Partial differential equations, Hamiltonian dynamical systems, and their applications

本研究计划的总体目标是分析一类偏微分方程（PDEs）的行为并解决其解的奇异性，这些方程控制着涉及波传播的各种物理现象。描述基本物理过程的大多数主要方程都是微分方程；以演化方程形式描述物理系统未来状态的守恒定律，其初始状态是在可能状态的相空间中规定的。当一个过程涉及到波的传播时，这些方程通常是偏微分方程，而那些具有物理或几何意义的方程通常可以表述为哈密顿动力系统，其相空间自然是一个无限维的状态空间。例子包括非线性薛定谔方程，许多版本的非线性波动方程，无粘流体动力学的欧拉方程，海浪问题，以及流体动力学和等离子体中出现的无数非线性色散方程，如Korteweg deVries方程和Boussinesq系统。

The overall goals of this research proposal are to analyse the behavior and to address the singularities of solutions of a class of partial differential equations (PDEs) that govern a variety of physical phenomena involving wave propagation. Most of the principal equations which describe fundamental physical processes are differential equations; conservation laws in the form of evolution equations which describe the future state of a physical systems whose initial state is prescribed among a phase space of possible states. When a process involves wave propagation, these equations are generally PDEs, and those of physical or of geometric significance can often be formulated as Hamiltonian dynamical systems, whose phase space is naturally an infinite dimensional space of states. Examples include the nonlinear Schroedinger equation, many versions of nonlinear wave equations, the Euler equations for inviscid fluid dynamics, the problem of ocean waves, and the myriad of nonlinear dispersive equations such as the Korteweg deVries equation and the Boussinesq system that arise in fluid dynamics and plasmas. The first component of this research program focuses on extensions of analytic techniques of Hamiltonian mechanics to the setting of an infinite dimensional phase space, addressing problems such as KAM theory for PDEs, normal forms transformations, Nekhoroshev stability theorems, and possibly exploiting resonant situations to construct diffusing orbits. The second component is to study the equations for surface water waves, which are themselves Hamiltonian PDEs, which are in addition of serious physical and oceanographic significance. The third component of this proposal addresses several fundamental analytic questions about the Navier - Stokes equations, including to pursue the ramifications of our recent work on lower bounds on the microlocal singular set. The fourth component is to continue our extended collaborations with other physical scientists, which in the past have led to interesting and sometimes unexpected connections between fields.