Microlocal analysis in nonlinear PDE and PDE on manifolds

非线性 PDE 和流形 PDE 中的微局域分析

• 批准号: 0900524
• 负责人: Hans Christianson
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2010-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900524&HistoricalAwards=false
• 关键词: Microlocal; analysis; nonlinear; PDE; manifolds

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.

本专题将探讨偏微分方程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的界面的时间演化，其自然地被设定为初始值PDE。 通过以正确的方式制定的问题，界面的演变是由一个准线性色散方程弱耦合到一个非线性输运方程。 本提案中概述的水波项目侧重于从色散关系得出的该系统的解决方案的性质。 具体而言，解决方案的非线性水波问题的表面张力，平均时间，比初始数据更平滑。此外，研究混合时空色散性质（Escherichartz估计）、高阶光滑性质和初值问题的长时间适定性也是有趣的。 除了水波问题外，这个项目还关注光滑流形上拉普拉斯算子的谱性质：紧情况下的本征函数疤痕，以及非紧情况下薛定谔方程解的局部光滑。 贯穿所有这些项目的共同主线是微局部分析的广泛使用，其大致思想是将方程的解在空间和频率上局部化（在不确定性原则允许的范围内），在此之后，这些解就更容易理解了。色散方程和弯曲空间上的方程（流形）提供了数学的各个分支之间以及不同科学之间的相互作用的丰富领域。 水波问题是一个来自数学流体力学的问题，它代表了纯数学、应用数学和物理学之间的跨学科合作。 该项目的主要目标是提供物理直观思想的数学陈述，例如“表面张力是一种正则化效应。弯曲空间上方程的研究对数学中的几何学家、分析家和数论家，以及从事量子混沌和广义相对论研究的理论物理学家都很感兴趣。