近可积理论在哈密顿系统经典模型上的应用

By using the two main theories of the Hamiltonian dynamics, the Arnold diffusion and the KAM method, I'm exploring some classical nearly integrable models, e.g. the celestial mechanics, the billiard map and the optimal transportation etc. It's expectable that for restricted N body problem, I will prove the existence of certain erratic orbits (the oscillatory orbits, diffusion orbits, collision orbits and so on), and give quantitative estimate on the measure, or topological structure of these orbits. As for the billiard map, I'm doing research on the relationship between the persistence of some characteristic caustics and the shape of the billiard boundaries, the so called ‘geometric rigidity' problem. In the optimal transport equation, I try to connect the control term with the quasi-convexity of the Hamiltonian functions, and verify the uniqueness of the general characteristics (namely the path of transportation). Partial results have been got in the aforementioned 3 branches, especially the first topic. I've gave the estimate of the density of the collision orbits for the restricted planar 3 body problem, which is the first result towards the Alexseev's Conjecture, after which has been proposed since 1970s. This result has been accepted by the journal Archive for Rational Mechanics and Analysis and will be published soon. The progress of other topics can be found from my newest posting on the arXiv.org, or from my old publications.

借助于哈密顿系统的两大传统理论：Arnold扩散与KAM方法，我将其运用到经典的近可积模型上，诸如天体力学、billiard映射以及最优传输方程等。预期将对于限制性多体问题验证特殊轨道（振荡轨，扩散轨、碰撞轨等）的存在性，以及给出有关的量级估计。在billiard映射上，将揭示某些特征不变换面的存在性与billiard边界之间的互相制约关系，即所谓的‘几何刚性’问题。在最优传输方程上，建立控制项与Hamilton函数的拟凸性之间的联系，并验证广义特征线，也即是传输路径的唯一性。目前已在上述三方面得到了一些成果，尤其是给出了限制性平面3体问题的碰撞轨道的密度估计，是Alexseev猜测50年来首个正面的结果。其论文已被ARMA杂志接收，其它方面的进展也已部分公布于arXiv网站，并稳步的向前推进。

该项目立足于若干哈密顿动力系统的经典问题，如天体力学、Billiard映射以及Hamilton-Jacobi PDE等，将若干前沿的理论方法进行大胆的结合，以揭示全新的动力学现象。近年来，国际上应用哈密顿摄动理论、Aubry Mather变分法、粘性解方法等对于哈密顿系统的定性研究取得了一系列突破性进展，多名数学家因这些课题上的突出工作而获得菲尔茨奖。该项目是针对哈密顿系统经典问题的技术性创新，将这些前沿方法进行有机结合以揭示经典问题中的新现象，在国际同领域属于方法论的先例。具体成果为限制性四体问题的振荡轨道构造、凸billiard映射的beta函数推导以及性质分析、切触消失的粘性解渐进性分析以及扭转映射的粘性解全局性质分析等， 相关成果分别发表于Arch.Ratio.Mech.Anal、JDE、Nonlinearity、JDDE、Sci.China.Ser.A等主流SCI期刊（均为通讯作者或独立作者身份）。这些成果不仅揭示了哈密顿经典问题的全新现象，更在理论层面验证了方法论创新的可行性与广阔前景， 受到了国内外专家的积极且广泛的引用评价。