Hamiltonian Partial Differential Equations

物理环境中的连续振动，如湖面的运动，光纤中的光波 双星系统的衰变引起的通道或时空度规振荡可以用欧拉和麦克斯韦系统、非线性薛定谔方程和爱因斯坦方程等方程进行数学建模。尽管它们出现在对各种物理现象的描述中，但模拟无摩擦连续介质振动的方程具有丰富的数学相似性(例如，能量和质量等守恒量、哈密顿公式、辛相空间等)。它们定义了哈密顿偏微分方程组领域的概念框架。所提出的研究目的是有助于对哈密顿偏微分方程初值问题解的最大时间性态的一般理解。其目的是为关于进化方程的主要问题提供新的答案：会发生什么？

Continuum vibrations in physical settings, like the motions of a lake's surface, light waves in a fiber optic channel or spacetime metric oscillations caused by the ringdown of a binary star system are mathematically modeled by equations like the Euler and Maxwell systems, the nonlinear Schrödinger and Einstein equations. Even though they arise in descriptions of wildly diverse physical phenomena, the equations that model continuum vibrations without friction enjoy rich mathematical similarities (e.g. conserved quantities like energy and mass, Hamiltonian formulation, symplectic phase space, ...) which define the conceptual framework for the field of Hamiltonian partial differential equations. The goal of the proposed research is to contribute to the general understanding of the maximal-in-time behavior of solutions of the initial value problem for Hamiltonian PDE. The aim is to produce new answers to the main question about evolution equations: what happens?