Microlocal analysis in nonlinear PDE and PDE on manifolds

本课题将探讨偏微分方程（PDE）解的定性性质的几个相关研究领域。具有表面张力的水波问题涉及两种不同流体之间界面的时间演化，它自然地被设定为初始值PDE。通过以正确的方式表述问题，界面的演化由一个弱耦合到非线性输运方程的准线性色散方程控制。本提案中概述的水波项目侧重于根据色散关系得出的该系统的解的性质。具体而言，考虑表面张力的非线性水波问题的解在平均时间上比初始数据更平滑。此外，研究混合时空色散特性（Strichartz估计）、高阶平滑特性以及初值问题的长时间适定性也将是有趣的。除了水波问题外，本项目还研究了光滑流形上拉普拉斯算子的谱性质：紧致情况下的特征函数疤痕，以及非紧致情况下薛定谔方程解的局部平滑。贯穿所有这些项目的共同线索是广泛使用微局部分析，其大致思想是将方程的解在空间和频率上（在不确定性原理允许的范围内）局部化，之后这些解更容易理解。色散方程和曲线空间（流形）上的方程为不同数学分支和不同科学之间的相互作用提供了丰富的领域。水波问题是一个来自数学流体力学的问题，是纯数学、应用数学和物理学的跨学科合作。该项目的主要目标是提供物理直观概念的数学陈述，例如“表面张力是一种正则化效应”。弯曲空间方程的研究对几何学家、分析学家、数论学家以及研究量子混沌和广义相对论的理论物理学家都很有兴趣。

This project will explore several related areas of research into the qualitative properties of solutions to partial differential equations (PDE). The water-wave problem with surface tension concerns the time evolution of the interface between two different fluids, which is naturally posed as an initial value PDE. By formulating the problem in the right fashion, the evolution of the interface is governed by a quasi-linear dispersive equation weakly coupled to a nonlinear transport equation. The water-wave projects outlined in this proposal focus on properties of solutions to this system that follow from the dispersion relation. Specifically, solutions to the nonlinear water-wave problem with surface tension are, on average in time, smoother than the initial data. In addition, it will be interesting to study mixed space-time dispersive properties (Strichartz estimates), higher-order smoothing properties, and the long-time well-posedness of the initial value problem. In addition to the water-wave problem, this project is concerned with spectral properties of the Laplacian on smooth manifolds: eigenfunction scarring in the compact case, and local smoothing for solutions to the Schroedinger equation in the noncompact case. The common thread running through all of these projects is the extensive use of microlocal analysis, the rough idea of which is to localize solutions of equations in space and in frequency (to the extent allowed by the uncertainty principle), after which these solutions are simpler to understand.Dispersive equations and equations on curved spaces (manifolds) provide a rich area of interaction between various branches of mathematics as well as between different sciences. The work on the water-wave problem, a problem coming from mathematical hydrodynamics, represents a cross-disciplinary collaboration between pure math, applied math, and physics. The major goal of the project is to provide mathematical statements of physically intuitive ideas, such as "surface tension is a regularizing effect." The study of equations on curved spaces is of interest to geometers, analysts, and number theorists in mathematics, as well as to theoretical physicists working in quantum chaos and general relativity.