Nonlinear Hyperbolic Equations

非线性双曲方程

基本信息

批准号：
9970297
负责人：
Daniel Tataru
金额：
$ 9.45万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1999
资助国家：
美国
起止时间：
1999-06-01 至 2002-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970297&HistoricalAwards=false
关键词：
Nonlinear Hyperbolic Equations

项目摘要

The project is devoted to two related topics, namely nonlinear hyperbolicequations and partial differential equations with nonsmooth coefficients.The hyperbolic equations we consider are either semilinear (such asthe critical Wave-Maps and Yang Mills equations), for which one hopes toobtainboth local and global results, or nonlinear, for which the aim is toimprovethe local theory. The problems related to partial differential equationswith nonsmooth coefficients are mostly inspired by nonlinear pde's; thisincludes various Strichartz and Carleman type estimates, as well ashyperbolicboundary value problems.The equations of interest for this project can be loosely described asmodels for wave-like phenomena, such as generalized relativity andnonlinear ellasticity. While many physical models are genuinely nonlinear, their mathematical analysisis extremely difficult, often beyond the current mathemathical theory;thus,in most cases we contend ourselves with simplified models which are linearor only mildly nonlinear. The proposed work is part of a larger programwhose aim is to understand such fully nonlinear equations. The strategy isto study a sequence of increasingly difficult model problems which getcloser andcloser to the relevant physical model, so that at each step some newnonlinear feature is understood.

该项目致力于两个相关的主题，即具有非平滑系数的非线性高音和部分微分方程。我们认为的双曲线方程要么是半线性（例如，关键波浪映射和Yang Mills方程），因此，人们希望这是toobtaint toobtaint toobtain toobtaint toobtain and toobtaint toobtain and toobtain and to toobtain and to the nonlineal的结果，或者是本地理论的，是本地理论。与非平滑系数的部分微分方程相关的问题主要是受非线性PDE的启发。这包括各种Strichartz和Carleman类型的估计，以及灰烬刺激价值问题。该项目的兴趣方程可以松散地描述为波浪样现象的Asmodels，例如广义的相对性和nononlineleareareareality。尽管许多物理模型是真正的非线性，但它们的数学分析非常困难，通常超出了当前的数学理论；因此，在大多数情况下，我们以简化的模型来争夺自己，这些模型仅是线性的，仅是轻度非线性。拟议的工作是更大的计划的一部分，其目的是了解这种完全非线性方程。该策略是研究一系列日益困难的模型问题，这些序列是将getcloser和closer符合到相关的物理模型，因此在每个步骤中都可以理解一些纽诺的特征。