Low Regularity and Long Time Dynamics in Nonlinear Dispersive Flows

这个项目的主要目标是研究可以被描述为非线性波的一类广泛的方程的解。这些数学方程模拟了流体力学(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。所研究的方程从半线性到完全非线性，从局部方程到非局部方程，我们的目标是在时间上以局部和全局的最优方式研究它们。这项研究发展和联系了偏微分方程组的思想和方法，在某些情况下也为几何、调和分析、复分析和微局部分析等领域的其他问题开辟了一条清晰的道路。该项目为研究生提供了研究培训机会。非线性波相互作用的强度是本方案所考虑的模型的共同特征，它显著影响了他们的短期行为和长期行为。该项目解决了与几类非线性色散方程有关的一系列非常有趣的问题：(I)低正则性背景下的短时存在理论；(Ii)波的破裂，这里由水波模型提供了一类特定的方程；以及(Iii)波的长时间持续和/或频散和衰减，这将涉及到与之相关的定性方面，即非线性解的渐近描述，或其定量描述，例如提供全局时间色散界的非传统散射声明。所有这一切也强烈依赖于初始数据属性，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智力优势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.