哈密顿偏微分方程中的小分母问题

A large number of partial differential equations of Physics share the structure of infinite-dimensional Hamiltonian system. The Schr?dinger equation, the wave equation, the Euler equations of hydrodynamics are all among this class. The investigation of periodic, quasi-periodic and almost periodic solution, the growth of sobolev norm leads to the well known "small divisor problem". We aim to get quasi periodic orbits with any kind of frequency structure, and develop a frame adapt to nonlinear perturbation problem with Liouville frequency. To enrich the result on existence of quasi periodic solution,almost periodic solution and so on, more special structure of given equation are needed and also the hidden nondegenerate condition of frequency drift. We also interested on the growth of sobolev norm, which gives more information on difussion of energy. In all these field there are many problems need to solve, our group have many good ideas now. Our approach will combine Nash-Moser Iteration, KAM theorem, normal form technical and so on.

物理中如薛定谔方程，波动方程、流体动力学中的欧拉方程等大量的偏微分方程都具有无穷维哈密顿系统结构。对它们的拟周期解与概周期解的存在性、索伯列夫范数的增长性等问题的研究最终都需要解决著名的"小分母"问题。我们计划研究具有各种频率特征的拟周期解，发展适用于刘维尔频率下非线性问题的迭代框架，对概周期解现有的结果进一步完善，对解的索伯列夫范数进行更深入的研究。这些问题的解决依赖于对频率漂移更加深刻的认识，挖掘其中所隐含的关于参数的非退化性；对方程本身的特定结构分析等；在这些方面我们已有比较好的结论与想法。我们的研究将涉及Nash-Moser迭代、KAM理论、正规形方法等各个方面。

物理中如薛定谔方程，波动方程、流体动力学中的欧拉方程等大量的偏微分方程都具有无穷维哈密顿系统结构。对它们的拟周期解与概周期解的存在性、索伯列夫范数的增长性等问题的研究最终都需要解决著名的"小分母"问题。我们计划研究具有各种频率特征的拟周期解，发展适用于刘维尔频率下非线性问题的迭代框架，对概周期解现有的结果进一步完善。这些问题的解决依赖于对频率漂移更加深刻的认识，挖掘其中所隐含的关于参数的非退化性；对方程本身的特定结构分析等；在这些方面我们已有比较好的结论与想法。我们的研究将涉及Nash-Moser迭代、KAM理论、正规形方法等各个方面。