Research in Harmonic Analysis and Partial Differential Equations

该项目涉及谐波分析和偏微分方程分析的研究。自从傅里叶在热扩散理论方面的工作以来，谐波分析在纯科学和应用科学中发挥了重要作用，并继续在量子力学中Schrödinger方程的成功。它是科学和工程中广泛使用的各种工具的基础，并为未来的进一步应用提供了希望。这项研究是为了解决基础问题，这可能有助于支撑未来的应用。在偏微分方程中，重点将是研究色散偏微分方程的长期动力学性质，如衰减和平滑，包括描述不同物理现象的几个基本方程。特别是，狄拉克方程是石墨烯的模型，在科学和工程中具有重要的应用。引入四阶Schrödinger方程对强激光束在体介质中的传播进行克尔非线性建模；此外，它还可用于水波相互作用的研究。在调和分析中，重点在于欧几里得空间中围绕勒贝格范数不等式的问题。傅里叶限制现象及其在偏微分方程和几何测量理论问题中的应用是一个正在进行的研究课题。该项目将通过伊利诺伊几何实验室的数值项目和研究生的指导，让本科生参与研究活动。更具体地说，研究包括色散PDE的色散衰减和平滑估计以及波算子的有界性，如高阶Schrödinger方程和Dirac方程，并研究在非线性对应的正则性和长时间动力学中的应用。所涉及的方法将包括自伴随算子的谱理论和傅立叶分析中的振荡积分估计。另一个研究领域是色散偏微分方程解图的分形维数，即塔尔博特效应。以前，这些问题是在周期边界条件下用非线性方程的指数和估计和平滑估计研究的；作为这个项目的一部分，更一般的几何图形，如球面和环面在更高的维度将被研究。在谐波分析中，该项目将需要加权限制估计，部分依赖于解耦理论的最新发展，以及加权限制估计在几何测量理论和色散偏微分方程中的应用，如Schrödinger方程的分形测量作为势。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project concerns research in harmonic analysis, and in the analysis of partial differential equations (PDE). Harmonic analysis has played major roles in pure and applied sciences since Fourier's work on the theory of heat diffusion, continuing with the success of Schrödinger’s equation in quantum mechanics. It underlies a diverse array of tools widely used in sciences and engineering and offers the promise of further applications in the future. The research is to deal with foundational issues, which may help to underpin future applications. In PDE, the focus will be to study long-time dynamical properties, such as decay and smoothing of dispersive PDE including several fundamental equations describing diverse physical phenomena. In particular, the Dirac equation is a model for graphene, which has important applications in science and engineering. The fourth order Schrödinger equation was introduced to model the propagation of intense laser beams in a bulk medium with Kerr nonlinearity; in addition it is useful in the study of interaction of water waves. In harmonic analysis, the focus lies on questions in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research is the Fourier restriction phenomenon and its applications on questions in PDE and geometric measure theory. The project will involve undergraduate students in research activities through numerical projects in Illinois Geometry Lab and the mentoring of graduate students. More specifically, the research is to encompass dispersive decay and smoothing estimates and the boundedness of wave operators for dispersive PDE such as higher order Schrödinger’s equations and Dirac equations, and to study applications to the regularity properties and long-time dynamics of the nonlinear counterparts. The methods involved will include the spectral theory of self-adjoint operators and oscillatory integral estimates in Fourier analysis. Another area of research is on the fractal dimension of solution graphs of dispersive PDE, or the Talbot effect. Previously, these questions were studied in the case of periodic boundary conditions using exponential sum estimates and smoothing estimates for nonlinear equations; as part of this project, more general geometries such as the sphere and tori in higher dimensions will be investigated. In harmonic analysis the project will entail weighted restriction estimates partly relying on recent developments in decoupling theory, as well as the applications of weighted restriction estimates on questions in geometric measure theory and in dispersive PDE such as the Schrödinger’s equation with a fractal measure as potential.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.