Low Regularity and Long Time Dynamics in Nonlinear Dispersive Flows

非线性弥散流中的低规律性和长时间动态

• 批准号: 2348908
• 负责人: Mihaela Ifrim
• 金额: $ 34.34万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348908&HistoricalAwards=false
• 关键词: Low; Regularity; Long; Time; Dynamics

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.

该项目的主要目的是检查可以描述为非线性波的广泛方程式的解决方案。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，非线性光学和一般相对论的广泛物理现象。所研究的方程式从半线性到完全非线性，以及从局部到非局部方程式，我们的目标是在本地和全球范围内以最佳的方式调查它们。这项研究开发并连接部分微分方程中的思想和方法，在某些情况下，也为诸如几何，谐波分析，复杂分析和微局部分析等领域的其他问题迈出了清晰的途径。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提案中考虑的模型中的共同特征，并且显着影响其短期和长期行为。该项目解决了一系列关于几类非线性分散方程的非常有趣的问题：（i）在低规律性设置中的短期存在理论； （ii）波浪的崩溃，在这里，水浪模型提供了特定的方程； （iii）波浪的长期持久性和/或分散性和衰减，这将涉及其附加的定性方面，即对非线性溶液的渐近描述，或对其的定量描述，例如，非传统散射陈述在时间分散范围内提供全球。所有这些都在很大程度上取决于最初的数据属性，例如规模，规律性和本地化。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。