Bilinear Estimates in Analysis and Partial Differential Equations

分析和偏微分方程中的双线性估计

• 批准号: 2154113
• 负责人: Virginia Naibo
• 金额: $ 22.26万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154113&HistoricalAwards=false
• 关键词: Bilinear; Estimates; Analysis; Partial; Differential

This research project concerns bilinear Fourier analysis. Broadly speaking, Fourier analysis is a mathematical discipline for the study of signals, such as sound and images. The study of signals by way of Fourier analysis involves breaking them down into fundamental pieces that are less complex and, therefore, easier to examine. Information obtained from the individual pieces is then synthesized to obtain information about the original signal. Fourier analysis has had far-reaching applications in other areas of mathematics, physics, engineering, medicine, industry, and the applied sciences. This project will investigate central questions in the field of bilinear Fourier analysis, where a pair of signals are analyzed simultaneously. The outcomes are anticipated to have applications in the theory of partial differential equations, to topics as diverse as fluid dynamics, quantum mechanics, and optics. The project will also contribute to the integration of research and education at the graduate and undergraduate levels.The project aims to contribute to new developments in bilinear Fourier analysis through the investigation of a suite of interrelated questions motivated by applications to analysis and partial differential equations. The project will investigate several approaches, based on tools including representations of functions, Littlewood-Paley techniques, and symbolic calculus, to generate a host of new bilinear estimates and boundedness properties of bilinear pseudodifferential operators. The results are expected to apply to the pointwise multiplication properties of function spaces, local well-posedness results for the Euler equations and the ideal magnetohydrodynamic equations, and scattering properties of solutions of systems of partial differential equations associated to local and nonlocal operators.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.

本研究项目涉及双线性傅立叶分析。广义上讲，傅立叶分析是一门研究声音和图像等信号的数学学科。用傅立叶分析的方法研究信号包括将它们分解成不那么复杂的基本部分，因此更容易检查。从各个片段获得的信息然后被合成以获得关于原始信号的信息。傅立叶分析在数学、物理、工程、医学、工业和应用科学的其他领域有着深远的应用。这个项目将研究双线性傅立叶分析领域的中心问题，在双线性傅立叶分析中，一对信号被同时分析。这些结果有望在偏微分方程组理论、流体动力学、量子力学和光学等不同的主题中得到应用。该项目还将有助于研究生和本科生水平的研究和教育的整合。该项目旨在通过研究一系列由分析和偏微分方程式的应用所激发的相互关联的问题，促进双线性傅立叶分析的新发展。该项目将研究几种基于工具的方法，包括函数表示、Littlewood-Paley技巧和符号演算，以生成一系列新的双线性估计和双线性伪微分算子的有界性。这些结果有望应用于函数空间的逐点乘法性质，欧拉方程和理想磁流体动力学方程的局部适定性结果，以及与局部和非局部算子相关的偏微分方程组的解的散射性质。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。