Long time dynamics of Hamiltonian PDEs

在许多物理问题中，耗散非常小，可以使用保守模型进行研究。这包括大气和海洋中的流体运动、量子力学中的玻色-爱因斯坦凝聚、受控聚变装置中的等离子体等。由于缺乏强耗散，这些模式的动力行为非常复杂。一个重要的课题是理解在实验和自然界中观察到的相干结构的形成和持续。一些例子包括大气和海洋中的大涡结构，海浪中的行波模式，以及观测到的星系结构。另一个主题是了解和控制许多应用程序中的不稳定性。其中一个例子是控制核聚变装置中等离子体的不稳定性，以实现用于能源生产的受控核聚变的目标。数学分析的方法是在拟议的调查中使用的主要工具。严格的数学计算使得稳定的数值计算和更好地理解数值和实验研究中发现的现象成为可能。该项目的一个重点是理解不变结构，包括准周期解和不变流形，接近几种PDE模型的物理有趣平衡，包括Vlasov方程，不可压缩Euler和Naiver-Stokes方程，以及Gross-Pitaevskii方程。这些不变结构构成了物理相空间的骨架，为研究长时间动力学行为提供了重要线索。研究的主题包括：非线性阻尼的正则性阈值和非平凡不变结构在非耗散模型（包括Vlasov模型和二维欧拉方程）的更简单平衡点附近的存在性，许多保守连续体模型和哈密顿偏微分方程中不变流形的构造，以及小耗散下大时间尺度不变结构的持久性。此外，还将开发新的方法来寻找Vlasov-Poisson系统的稳定BGK波和静电孤立波。

In many physical problems, the dissipation is very small and conservative models can be used for their study. These include the fluid motions in atmosphere and ocean, the Bose-Einstein condensate in quantum mechanics, the plasmas in controlled fusion devices, etc. The dynamical behavior of these models is very complex due to the lack of strong dissipation. One important topic is to understand the formation and persistence of coherent structures observed in experiments and in nature. Some examples include the large vortex structures in the atmosphere and ocean, the traveling wave patterns in ocean waves, and the observed galaxy structures. Another topic is to understand and control the instability in many applications. One such example is to control the instability of plasmas in fusion devices to achieve the goal of controlled nuclear fusion for energy production. Methods of mathematical analysis are the primary tools employed in the proposed investigations. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies. One focus of the project is to understand invariant structures including quasi-periodic solutions and invariant manifolds, near physically interesting equilibria of several PDE models, including Vlasov equations, incompressible Euler and Naiver-Stokes equations, and Gross-Pitaevskii equation. These invariant structures form a skeleton in the physical phase space and provide important clues to study long time dynamical behaviors. Topics to be studied include: the regularity threshold for nonlinear damping and the existence of nontrivial invariant structures near simpler equilibria of non-dissipative models including Vlasov models and 2D Euler equations, the construction of invariant manifolds in many conservative continuum models and Hamiltonian PDEs, and the persistence of invariant structures for large time scale under small dissipation. Also, new approaches will be developed to find stable BGK waves and electrostatic solitary waves of Vlasov-Poisson systems.