Critical Nonlinear Dispersive Equations

该项目考虑了不同类型的波动方程的解的行为，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。 这些方程在其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，并且可以以富有成效的方式综合数学各个领域的技术和见解。 这些方程的解至少在短时间内存在，但除此之外，可以在某个有限时间达到奇点，或者扩展到所有时间，在这种情况下，它们被称为整体解。 对于奇异解，中心问题是寻找解的崩溃机制和解的结构随时间向奇异性演化。 对于全局解，目标是将解随着时间延伸到无穷大而可能出现的不同行为进行分类，这些行为可以包括分散、产生内聚静态波形或集中波形。这些方程的数学分析应用了几何学、傅立叶分析和频谱理论的技术。 对于某些方程，局部时间理论是非常好理解的，并且设置在尺度不变函数空间中，该函数空间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展，但在许多情况下，特定的初始条件是否会产生奇异或全局解仍然是一个悬而未决的问题。 可以预期，在许多情况下，解的行为类似于方程的线性化版本的扰动，尽管存在其他已知的现象。 一些分析技术共同的这些问题，或其中一个希望适用于这些问题，包括Morawetz估计和能源估计。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的知识价值和更广泛的影响审查标准。

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.