Low Regularity and Long Time Dynamics in Nonlinear Dispersive Flows

这个项目的主要目标是检验一类可以被描述为非线性波的方程的解。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、非线性光学和广义相对论中出现的各种物理现象。所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，我们的目标是在局部和全局时间上以最优方式研究它们。本研究发展和连接了偏微分方程的思想和方法，在某些情况下也为几何、谐波分析、复分析和微局部分析等领域的其他问题提供了清晰的路径。本项目为研究生提供研究训练机会。非线性波相互作用的强度是本文所考虑的模型的共同特征，它对模型的短期和长期行为都有显著影响。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：(i)低正则性设置下的短时间存在理论；（ii）波浪的分解，在这里，水波模型提供了一类特殊的方程；（iii）波的长时间持续和/或色散和衰减，这将涉及与之相关的定性方面，即非线性解的渐近描述，或对其进行定量描述，例如提供全局时间色散边界的非传统散射陈述。所有这些都很大程度上取决于初始数据属性，如大小、规则性和定位。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。

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.