Research in Harmonic Analysis and Partial Differential Equations

该项目涉及调和分析和偏微分方程(PDE)分析的研究。自从傅里叶关于热扩散理论的工作，以及薛定谔方程在量子力学中的成功之后，调和分析在纯科学和应用科学中发挥了重要作用。它是在科学和工程中广泛使用的各种工具的基础，并提供了在未来进一步应用的前景。这项研究是为了解决基础性问题，这可能有助于支持未来的应用。在PDE中，重点将是研究色散PDE的长期动力学性质，如衰变和光滑化，包括描述各种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，在科学和工程中有重要的应用。引入了四阶薛定谔方程来模拟强激光在具有克尔非线性的块体介质中的传输，并将其用于研究水波相互作用。在调和分析中，焦点集中在欧几里得空间中以勒贝格范数不等式为中心的问题。正在进行的研究课题之一是傅里叶限制现象及其在偏微分方程组和几何测度论问题上的应用。该项目将通过伊利诺伊州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，研究内容包括色散偏微分方程色散衰减和光滑化估计以及色散偏微分方程波算子的有界性，如高阶薛定谔方程和狄拉克方程，并研究其在非线性方程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算子的谱理论和傅里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组的解图的分维，即Talbot效应。以前，这些问题是在周期边界条件下利用非线性方程的指数和估计和光顺估计来研究的；作为本项目的一部分，我们将研究更一般的几何，如高维的球面和环面。在调和分析方面，该项目将需要部分依赖于解耦理论的最新发展的加权限制估计，以及加权限制估计在几何测量理论和离散PDE中的应用，例如以分形量为势的薛定谔方程。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。

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.