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Singularities and Long Time Dynamics in Nonlinear Dispersive Flows

非线性色散流中的奇点和长时间动力学

基本信息

批准号：
1800294
负责人：
Daniel Tataru
金额：
$ 27万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800294&HistoricalAwards=false
关键词：
Singularities Long Time Dynamics Nonlinear

项目摘要

The main goal of this project is to study the solutions to a wide class of equations that can best be described as nonlinear waves. These mathematical equations model a broad range of physical phenomena in our universe and even in our daily lives, including the water waves at the surface of our oceans and the quantum-mechanical interaction of multi-particle systems, as well as, on a broader scale, gravitational waves and their evolution in the proximity of black holes. The common feature in these models lies in the strength of the wave interactions, which drastically affects both their short-time and long-time behavior. Phenomena to be studied include both short-time breakdown of waves, as well as long-time persistence and/or dispersion and decay of waves. The objective of this project is to attack a broad array of challenging problems within the field of nonlinear dispersive equations. The equations to be considered range from semilinear to fully nonlinear, and the types of properties to be investigated span from short-range to long-range behavior, from scattering to blow-up questions. The research connects ideas and methods in partial differential equations to geometry, harmonic analysis, complex analysis, and microlocal analysis.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.

该项目的主要目标是研究可以最好地描述为非线性波的一类方程的解。这些数学方程模拟了我们宇宙甚至日常生活中的各种物理现象，包括海洋表面的水波和多粒子系统的量子力学相互作用，以及更广泛的引力波及其在黑洞附近的演化。这些模型的共同特征在于波相互作用的强度，这极大地影响了它们的短期和长期行为。要研究的现象包括波的短时击穿，以及波的长时间持续和/或色散和衰减。这个项目的目标是攻击一系列具有挑战性的问题在非线性色散方程领域。要考虑的方程范围从半线性到完全非线性，和类型的属性要调查的跨度从短期到长期的行为，从散射爆破问题。该研究将偏微分方程中的思想和方法与几何、调和分析、复分析和微局部分析联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。