Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 2054424
• 负责人: Jared Wunsch
• 金额: $ 27万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054424&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

Many seemingly different physical phenomena such as light, sound propagation, and the motion of quantum particles are described in a mathematically unified manner as propagating waves. It is therefore of considerable practical importance to describe how waves oscillate and die off depending on the source producing them and the medium in which they propagate. The goal of this project is to investigate several questions surrounding the rate of wave decay, and in particular how it is influenced by the effects of diffraction, which occurs when the waves encounter sharp discontinuities in the medium. The questions of interest include those motivated by very small-scale physics (the hydrogen atom) and very large-scale physics (decay of waves on backgrounds arising in cosmology). The PI’s prior work on analyzing the effectiveness of computational methods used in modeling solutions to these equations in practical settings will be continued. Training of graduate students and postdoctoral fellows will be incorporated throughout the project.This project revolves around questions involving the decay rate of waves near their source in several settings. Of particular interest will be understanding the role of wave diffraction by rough media in the qualitative behavior and long-time decay rates of solutions to wave and Schrödinger equations. When studying the effects of the singularity of the Coulomb potential on the structure of the Dirac propagator for the hydrogen atom, diffractive effects will again play an important role. In spacetimes of interest in general relativity, the large-scale structure of spacetime effects on the decay of waves and the structure of their radiation patterns will be investigated. This project will further entail a study of the performance of numerical algorithms for computation of the scattering of waves, bringing to bear techniques of phase space analysis that have not previously been employed in these problems.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.

许多看似不同的物理现象，例如光，声音传播和量子颗粒的运动，以数学统一的方式描述为传播波。因此，考虑到波浪如何根据产生的来源以及它们传播的媒介而振荡和死亡是很重要的。该项目的目的是研究围绕波浪衰减速率的几个问题，尤其是它如何受到衍射的影响，当波在介质中遇到急剧不连续性时发生。感兴趣的问题包括由非常小的物理学（氢原子）和非常大的物理学（在宇宙学中引起的波浪衰减）动机的问题。 PI先前在分析用于在实践环境中为这些等式建模的计算方法的有效性的工作将继续进行。对研究生和博士后研究员进行培训，这个项目围绕着涉及在多种情况下源附近的波浪衰减率的问题。特别有趣的是，粗糙介质在定性行为和wave和schrödinger方程解决方案的长期衰减率中的波衍射的作用。在研究库仑电势对氢原子狄拉克传播剂结构的奇异性的影响时，衍射作用将再次发挥重要作用。在一般相对论中感兴趣的空间中，将研究时空对波衰减的大规模结构及其辐射模式的结构。该项目将进一步研究用于计算波浪散射的数值算法的性能，使这些问题以前尚未雇用的相位空间分析技术。该奖项反映了NSF的法定任务，并通过使用该基金会的智力功能和广泛的影响来评估Criteria criteria criteria诚实地认为，通过评估诚实的支持。

项目成果

Baroclinic tidal conversion: note on a paper of L.R.M. Maas

• DOI: 10.1017/jfm.2022.637
• 发表时间: 2022-08
• 期刊: Journal of Fluid Mechanics
• 影响因子: 3.7
• 作者: C. Wunsch;J. Wunsch