Research in Harmonic Analysis and Partial Differential Equations

主要研究者将从事谐波分析和偏微分方程（PDE）分析方面的研究。 调和分析在纯科学和应用科学中发挥了重要作用，因为傅立叶的开创性工作的理论热扩散，继续与薛定谔方程在量子力学。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。拟议的研究涉及可能有助于支持未来应用的基础问题。在偏微分方程中，该项目将研究描述各种物理现象的几个基本方程的长时间动力学特性。 特别地，非线性薛定谔方程（NLS）对光纤通信系统中的数据传输进行建模，并且Korteweg-de弗里斯方程（KdV）对表面水波以及冷等离子体中的离子声波进行建模。所谓的分数NLS被用作描述生物聚合物如DNA中的电荷传输的模型。近线性行为和平滑提出的问题是直接由真实的世界工程问题在光纤通信systems，和所使用的方法可能是有用的在一系列的applications.In谐波分析这个项目的重点是在欧几里德空间围绕勒贝格范数不等式为中心的问题。正在进行的研究的一个主题是傅立叶限制现象及其在偏微分方程和几何测度理论中的应用。在该项目的PDE部分，重点是线性和非线性色散方程的动力学性质。这里感兴趣的主题是色散衰减和平滑估计薛定谔方程和波动方程，及其应用的稳定性问题，其非线性对应。另一个主题是非线性色散偏微分方程，如KdV方程，Zakharov系统和分数阶NLS的解的正则性。首席研究员将继续探索有界域上的色散线性群的平滑效应，他将研究非线性解的正则性和长时间动力学的应用。建议的应用程序的存在性和规律性的整体吸引子，分散量化/塔尔博特效应，高阶Sobolev规范的界限，可控性。

The principal investigator will undertake research in harmonic analysis and in the analysis of partial differential equations (PDE). Harmonic analysis has played major roles in the pure and applied sciences since Fourier's seminal work on the theory of heat diffusion, continuing on with Schrodinger's equation in quantum mechanics. It underlies a diverse array of tools widely used in science and engineering, and it offers the promise of further applications in the future. The proposed research deals with foundational issues that may help to underpin future applications. In PDE, the project will study the long-time dynamical properties of several fundamental equations describing diverse physical phenomenon. In particular, the nonlinear Schrodinger equation (NLS) models the transmission of data in fiber optic communication systems, and the Korteweg-de Vries equation (KdV) models surface water waves as well as ion-acoustic waves in a cold plasma. The so-called fractional NLS is used as a model describing charge transport in bio polymers like DNA. Proposed problems on near-linear behavior and smoothing are directly motivated by real world engineering problems in fiber optic communication systems, and the methods used are likely to be useful in a range of applications.In harmonic analysis this project focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research is the Fourier restriction phenomenon and its applications to problems in PDE and geometric measure theory. In the PDE component of the project, the focus is on the dynamical properties of linear and nonlinear dispersive equations. Subjects of interest here are dispersive decay and smoothing estimates for Schrodinger and wave equations, and their applications to the stability problem for their nonlinear counterparts. Another topic is the regularity properties of the solutions of nonlinear dispersive PDE such as the KdV equation, the Zakharov system, and the fractional NLS. The principal investigator will continue to explore the smoothing effect of the dispersive linear group on bounded domains, and he will study applications to the regularity properties and long-time dynamics of the nonlinear solutions. Proposed applications are on the existence and regularity of global attractors, dispersive quantization/Talbot effect, bounds for higher order Sobolev norms, and controllability.