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Perturbation estimates of weak KAM solutions and minimal invariant sets for nearly integrable Hamiltonian systems


Authors: Qinbo Chen and Min Zhou
Journal: Proc. Amer. Math. Soc. 145 (2017), 201-214
MSC (2010): Primary 37Jxx, 70Hxx
DOI: https://doi.org/10.1090/proc/13193
Published electronically: June 30, 2016
MathSciNet review: 3565373
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Abstract | References | Similar Articles | Additional Information

Abstract: For nearly integrable and Tonelli system

$\displaystyle H_{\epsilon }=H_0(p)+\epsilon H_1(q,p,t). \quad (q,p,t)\in \mathbb{T}^n\times \mathbb{R}^n\times \mathbb{T},$

we give the perturbation estimates of weak KAM solution $ u_{\epsilon }$ with respect to parameter $ \epsilon $ and prove the stability of the Mather set $ \tilde {\mathcal {M}}_\epsilon $, Aubry set $ \tilde {\mathcal {A}}_\epsilon $, Mañé set $ \tilde {\mathcal {N}}_\epsilon $ and even the backward (forward) calibrated curves under the perturbation.

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  • [1] Diogo Aguiar Gomes, Regularity theory for Hamilton-Jacobi equations, J. Differential Equations 187 (2003), no. 2, 359-374. MR 1949445, https://doi.org/10.1016/S0022-0396(02)00013-X
  • [2] V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13-40. MR 0163025 (29 #328)
  • [3] P. Bernard, V. Kaloshin, and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders, preprint, arXiv:1112.2773 [math.DS] (2011).
  • [4] Patrick Bernard, Homoclinic orbits to invariant sets of quasi-integrable exact maps, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1583-1601. MR 1804946, https://doi.org/10.1017/S0143385700000870
  • [5] Patrick Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc. 21 (2008), no. 3, 615-669. MR 2393423, https://doi.org/10.1090/S0894-0347-08-00591-2
  • [6] David Bernstein and Anatole Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987), no. 2, 225-241. MR 880950, https://doi.org/10.1007/BF01388907
  • [7] Piermarco Cannarsa, Wei Cheng, and Qi Zhang, Propagation of singularities for weak KAM solutions and barrier functions, Comm. Math. Phys. 331 (2014), no. 1, 1-20. MR 3231994, https://doi.org/10.1007/s00220-014-2106-x
  • [8] Chong-Qing Cheng, Arnold diffusion in nearly integrable Hamiltonian systems, preprint, arXiv:1207.4016 [math.DS] (2012).
  • [9] Chong-Qing Cheng and Jinxin Xue, Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom, preprint, arXiv:1503.04153 [math.DS] (2015).
  • [10] Chong-Qing Cheng and Min Zhou, Global normally hyperbolic cylinders in Lagrangian systems, to appear in Math. Res. Lett.
  • [11] G. Contreras, R. Iturriaga, and H. Sánchez-Morgado, Weak solutions of the Hamilton-Jacobi equation for time periodic Lagrangians, preprint, arXiv:1307.0287 [math.DS] (2000).
  • [12] L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal. 157 (2001), no. 1, 1-33. MR 1822413, https://doi.org/10.1007/PL00004236
  • [13] Albert Fathi, Weak KAM theorem in Lagrangian dynamics, to be published by Cambridge University Press.
  • [14] Diogo Aguiar Gomes, Perturbation theory for viscosity solutions of Hamilton-Jacobi equations and stability of Aubry-Mather sets, SIAM J. Math. Anal. 35 (2003), no. 1, 135-147. MR 2001468, https://doi.org/10.1137/S0036141002405960
  • [15] V. Kaloshin and K. Zhang, A strong form of Arnold diffusion for two and a half degrees of freedom, preprint, arXiv:1212.1150 [math.DS] (2012).
  • [16] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527-530 (Russian). MR 0068687
  • [17] Zhenguo Liang, Jun Yan, and Yingfei Yi, Viscous stability of quasi-periodic tori, Ergodic Theory Dynam. Systems 34 (2014), no. 1, 185-210. MR 3163030, https://doi.org/10.1017/etds.2012.120
  • [18] Ricardo Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 (1996), no. 2, 273-310. MR 1384478, https://doi.org/10.1088/0951-7715/9/2/002
  • [19] John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169-207. MR 1109661, https://doi.org/10.1007/BF02571383
  • [20] John N. Mather and Giovanni Forni, Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991) Lecture Notes in Math., vol. 1589, Springer, Berlin, 1994, pp. 92-186. MR 1323222, https://doi.org/10.1007/BFb0074076
  • [21] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1-20. MR 0147741
  • [22] N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk 32 (1977), no. 6(198), 5-66, 287. MR 0501140 (58 #18570)
  • [23] Kaizhi Wang and Jun Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Comm. Math. Phys. 309 (2012), no. 3, 663-691. MR 2885604, https://doi.org/10.1007/s00220-011-1375-x
  • [24] Min Zhou, Hölder regularity of weak KAM solutions in a priori unstable systems, Math. Res. Lett. 18 (2011), no. 1, 75-92. MR 2770583, https://doi.org/10.4310/MRL.2011.v18.n1.a6

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Additional Information

Qinbo Chen
Affiliation: Department of Mathematics, Nanjing University, Nanjing, Jiangsu, People’s Republic of China 210093
Email: qinboChen1990@gmail.com

Min Zhou
Affiliation: School of Information Management, Nanjing University, Nanjing, Jiangsu, People’s Republic of China 210093
Email: minzhou@nju.edu.cn

DOI: https://doi.org/10.1090/proc/13193
Keywords: Weak KAM solutions, perturbation estimates, Mather theory, minimal invariant sets
Received by editor(s): December 7, 2015
Received by editor(s) in revised form: March 7, 2016
Published electronically: June 30, 2016
Additional Notes: The authors were supported by the National Basic Research Program of China (973 Program) (Grant No. 2013CB834100), the National Natural Science Foundation of China (Grant No. 11171146, Grant No. 11201222) and a program PAPD of Jiangsu Province, China.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society

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