Critical Nonlinear Dispersive Equations

本项目考虑了不同类型的波方程的解的行为，包括超双曲薛定谔方程、薛定谔映射问题、哈特里方程、Skyrme模型和爱因斯坦波映射系统。这些方程在物理和历史起源上是广泛分离的，然而，相关的数学是密切相关的，各个数学领域的技术和见解可以以富有成效的方式综合起来。这些方程的解至少在短时间内存在，但在此之后，它们要么在某一有限时间内达到奇点，要么延伸到所有时间，在这种情况下，它们被称为全局解。对于奇异解，中心问题是找到解的分解机制和解的结构随着时间向奇点演化。对于全局解决方案，目标是在时间延伸到无穷大时对解决方案的不同可能行为进行编目，其中可以包括色散，内聚静态波形或集中波形的产生。这些方程的数学分析应用了几何学、傅立叶分析和光谱理论的技术。对于一些方程，时间局域理论被很好地理解，并且被设置在一个尺度不变的函数空间中，这在解的长时间分析中也起着至关重要的作用。尽管在过去的几十年里取得了许多进步，但在许多情况下，某个初始条件是否会产生一个单一的或全局的解决方案仍然是一个悬而未决的问题。预计在许多情况下，解将表现为方程线性化版本的扰动，尽管还有其他已知现象发生。这些问题常见的一些分析技术，或者人们希望应用于这些问题的分析技术，包括Morawetz估计和能量估计。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project considers the behavior of solutions of different types of wave equations, including the ultra-hyperbolic Schrodinger equation, the Schrodinger map problem, the Hartree equation, the Skyrme model, and the Einstein wave map system. These equations are widely separated in their physical and historical origins, however, the relevant mathematics is closely related, and techniques and insights in various areas of mathematics may be synthesized in fruitful ways. Solutions to these equations exist at least for a short time, but beyond that, can either reach a singularity at some finite time, or extend to all time in which case they are called global solutions. For singular solutions, the central questions are to find the mechanisms for the breakdown of the solutions and the structure of the solutions as time evolves toward the singularity. For global solutions, the goal is to catalogue the different possible behaviors of the solution as time extends to infinity, which can include dispersion, the production of cohesive static waveforms or concentrating waveforms. The mathematical analysis of these equations applies techniques from geometry, Fourier analysis, and spectral theory. For some of the equations, the local in time theory is very well understood and is set in a scale-invariant function space that plays a crucial role in the long time analysis of solutions as well. Despite numerous advances in the last several decades, in many cases it remains an open problem whether a certain initial condition will produce a solution that is singular or global. It is expected that the solution will behave like a perturbation of a linearized version of the equation in many cases, although there are other known phenomena which occur. Some of the analytical techniques common to these problems, or which one hopes to apply to these problems, include Morawetz estimates and energy estimates.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.