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Fourier analysis and partial differential equations

傅里叶分析和偏微分方程

基本信息

批准号：
1700180
负责人：
Charles Fefferman
金额：
$ 45万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700180&HistoricalAwards=false
关键词：
Fourier analysis partial differential equations

项目摘要

The principal investigator on this project is Charles Fefferman, and the co-principal investigator is Elias Stein. The project involves research in harmonic analysis, theoretical fluid dynamics, materials science, and statistics. Harmonic analysis is a basic tool in science and engineering. Fluids also arise in many problems of science and technology, but our fundamental understanding of their behavior is very primitive. The materials that will be studied include graphene and its photonic analogues, which are widely regarded as having great promise for technology. The statistics problems to be studied entail fitting to a given data set a smooth object (say, a graph or a "manifold"), which has become a standard strategy for studying big data.Fefferman hopes to understand extension and interpolation of functions in the context of Sobolev spaces; to provide efficient algorithms for selection problems geared to m-times smoothly differentiable functions; to demonstrate new mechanisms for singularity formation for fluid equations; and to obtain asymptotic formulas for the lifetimes of edge resonances in graphene-like materials. Stein seeks to produce counterexamples to demonstrate the optimality of his previous estimates in Lebesgue spaces for Cauchy-type integrals in several complex variables; to obtain dimension-free estimates by techniques he had developed earlier to study discrete analogues; and to obtain analogous results in the setting of nilpotent Lie groups.

该项目的首席研究员是查尔斯·费弗曼，联合首席研究员是伊莱亚斯·斯坦。该项目涉及谐波分析、理论流体动力学、材料科学和统计学的研究。调和分析是科学和工程中的一种基本工具。流体也出现在许多科学和技术问题中，但我们对它们行为的基本理解非常原始。将被研究的材料包括石墨烯及其光子类似物，它们被广泛认为具有巨大的技术前景。要研究的统计学问题需要将光滑对象(如图或流形)拟合到给定的数据集，这已成为研究大数据的标准策略。Fefferman希望在Sobolev空间的背景下理解函数的延拓和内插；为m次光滑可微函数的选择问题提供有效的算法；演示流体方程奇点形成的新机制；以及获得类石墨烯材料中边缘共振寿命的渐近公式。Stein试图给出反例，以证明他以前在勒贝格空间中对多个复变量的柯西型积分估计的最优性；通过他早先发展的研究离散类似物的方法获得无量纲估计；并在幂零李群的设置中获得类似的结果。