Mather理论与弱KAM理论中的若干问题
|
摘要
Mather理论是近年来Hamilton系统研究中的一个突破,它在研究著名的Arnold扩散问题中显示了巨大的威力,其理论自身也是十分漂亮和有趣的。用Mather理论来研究Arnold扩散,其基本思路是寻找对应于不同上同调类的Mane集之间的连接轨道。其连接轨道的构造依赖于Mane集的拓扑结构。障碍函数B_c(m),(?)m∈M关于参数c∈H~1(m,R)的连续性在Mane集的拓扑结构的研究中起着关键的作用。另一方面,Hamilton-Jacobi方程的粘性解理论为Mather理论的研究提供了丰富的方法来源,两者相结合必将产生更加强有力的工具。
本文主要包含以下几个方面的结果:
一,Barrier函数关于参数的连续性和Mather极小测度的遍历性存在着深刻的联系。本文给出了Barrier函数B_c(m),(?)m∈M关于参数c连续的一个充分条件一Maher集唯一遍历。
二,本文构造出了一个反例,通过对该反例的具体分析,对Barrier函数关于参数的不连续性做了深入探讨。我们的研究表明:当Mather极小测度有多于1个遍历分支时,Barrier函数关于平均作用量变量通常是不连续的。
三,将我们对Barrier函数的研究应用于Hamilton-Jacobi方程粘性解的研究,我们得到了Hamilton-Jacobi方程粘性解在相差一个常数的意义下关于平均作用量变量连续性的充分条件。
Mather theory as a great breakthrough in Hamiltonian systems has shown significant influence in study of Arnold diffusion. It is also an interesting and elegant theory. The basic approach for studying Arnold diffusion is to find out the orbits which connect different Mane sets associating to different cohomology classes. The existence of connecting orbits depend upon the topological structure of the Mane sets. The continuity of the barrier function B_c(m) ((?) m∈M) with respect to c plays the key role in studying of the topological properties of Mane sets. On the other hand, the viscosity solution theory on the Hamilton-Jacobi equation provides us plenty of sources of methods to study the Mather theory. Combining these two methods will surely give us more powerful tools.
The main results of this thesis are as follows:
1, There are close relationships between the barrier function's continuity with respect to the parameter and the ergodic property of Mather's minimal measures. In this thesis we get a sufficient condition for barrier function's continuity with respect to the parameter------Mather set is uniquely ergodic.
2, We construct a counterexample and discuss deeply inside the discontinuity of barrier function with respect to the parameter through analyzing the counterexample carefully. The study indicates that the barrier function is discontinuity with respect to the average action variable when Mather's minimal measure has more than one ergodic components in general cases.
3, Applying the results of barrier function to the viscosity solution of Hamilton- Jacobi equation, we get a sufficient condition of the viscosity solution's continuity with respect to the average action variable in the sense of a constant difference.
本文主要包含以下几个方面的结果:
一,Barrier函数关于参数的连续性和Mather极小测度的遍历性存在着深刻的联系。本文给出了Barrier函数B_c(m),(?)m∈M关于参数c连续的一个充分条件一Maher集唯一遍历。
二,本文构造出了一个反例,通过对该反例的具体分析,对Barrier函数关于参数的不连续性做了深入探讨。我们的研究表明:当Mather极小测度有多于1个遍历分支时,Barrier函数关于平均作用量变量通常是不连续的。
三,将我们对Barrier函数的研究应用于Hamilton-Jacobi方程粘性解的研究,我们得到了Hamilton-Jacobi方程粘性解在相差一个常数的意义下关于平均作用量变量连续性的充分条件。
Mather theory as a great breakthrough in Hamiltonian systems has shown significant influence in study of Arnold diffusion. It is also an interesting and elegant theory. The basic approach for studying Arnold diffusion is to find out the orbits which connect different Mane sets associating to different cohomology classes. The existence of connecting orbits depend upon the topological structure of the Mane sets. The continuity of the barrier function B_c(m) ((?) m∈M) with respect to c plays the key role in studying of the topological properties of Mane sets. On the other hand, the viscosity solution theory on the Hamilton-Jacobi equation provides us plenty of sources of methods to study the Mather theory. Combining these two methods will surely give us more powerful tools.
The main results of this thesis are as follows:
1, There are close relationships between the barrier function's continuity with respect to the parameter and the ergodic property of Mather's minimal measures. In this thesis we get a sufficient condition for barrier function's continuity with respect to the parameter------Mather set is uniquely ergodic.
2, We construct a counterexample and discuss deeply inside the discontinuity of barrier function with respect to the parameter through analyzing the counterexample carefully. The study indicates that the barrier function is discontinuity with respect to the average action variable when Mather's minimal measure has more than one ergodic components in general cases.
3, Applying the results of barrier function to the viscosity solution of Hamilton- Jacobi equation, we get a sufficient condition of the viscosity solution's continuity with respect to the average action variable in the sense of a constant difference.
引文
[1]A.Fathi.Theoreme KAM faible et theorie de Mather sur les systemes lagrangiens.C.R.Acad.Sci.Paris Ser.I Math.1997,324:1043-1046.
[2]A.Fathi.Solutions KAM faibles conjuguees et barrieres de Peierls,C.R.Acad.Sci.Paris Ser.I Math.1997,325:649-652.
[3]A.Fathi.Sur la convergence du semi-groupe de Lax-Oleinik,C.R.Acad.Sci.Paris Ser.I Math.1998,327:267-270.
[4]Albert Fathi and Antonio Siconolfi.Existence of C~1 critical subsolutions of the Hamilton-Jacobi equation.Invent.Math..2003.
[5]Albert Fathi and Antonio Siconolfi.PDE aspects of Aubry Mather theory for continuous convex Hamiltonians.Calc.Var..2004.
[6]Albert Fathi.Weak KAM Theorem in Lagrangian Dynamics.The fifth Preliminary Version.
[7]Arnold V.I..Proof of a theorem of A.N.Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian.Russ Math Surv.1963,18:9-36.
[8]Arnold V.I..Instability of dynamical several degrees of freedom.Sovie Math Dokl.1964,5:581-585.
[9]Arnold V.I..Mathematical Methods of Classical Mechanics.Second edition.Springer-Verlag.1999.
[10]Aubry S..The twist map,the extended Frenkel-Kontorova model and the devil's staircase.Physica 7D 1983:240-158.
[11]Aubry S.,LeDaeron P.Y..The discrete Frenkel-Kontorova model and its extensions Ⅰ:exact results for ground states.Physica 8D 1983:381-422.
[12]Ball J.,Mizel V..One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation.Arch.Ration.Mech.Anal..1985,90:325-388.
[13]Bangert V..Minimal Geodesics.Ergod.Th.& Dynam.Sys..1989,10:263-286.
[14]Bernard Patrick.Connecting orbits of time dependent Lagrangeian systems.Ann.Inst.Fourier.2002,52(5):1533-1568.
[15]Bessi U..An approach to Aenol'd diffusion through calculus of variations.Nonlinear Analysis TMA.1996,26:1115-1135.
[16]Bessi U.,Chierchia L.,Valdinoci E..Upper bounds on Arnol'd diffusion time via Mather theory.J.Math Pure Appl..2001,80:105-129.
[17]Birkhoff.Surface transformations and their dynamical applications.Acta Math..1992,43:1-119.
[18]程崇庆.KAM理论与Arnold扩散:Hamilton系统的动力学稳定性问题.中国科学A辑,2004,34(3):257-367.
[19]程崇庆,孙义燧.哈密顿系统中的有序与无序运动.上海科技教育出版社.1996.
[20]Chong-Qing Cheng,Jun Yan.Existence of Diffusion Orbits in a Priori Unstable Hamiltonian Systems.Journal of Differential Geometry.2004,67:1-61.
[21]Cheng C.-Q.,Yan J..Arnold diffusion in Hamiltonian systems Ⅰ:a priori unatable case.Preprint.
[22]Cheng Wei.Birkhoff's Theorem and Viscosity Solution of Hamilton-Jacobi Equations,Science in China,Series A:Mathematics,Accepted.
[23]Chierchia L.,Gallavotti G..Drift and diffusion in phase space.Ann De I'IHP Section Phys Theorique.1994,60:1-144.
[24]Crandall M.G.,Evans L.C.and P.-L.Lions.Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans.Amer.Math.Soc..1984,282(2):487-502.
[25]Christian Bonatti,Lorenzo J.Diaz,Marcelo Viana.Dynamics Beyond Uniform Hyperbolicity—A Global Geometric and Probabilistic Perspective.国外数学名著系列(影印版),科学出版社2007.
[26]Delshams A.,de la Llave R,Seara T.M..Geometric mechanism for diffusion in Hamiltonian sysytems overcoming the large gap problem:heuristic and rigorous verification of a model.MP-ARC 03-182,2003.
[27]丁同仁,李承志.常微分方程教程.高等教育出版社,2004.
[28] E, W.. Aubry-Mather Theory and Periodic Solutions of the Forced Burgers Equation. Commun. on Pure and Appl. Math. Vol.LII. 1999, 811-828.
[29] Guy Barles. Solutions de viscosit(?) des equations de Hamilton-Jacobi. Springer-Verlag. Paris.1994.
[30] Herman M.. some open problems in dynamical systems. Doc. Math. J. DMV Extra Volume ICM. 1998, 2: 797-808.
[31] Hirsch M. W., C. C. Pugh, M. Shub. Invariant Manifolds, Lecture Notes in Mathematics (583). Berlin, Heidelberg, New York, Springer-Verlag. 1977.
[32] Kennan T. Smith. Primmer of modern analysis. New York, Springer-Verlag, second edition.1983.
[33] Kolmogorov A. N.. Th(?)rie g(?)n(?)rale des syst(?)mes dynamiques et m(?)canique classique. ICM Proceedings. 1954: 315-333.
[34] Lanford O.. Selected Topics in Functional Analysis. In: Dewitt, Stora (eds.): Statistical Mechanics and Quantum Field Theory. Proceedings of the Summer School of Theretical Physics, Les Houches 1970: 109-214.
[35] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions.CRC Press. Boca Raton. FL. 1992.
[36] Lawrence C. Evans. Partial differential equations. volume 19 of Fraduate Studies in Mathematics. American Mathematical Society. Providence. RI. 1998.
[37] L.C. & Gemes, D.. Effective Hamiltonians and Averaging for Hamiltonian Dynamics I, Arch.Rational Mech. Anal.. 2001, 157: 1-33.
[38] L.C. & Gemes, D.. Effective Hamiltonians and Averaging for Hamiltonian Dynamics II, Arch.Rational Mech. Anal.. 2002, 161: 271-305.
[39] Li X.X. & Yan J.. The continuity of Barrier function with respect to the parameter. Chinese Annals of Mathematics, Series B. 2009, 30(2): 145-153.
[40] Liang Zhenguo & Yan Jun. Regularity of Viscosity solutions near KAM torus. Science in China Series A. 2008, 51(3): 361-368.
[41]Liang Zhenguo & Yan Jun.Strong stability of KAM tori.accepted by Journal of Dynamics and Differential equations.
[42]Lochak P.In:Simo C,ed.Arnold diffusion;A Compendium of Remarks and Questions,Hamiltonian systems with three or more degrees of freedom.Dordrech,Boston,London:Kluwer Academic Publishers.1999:168-183.
[43]Martino Bardi and Italo Capuzzo-Dolcetta.Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations.Systems & Contral:Foundations & Applications.Birkh(a|¨)ser Boston Inc.,Boston.MA.1997.With appendices by Maurizio Falcone and Pierpaolo Soravia.
[44]Mather J.N..Action minimizing invariant measures for positive definite Lagrangian systems.Math.Z..1991,207:169-207.
[45]Mather J.N.,Forni G..Action minimizing orbits in Hamiltonian systems.In:Graffi(ed)Transition to Chaos in Classical and Quantum Mechanics.Springer LNM 1589,(1992) 92-186.
[46]Mather J.N..Variational construction of connecting orbits.ANNALES DE L'INSTITUT FOURIER.1993,43(5):1349-1386.
[47]Mather J.N..A property of compact,connected,laminated subsets of manifolds.Erg.Th.&Dynam.Sys..2002,22:1507-1520.
[48]Mather J.N..Total disconnectedness of the quotient Aubry set in low dimensions.Comm.Pure Appl.Math..2003,56:1507-1183.
[49]Mather J.N..Arnold diffusion,I:announcement of results.J.Maht.Sci..2004,124(5):169-207.
[50]Mather J.N..Examples of Aubry sets.Erg.Th.& Dynam.Sys..2004,24:1667-1723.
[51]Mather J.N..Variational construction of trajectories for time periodic Lagrangian systems on th etwo torus.131pp.,unfinished manuscript.
[52]Mane R..Generic properties and problems of minimizing measures of Lagrangian systems.Nonlinearity.1996,9:273-310.
[53]Mane R..Lagrangian flows:the dynamics of globally minimmizing orbits.Bol.Soc.Bras.Mat..1997,28:141-153.
[54]Michael G.Crandall and Pierre-Louis Lions.Viscosity solutions of Hamilton-Jacobi equations.Tans.Amer.Math.Soc..1983,277(1):1-42.
[55]Moser J..On invariant curves of area-preserving mappings of annulus.Nachr Akad Wiss G(o|¨)tt Math Phys.1962,K1:1-20.
[56]Nekhoroshev N.N..An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems.Russian Math.Surveys.1977,32(6):5-66.
[57]Walters P..An introduct to ergodic theory.GTM 79.1982.
[58]Xia Z..Arnold diffusion:A variatioal construction.Doc Math J.DMV Extra Volume ICM.1998,2:867-877.
[59]张锦炎,钱敏.微分动力系统导引.北京大学出版社,2000.
[2]A.Fathi.Solutions KAM faibles conjuguees et barrieres de Peierls,C.R.Acad.Sci.Paris Ser.I Math.1997,325:649-652.
[3]A.Fathi.Sur la convergence du semi-groupe de Lax-Oleinik,C.R.Acad.Sci.Paris Ser.I Math.1998,327:267-270.
[4]Albert Fathi and Antonio Siconolfi.Existence of C~1 critical subsolutions of the Hamilton-Jacobi equation.Invent.Math..2003.
[5]Albert Fathi and Antonio Siconolfi.PDE aspects of Aubry Mather theory for continuous convex Hamiltonians.Calc.Var..2004.
[6]Albert Fathi.Weak KAM Theorem in Lagrangian Dynamics.The fifth Preliminary Version.
[7]Arnold V.I..Proof of a theorem of A.N.Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian.Russ Math Surv.1963,18:9-36.
[8]Arnold V.I..Instability of dynamical several degrees of freedom.Sovie Math Dokl.1964,5:581-585.
[9]Arnold V.I..Mathematical Methods of Classical Mechanics.Second edition.Springer-Verlag.1999.
[10]Aubry S..The twist map,the extended Frenkel-Kontorova model and the devil's staircase.Physica 7D 1983:240-158.
[11]Aubry S.,LeDaeron P.Y..The discrete Frenkel-Kontorova model and its extensions Ⅰ:exact results for ground states.Physica 8D 1983:381-422.
[12]Ball J.,Mizel V..One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation.Arch.Ration.Mech.Anal..1985,90:325-388.
[13]Bangert V..Minimal Geodesics.Ergod.Th.& Dynam.Sys..1989,10:263-286.
[14]Bernard Patrick.Connecting orbits of time dependent Lagrangeian systems.Ann.Inst.Fourier.2002,52(5):1533-1568.
[15]Bessi U..An approach to Aenol'd diffusion through calculus of variations.Nonlinear Analysis TMA.1996,26:1115-1135.
[16]Bessi U.,Chierchia L.,Valdinoci E..Upper bounds on Arnol'd diffusion time via Mather theory.J.Math Pure Appl..2001,80:105-129.
[17]Birkhoff.Surface transformations and their dynamical applications.Acta Math..1992,43:1-119.
[18]程崇庆.KAM理论与Arnold扩散:Hamilton系统的动力学稳定性问题.中国科学A辑,2004,34(3):257-367.
[19]程崇庆,孙义燧.哈密顿系统中的有序与无序运动.上海科技教育出版社.1996.
[20]Chong-Qing Cheng,Jun Yan.Existence of Diffusion Orbits in a Priori Unstable Hamiltonian Systems.Journal of Differential Geometry.2004,67:1-61.
[21]Cheng C.-Q.,Yan J..Arnold diffusion in Hamiltonian systems Ⅰ:a priori unatable case.Preprint.
[22]Cheng Wei.Birkhoff's Theorem and Viscosity Solution of Hamilton-Jacobi Equations,Science in China,Series A:Mathematics,Accepted.
[23]Chierchia L.,Gallavotti G..Drift and diffusion in phase space.Ann De I'IHP Section Phys Theorique.1994,60:1-144.
[24]Crandall M.G.,Evans L.C.and P.-L.Lions.Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans.Amer.Math.Soc..1984,282(2):487-502.
[25]Christian Bonatti,Lorenzo J.Diaz,Marcelo Viana.Dynamics Beyond Uniform Hyperbolicity—A Global Geometric and Probabilistic Perspective.国外数学名著系列(影印版),科学出版社2007.
[26]Delshams A.,de la Llave R,Seara T.M..Geometric mechanism for diffusion in Hamiltonian sysytems overcoming the large gap problem:heuristic and rigorous verification of a model.MP-ARC 03-182,2003.
[27]丁同仁,李承志.常微分方程教程.高等教育出版社,2004.
[28] E, W.. Aubry-Mather Theory and Periodic Solutions of the Forced Burgers Equation. Commun. on Pure and Appl. Math. Vol.LII. 1999, 811-828.
[29] Guy Barles. Solutions de viscosit(?) des equations de Hamilton-Jacobi. Springer-Verlag. Paris.1994.
[30] Herman M.. some open problems in dynamical systems. Doc. Math. J. DMV Extra Volume ICM. 1998, 2: 797-808.
[31] Hirsch M. W., C. C. Pugh, M. Shub. Invariant Manifolds, Lecture Notes in Mathematics (583). Berlin, Heidelberg, New York, Springer-Verlag. 1977.
[32] Kennan T. Smith. Primmer of modern analysis. New York, Springer-Verlag, second edition.1983.
[33] Kolmogorov A. N.. Th(?)rie g(?)n(?)rale des syst(?)mes dynamiques et m(?)canique classique. ICM Proceedings. 1954: 315-333.
[34] Lanford O.. Selected Topics in Functional Analysis. In: Dewitt, Stora (eds.): Statistical Mechanics and Quantum Field Theory. Proceedings of the Summer School of Theretical Physics, Les Houches 1970: 109-214.
[35] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions.CRC Press. Boca Raton. FL. 1992.
[36] Lawrence C. Evans. Partial differential equations. volume 19 of Fraduate Studies in Mathematics. American Mathematical Society. Providence. RI. 1998.
[37] L.C. & Gemes, D.. Effective Hamiltonians and Averaging for Hamiltonian Dynamics I, Arch.Rational Mech. Anal.. 2001, 157: 1-33.
[38] L.C. & Gemes, D.. Effective Hamiltonians and Averaging for Hamiltonian Dynamics II, Arch.Rational Mech. Anal.. 2002, 161: 271-305.
[39] Li X.X. & Yan J.. The continuity of Barrier function with respect to the parameter. Chinese Annals of Mathematics, Series B. 2009, 30(2): 145-153.
[40] Liang Zhenguo & Yan Jun. Regularity of Viscosity solutions near KAM torus. Science in China Series A. 2008, 51(3): 361-368.
[41]Liang Zhenguo & Yan Jun.Strong stability of KAM tori.accepted by Journal of Dynamics and Differential equations.
[42]Lochak P.In:Simo C,ed.Arnold diffusion;A Compendium of Remarks and Questions,Hamiltonian systems with three or more degrees of freedom.Dordrech,Boston,London:Kluwer Academic Publishers.1999:168-183.
[43]Martino Bardi and Italo Capuzzo-Dolcetta.Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations.Systems & Contral:Foundations & Applications.Birkh(a|¨)ser Boston Inc.,Boston.MA.1997.With appendices by Maurizio Falcone and Pierpaolo Soravia.
[44]Mather J.N..Action minimizing invariant measures for positive definite Lagrangian systems.Math.Z..1991,207:169-207.
[45]Mather J.N.,Forni G..Action minimizing orbits in Hamiltonian systems.In:Graffi(ed)Transition to Chaos in Classical and Quantum Mechanics.Springer LNM 1589,(1992) 92-186.
[46]Mather J.N..Variational construction of connecting orbits.ANNALES DE L'INSTITUT FOURIER.1993,43(5):1349-1386.
[47]Mather J.N..A property of compact,connected,laminated subsets of manifolds.Erg.Th.&Dynam.Sys..2002,22:1507-1520.
[48]Mather J.N..Total disconnectedness of the quotient Aubry set in low dimensions.Comm.Pure Appl.Math..2003,56:1507-1183.
[49]Mather J.N..Arnold diffusion,I:announcement of results.J.Maht.Sci..2004,124(5):169-207.
[50]Mather J.N..Examples of Aubry sets.Erg.Th.& Dynam.Sys..2004,24:1667-1723.
[51]Mather J.N..Variational construction of trajectories for time periodic Lagrangian systems on th etwo torus.131pp.,unfinished manuscript.
[52]Mane R..Generic properties and problems of minimizing measures of Lagrangian systems.Nonlinearity.1996,9:273-310.
[53]Mane R..Lagrangian flows:the dynamics of globally minimmizing orbits.Bol.Soc.Bras.Mat..1997,28:141-153.
[54]Michael G.Crandall and Pierre-Louis Lions.Viscosity solutions of Hamilton-Jacobi equations.Tans.Amer.Math.Soc..1983,277(1):1-42.
[55]Moser J..On invariant curves of area-preserving mappings of annulus.Nachr Akad Wiss G(o|¨)tt Math Phys.1962,K1:1-20.
[56]Nekhoroshev N.N..An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems.Russian Math.Surveys.1977,32(6):5-66.
[57]Walters P..An introduct to ergodic theory.GTM 79.1982.
[58]Xia Z..Arnold diffusion:A variatioal construction.Doc Math J.DMV Extra Volume ICM.1998,2:867-877.
[59]张锦炎,钱敏.微分动力系统导引.北京大学出版社,2000.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |