可换系统动力学与格点系统的扩散轨道
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摘要
近年来,许多数学家开始投身于Hamilton PDEs中的扩散的研究,这类方程可看成是具有无穷多个自由度的Hamilton方程,典型的是Schrodinger方程,波方程,Kdv方程等。最近由Tao等五人在[]一文中解决了色散方程中的扩散问题,这里“扩散”是指一类光滑解其能量从低频率态转移到高频率态,但是我们并不知道这种能量转换是否随着时间趋于无穷仍然被保持。对于其它类型的偏微分方程,至今尚不知道是否有类似的结果。研究偏微分方程中的扩散问题,一个自然的思路便是先研究相应的离散系统是否存在扩散现象。在[]一文中,他证明了对于Schrodinger方程的一种离散化方程,存在Arnold扩散。但是,此文中所构造的扰动非常特殊,所有的周期轨都保存了下来,从而形成了转移链,本质上,仍然是Arnold的例子。在[]一文中,他们证明了一个周期自由连接的单摆扰动系统存在扩散轨道,并且估计了扩散速度,该模型可看成是sine-Gorden方程的一种离散化。
本文的第三部分研究了可交换Hamilton系统在变分框架下动力学上的一些性质,证明了相应的Aubry集,Mane集分别为一致的,相应的Lax-Oleinik半群可交换,且具有公共的C1,1粘性下解,α函数具有拟线性性质等。
本文的第四部分考虑了一类周期格点系统,该系统为N个质点通过弹簧连成一圈,在未扰时,这N个质点并不是处于自由状态的,我们通过对该系统Mather集,Aubry集,Mane集的详尽分析,以及对不变柱面的刻画,寻找了一条合理的扩散路径,用变分法证明了扩散轨道的存在性,由于有[]一文中的正则性保证,我们在构造扩散轨道时,不需要通过c-等价来构造局部(全局)连接轨道,而是完全遵从Arnold机制,这与[]中所构造的扩散轨道图像是不一样的,他们所构造的扩散轨道有一部分是“贴着”柱面的。这样其实说明了Arnold扩散的机制可以为Arnold机制,也可不为Arnold机制,这两种方式可以并存。多自由度Hamilton系统动力学的复杂程度,由此也可窥见一斑。
Recently, many mathematicians begin to study the diffusion in Hamilto-nian PDEs, for example, Schrodinger equation, wave equation, Kdv equation, these all can be treated as infinite degrees of freedom Hamiltonian systems. A recent progress for the cubic deforcusing nonlinear Schrodinger equation has been made in [], where,"diffusion" is exhibited by smooth solutions for which the energy moves from lower order Fourier modes to higher order Fourier modes. However, one does not know whether such transition of energy persists as time goes to infinity. For other PDEs, we still don't know whether there exist similar diffusion phenomena. An effective way to study the diffusion in Hamiltonian PDEs is to study the lattice system first. In [], he proved in a discrete Schrodinger equation, there exists Arnold diffusion, but the pertur-bation is so special that all the periodic orbits are preserved, so the transition chain is preserved, essentially, it is of Arnold's type. In [], they proved that for a lattice of perturbation of couple pendulums, there exists diffusion and the speed is estimated. In this model, the lattice system can be considered as a discretness of the sine-Gorden equation, the pendulums are freely connected in the unperturbed system.
In the third section of this thesis, we studied the dynamics of two Hamil-tonians which are commutative from the variational point of view. We proved that the corresponding Aubry set, Mane set are the same separately, the corre-sponding Lax-Oleinik semigroup are commutative, they have a common viscosity subsolution of C1,1, and the a function is quasi-linear. In the fourth section of this thesis, we focus on studying a periodic lattice system which can be thought as N particals with springs connecting from the first one to the last one, and the last one is also connected with the first one. In the unperturbed system, these springs are not free. Through the concrete study of the Mather set, Aubry set, Mane set of this model, and the location of the invariant cylinder, we find a reasonable path which the diffusion can occur, using the variational method, we proved the existence of the diffusion orbit. By the regularity of the barrier functions in [], we needn't use the c-equivalence to construct the local(global) connecting orbits, the diffusion orbits we construct obey the Arnold mechanism, this is different from the one constructed in [], where part of each diffusion orbit walks along the invariant cylinder. In fact, the mechanism of Arnold diffusion can be Arnold mechanism, or non-Arnold mechanism, from this we can see that the dynamics of multi-degrees of freedom of Hamiltonian system are very complicated.
本文的第三部分研究了可交换Hamilton系统在变分框架下动力学上的一些性质,证明了相应的Aubry集,Mane集分别为一致的,相应的Lax-Oleinik半群可交换,且具有公共的C1,1粘性下解,α函数具有拟线性性质等。
本文的第四部分考虑了一类周期格点系统,该系统为N个质点通过弹簧连成一圈,在未扰时,这N个质点并不是处于自由状态的,我们通过对该系统Mather集,Aubry集,Mane集的详尽分析,以及对不变柱面的刻画,寻找了一条合理的扩散路径,用变分法证明了扩散轨道的存在性,由于有[]一文中的正则性保证,我们在构造扩散轨道时,不需要通过c-等价来构造局部(全局)连接轨道,而是完全遵从Arnold机制,这与[]中所构造的扩散轨道图像是不一样的,他们所构造的扩散轨道有一部分是“贴着”柱面的。这样其实说明了Arnold扩散的机制可以为Arnold机制,也可不为Arnold机制,这两种方式可以并存。多自由度Hamilton系统动力学的复杂程度,由此也可窥见一斑。
Recently, many mathematicians begin to study the diffusion in Hamilto-nian PDEs, for example, Schrodinger equation, wave equation, Kdv equation, these all can be treated as infinite degrees of freedom Hamiltonian systems. A recent progress for the cubic deforcusing nonlinear Schrodinger equation has been made in [], where,"diffusion" is exhibited by smooth solutions for which the energy moves from lower order Fourier modes to higher order Fourier modes. However, one does not know whether such transition of energy persists as time goes to infinity. For other PDEs, we still don't know whether there exist similar diffusion phenomena. An effective way to study the diffusion in Hamiltonian PDEs is to study the lattice system first. In [], he proved in a discrete Schrodinger equation, there exists Arnold diffusion, but the pertur-bation is so special that all the periodic orbits are preserved, so the transition chain is preserved, essentially, it is of Arnold's type. In [], they proved that for a lattice of perturbation of couple pendulums, there exists diffusion and the speed is estimated. In this model, the lattice system can be considered as a discretness of the sine-Gorden equation, the pendulums are freely connected in the unperturbed system.
In the third section of this thesis, we studied the dynamics of two Hamil-tonians which are commutative from the variational point of view. We proved that the corresponding Aubry set, Mane set are the same separately, the corre-sponding Lax-Oleinik semigroup are commutative, they have a common viscosity subsolution of C1,1, and the a function is quasi-linear. In the fourth section of this thesis, we focus on studying a periodic lattice system which can be thought as N particals with springs connecting from the first one to the last one, and the last one is also connected with the first one. In the unperturbed system, these springs are not free. Through the concrete study of the Mather set, Aubry set, Mane set of this model, and the location of the invariant cylinder, we find a reasonable path which the diffusion can occur, using the variational method, we proved the existence of the diffusion orbit. By the regularity of the barrier functions in [], we needn't use the c-equivalence to construct the local(global) connecting orbits, the diffusion orbits we construct obey the Arnold mechanism, this is different from the one constructed in [], where part of each diffusion orbit walks along the invariant cylinder. In fact, the mechanism of Arnold diffusion can be Arnold mechanism, or non-Arnold mechanism, from this we can see that the dynamics of multi-degrees of freedom of Hamiltonian system are very complicated.
引文
[Ch1]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34(3),(2004),257-267.
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[CCC]X.J. Cui, C.-Q. Cheng, W. Cheng, Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems,Volume 214(2005), 176 188.
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[LC]X.Li and C.-Q.Cheng, Connecting orbits of autonomous Lagrangian systems. Nonlinearity.23 (2010),119-141.
[Li]Y.Charles Li, Arnold diffusion of the discrete nonlinear Schrodinger equation. Dynamics of PDE.3 (2006),235-258.
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[Ar1]V.I.Arnold, Proof of A.N.Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian. Russian Math. Surveys.18 (1963),9-36.
[Ar2]V.I. Arnold, Small divisor problems in classical and celestial mechanics. Russian Math. Surveys.18 (1963),85-192.
[Ar3]V.I.Arnold, Instability of dynamical systems with several degrees of freedom. Soviet Math.5 (1964),581-585.
[Ar4]V.I.Arnold, Mathematical methods of classical mechanics. Springer-Verlag, New York.1978.
[Ar5]V.I. Arnold, Geometric methods in the theory of ordinary differential equations. Springer-Verlag, New York.1983.
[BP]M.Bialy and L.Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem. Math. Ann.292 (1992),619-627.
[Ba]V.Bangert, Mather sets for twist maps and geodesics on tori. Dynamics Reported 1 (1988),1-45.
[BC]P. Bernard and G. Contreras, A geniric property of families of Lagrangian systems, Ann. Math.167(2008) 1099-1108.
[Bel]U.Bessi, An approach to Arnold's diffusion through the calculus of variations. Nonlinear Analysis, T.M.A.26 (1996),1115-1135. Ergod.
[Be2]U.Bessi, An analytic counterexample to KAM theorem Th.& Dynam. Sys.20 (2000),317-333.
[Berl]P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier 52(2002),1533-1568.
[Ber2]P. Bernard, Homoclinic orbits to invariant sets of quasi-integrable exact maps, Ergodic Theory Dynamical System,20(2000),1583 1601.
[Bol]S. Bolotin, Homoclinic orbits to invariant tori of Hamilton systems,Amer. Math. Soc. Trans. Ser.2,168 (1995),21 90
[Bo2]S. Bolotin, Symbolic dynamics near minimal hyperbolic invariant tori of Lagrangian systems, Nonlinearity,14 (2001),1123-1140
[Bou]J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrodinger equations. Ann. Math.148(1998),363-439.
[Ca]M.J.Dias Carneiro, On the minimizing measures of the action of autonomous Lagrangians. Nolinearity.8(1995),1077-1085.
[CCC]X.J. Cui, C.-Q. Cheng, W. Cheng, Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems,Volume 214(2005), 176 188.
[CDI]G. Contreras, J.Delgado and R.Iturriaga, Lagrangian flows:the dynamics of globally minimizing orbits. Ⅱ. Bol. Soc. Brasil. Math.28 (1997),155-196.
[CG]L. Chierchia and G. Gallavotti, Drift and diffusion in phase space. Ann. Ins. H. Poincare, Physique Theorique 60(1994),1-144.
[CI]G.Contreras and R.Iturriaga, Global minimizers of autonomous Lagrangians. 22°Coloquio Brasileiro de Matematica. [22nd Brazilian Mathematics Collo-quium]. Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro,1999.
[CIPP]G.Contreras, R.Iturriaga, G.P.Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mane's critical values. Geom. Funct. Anal.8 (1998), 788-809.
[CKSTT]J.Colliander, M.Keel, G.Staffilani, H.Takaoka and T.Tao, Transfer of energy to high frequencies in the cubic defocousing nonlinear Schrodinger equation. Invent. Math.181 (2010),39-113.
[CP]G.Contreras and G.P.Paternain, Connecting orbits between static classes for generic Lagrangian systems. Topology.41 (2002),645-666.
[CS]C-Q. Cheng and Y-S. Sun,Existence of invariant tori in three dimensional measure-preserving mappings. Celestial Mechanics 47(1989/1990),275-292.
[CaSi]P. Cannarsa and C. Sinestrari,Semiconcave functions, Hamilton-Jacobi equa-
tions, and optimal control. Progress in nonlinera differential equations and their applications.vol 58.
[CW]W.Craig and C.E.Wanye, Newton's method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math.46(1993),1409-1498.
[CY1]C-Q. Cheng and J.Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differential Geom.67 (2004),457-517.
[CY2]C-Q. Cheng and J.Yan, Arnold diffusion in Hamiltonian Systems:a priori unstable case. J. Differential Geom.82 (2009),229-277.
[DLS]A.Delshams, R.de la Llave and T.M.Seara, Geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem:heuristic and rigorous verification of a model. Momoirs Amer. Math. Soc.179 (2006)
[Fo]G.Forni, Analytic destruction of invariant circles. Ergod. Th.& Dynam. Sys. 14 (1994),267-298.
[FGS]A.Fathi, A.Giuliant and A.Sorrentino. Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class. Ann. Sc. Norm. Super. Pisa Cl. Sci.8 (2009),659-680.
[Fa]A. Fathi, Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, Cambridge University Press,2009.
[FM]A.Fathi and J.N.Mather, Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case. Bull. Soc. Math. France.128 (2000),473-483.
[FoM]G.Forni and J.N.Mather, Action minimizing orbits in Hamiltonian systems. In Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), Lecture Notes in Math.1589 (1994),92-186.
[Hed]G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math.33(1932),719-739.
[HPS]M.W.Hirsh, C.C.Pugh & M.Shub, Invariant manifold. Lect. Notes Math.583 (1977), Springer- Verlag.
[It]R. Iturriaga, Minimizing measures for time dependent Lagrangiana Proc. London Math. Soc.73(1996),216-240.
[KLS]V.Kaloshin, M.Levi & M.Saprykina. Arnold diffusion in a pendulum lattice. preprint (2011).
[Kul]S.Kuksin, Analysis of Hamiltonian PDEs. Oxford Lect. Ser. Math. Appl. 19,2000.
[Ku2]S.Kuksin, Hamiltonian PDEs. Handbook of Dynamical systems 1B(2006), 1087-1133.
[LC]X.Li and C.-Q.Cheng, Connecting orbits of autonomous Lagrangian systems. Nonlinearity.23 (2010),119-141.
[Li]Y.Charles Li, Arnold diffusion of the discrete nonlinear Schrodinger equation. Dynamics of PDE.3 (2006),235-258.
[Lo1]P.Lochak, Canonical perturbation theorey via simultaneous approximation. Russian Math. Surveys 47(1992),57-133.
[Lo1]P.Lochak, 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.
[Ma1]J.N.Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus. Topology 21 (1982),457-467.
[Ma2]J.N.Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv.60(1985),508-557.
[Ma3]J.N.Mather, A criterion for the non-existence of invariant circle. Publ. Math. IHES 63 (1986),301-309.
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