Bilinear Estimates in Analysis and Partial Differential Equations

本研究计画系关于双线性傅立叶分析。广义上讲，傅里叶分析是一门研究信号（如声音和图像）的数学学科。通过傅立叶分析的方式研究信号涉及将它们分解为不太复杂的基本部分，因此更容易检查。然后将从各个片段获得的信息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。本项目将研究双线性傅立叶分析领域的核心问题，其中一对信号被同时分析。预计这些成果将应用于偏微分方程理论，以及流体动力学、量子力学和光学等各种主题。该项目还将促进研究生和本科生水平的研究和教育的一体化。该项目旨在通过调查一系列相互关联的问题，为双线性傅立叶分析的新发展做出贡献，这些问题的动机是分析和偏微分方程的应用。该项目将研究几种方法，基于工具，包括函数的表示，Littlewood-Paley技术和符号演算，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。这些结果可望应用于函数空间的点态乘性质、Euler方程和理想磁流体动力学方程的局部适定性结果，该奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值，更广泛的影响审查标准。

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.