Regularity Properties of Nonlinear Wave Equations

PI：Markus R.基尔大学明尼苏达州-双城DMS-0303704摘要：主要研究员将继续工作的规律性的某些非线性色散偏微分方程的发展，无论是紧凑的域与周期性的边界条件，在空间中的障碍物外，或在所有的空间。该项目将研究的方程的例子包括KdV、某些非线性薛定谔方程和非线性二阶双曲方程。许多物理理论--例如流体动力学、弹性和引力--最终都是数学方程，其解不能用任何已知公式精确地恢复。PI的研究采取了两条路线，以更定性地了解这些问题。 一种方法旨在将复杂系统理解为简单但相关理论的扰动。 一种补充方法将复杂的解视为恒定频率的简单波的总和，重点关注这些成分随着时间的推移的相互作用。 这种分析的目的是揭示未被识别的物理现象，并设计更好的原始物理理论的计算模型。

PI: Markus R. Keel, U. of Minnesota - Twin CitiesDMS-0303704Abstract: The Principal Investigator will continue work on the regularity properties of certain nonlinear dispersive PDE evolving on either compact domains with periodic boundary conditions, outside of an obstacle in space, or in all of space. Examples of the equations, which the project will study, include KdV, certain nonlinear Schroedinger equations, and nonlinear second order hyperbolic equations.Many physical theories - e.g. fluid dynamics, elasticity, and gravitation - culminate in mathematical equations whose solutions cannot be recovered precisely by any known formulae. The PI's research takes two routes to a more qualitative understanding of such problems. One approach aims to understand the complicated system as a perturbation of a simple, but related theory. A complementary approach views the complicated solution as a sum of simple waves of constant frequency, focusing on the interaction of these constituentsas time passes. The goal of this analysis is to uncover unrecognized physical phenomena, and to design better computational models of the original physical theory.