ann Phys, 1958, 3: 91-112
[32] Zhang Y.Global strong solution to a nonlinear Dirac type equation in one dimension. Nonlinear Anal, 2013, 80: 150-155
[33] Zhang Y, Zhao Q.Global solution to nonlinear Dirac equation for Gross-Neveu model in $1+1$ dimensions. Nonlinear Anal, 2015, 118: 82-96

CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS

CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN $1+1$ DIMENSIONS

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

Metrics

本文评价

推荐阅读 10

[1]	Youmin CHEN, Lei LIU, Miaomiao ZHU. BUBBLING ANALYSIS FOR A NONLINEAR DIRAC EQUATION ON SURFACES[J]. 数学物理学报(英文版), 2024, 44(6): 2073-2082.
[2]	于海波. THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(1): 233-246.
[3]	吴国春, 钟新. GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM[J]. 数学物理学报(英文版), 2021, 41(5): 1428-1444.
[4]	穆鹏程, 琚强昌. THE CAUCHY PROBLEM FOR THE TWO LAYER VISCOUS SHALLOW WATER EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(6): 1783-1807.
[5]	张永前, 赵勤. LARGE TIME BEHAVIOR OF SOLUTION TO NONLINEAR DIRAC EQUATION IN 1+1 DIMENSIONS[J]. 数学物理学报(英文版), 2019, 39(2): 597-606.
[6]	张健, 唐先华, 张文. ON GROUND STATE SOLUTIONS FOR SUPERLINEAR DIRAC EQUATION[J]. 数学物理学报(英文版), 2014, 34(3): 840-850.