Low Regularity and Long Time Dynamics in Nonlinear Dispersive Flows

这个项目的主要目标是研究可以描述为非线性波的广泛的一类方程的解决方案。这些数学方程模拟了流体动力学（海洋学）、量子力学、等离子体物理学、非线性光学和广义相对论中出现的各种物理现象。正在研究的方程范围从半线性到完全非线性，从局部到非局部方程，我们的目标是调查他们在本地和全球的时间在一个最佳的方式。这项研究发展和连接思想和方法在偏微分方程，并在某些情况下，也绘制了一个明确的路径，对其他问题的领域，如几何，调和分析，复杂的分析和微观分析。该项目为研究生提供了研究培训的机会。非线性波相互作用的强度是本提案中考虑的模型的共同特征，它显着影响它们的短期和长期行为。该项目解决了一系列非常有趣的问题，涉及几类非线性色散方程：（一）在低规则性设置的短时存在理论;（二）波的崩溃，在这里，一个特殊的一类方程是由水波模型提供的;以及（iii）波浪的长时间持续和/或分散和衰减，这将涉及附加于其的定性方面，即，非线性解的渐近描述或其定量描述，例如提供时间色散范围内的全局的非传统散射陈述。所有这一切都在很大程度上取决于初始数据的属性，如大小、规律性和本地化。该奖项反映了NSF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。

The primary objective of this project is to examine solutions to a broad class of equations that can be described as nonlinear waves. These mathematical equations model a wide range of physical phenomena arising in fluid dynamics (oceanography), quantum mechanics, plasma physics, nonlinear optics, and general relativity. The equations being studied range from semilinear to fully nonlinear, and from local to nonlocal equations, and we aim to investigate them in an optimal fashion both locally and globally in time. This research develops and connects ideas and methods in partial differential equations, and in some cases also draws a clear path towards other problems in fields such as geometry, harmonic analysis, complex analysis, and microlocal analysis. The project provides research training opportunities for graduate students.The strength of the nonlinear wave interactions is the common feature in the models considered in this proposal, and it significantly impacts both their short-time and their long-time behavior. The project addresses a series of very interesting questions concerning several classes of nonlinear dispersive equations: (i) short-time existence theory in a low regularity setting; (ii) breakdown of waves, and here a particular class of equations is provided by the water wave models; and (iii) long-time persistence and/or dispersion and decay of waves, which would involve either a qualitative aspect attached to it, that is, an asymptotic description of the nonlinear solution, or a quantitative description of it, for instance nontraditional scattering statements providing global in time dispersive bounds. All of this also depends strongly on the initial data properties, such as size, regularity and localization.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.