哈密顿偏微分方程的动力学行为研究

This project aims to study phenomena of dynamics of Hamiltonian Partial differential equation with KAM and Normal form method. The first problem we main focus is the preservation of response solution of schrodiner equation and harmonic equation with small perturbation. Based on these work, we will generalize the relevant result to high dimension liouvillean frequency in a suitable way, also we will try to study the phase transition of spectrum in suitable system. The second problem we are interested in is the growth of Sobolev norms, we will focus on also schrodinger equation and harmonic equation forced by Liouvillean frequency, the system we study will be linear and nolinear. The last problem we focus is the Infinite dimension KAM theory with dense normal frequency, we aim to obtain a general theory and apply it to partial differential equation.

本项目主要考虑利用KAM与正规形方法对哈密顿偏微分方程的动力学行为进行研究。首先我们将研究刘维尔频外力驱动下的薛定谔方程、调和振子方程等方程的Response解在小扰动下的保持性，以此为基础我们将把相关结果拓展到高维刘维尔频率驱动下的问题。与此同时我们将尝试考虑谱的相变的可能性。另外，我们将研究哈密顿偏微分方程解的索比列夫范数的增长性，我们主要侧重于线性与非线性的薛定谔方程和调和振子方程在刘维尔频率驱动下的相关问题。最后我们将关注稠法频情况下的无穷维KAM定理的研究，推动KAM理论的进一步发展和实际应用。

本项目主要利用KAM理论与正规形方法研究哈密顿偏微分方程的动力学行为。我们研究了刘维尔频外力驱动下的薛定谔方程方程的Response解在小扰动下的保持性，给出了刘伟尔频率下的无穷维KAM定理；考虑了1维拟周期驱动下的量子调和振子的可约性与Sobolev范数的增长性；证明了无穷维哈密顿系统在法向频率满足次线性增长性的条件下的KAM定理；给出了超可微条件下的形式正规化理论。我们的研究推动了KAM理论的进一步发展和实际应用。相关成果发表在Anal. PDE, J. Math. Pures Appl. ,Adv.Math., Math. Z. 和J. Funct. Anal.等国际知名期刊。