Arnol'd diffusion, Growth of Sobolev norms, Spectral rigidity for convex billiards

在数学和物理的许多领域中，分析系统的动力学行为是很重要的，这些系统保存了一些重要的物理量，如能量，质量或体积。 这些被称为保守系统，保守系统的一种自然类型是有限维或无限维的哈密顿系统（以W.汉密尔顿，谁研究他们在世纪）。 哈密顿系统源自经典力学，并且仍然是研究的基本对象。理解这些系统解的不稳定性的形成是至关重要的。这个提议的目标之一是研究几乎可积的哈密顿系统的动力学，其解可以用显式公式近似的系统，以及它们的不稳定性。这项研究将为理解经典问题提供重要的现代进展，具有许多当前应用，特别是天体物理学。PI希望在这个方向上取得进展，部分是通过将他以前的成功工作扩展到更高维的系统。该提案将集中在证明各种哈密顿系统的Arnol'd扩散的强形式和研究凸台球（弹性碰撞）的谱刚性。该提案的主要议题是：凸Hamilton系统的Arnol'd扩散，几乎可积Hamilton系统能量面上几乎稠密轨道的存在性。后者是拟遍历假设的弱形式。 我们研究的另一个方向是证明几乎可积系统的随机性和扩散行为，其中随机性是相对于初始条件。我们还计划研究Hamiltonian偏微分方程的Sobolev范数的增长性，并扩展我们以前在这个问题上的结果。

In many areas of mathematics and physics it is important to analyze the dynamical behavior of systems that conserve some important physical quantities such as energy, mass, or volume. These are called conservative systems, and one natural type of conservative system is a Hamiltonian system of finite or infinite dimension (named after W. Hamilton, who studied them in the 19th century). Hamiltonian systems are derived from classical mechanics and remain fundamental objects of study. It is critically important to understand the formation of instabilities of solutions for these systems. One of the goals of this proposal is to study dynamics of nearly integrable Hamiltonian systems, systems whose solutions can be approximated by explicit formulas, as well as their instabilities. This study will provide significant modern advances in understanding the classical problems, with many current applications, especially to astrophysics. The PI expects to make progress in this direction in part by extending previous successful work by him to higher dimensional systems. The proposal will be focused around proving strong forms of Arnol'd diffusion for a variety of Hamiltonian systems and studying spectral rigidity for convex billiards (elastic collisions). The major topics of the proposal are: Arnol'd diffusion for convex Hamiltonian systems, existence of an almost dense orbit on an energy surface for nearly integrable Hamiltonian systems. The latter is a weak form of quasi-ergodic hypothesis. Another direction of our research is to prove stochastic and diffusive behavior for nearly integrable systems, where randomness is with respect to initial conditions. We also plan to study growth of Sobolev norms for Hamiltonian PDEs and extend our previous results on this topic.