Microlocal Analysis and Nonlinear Waves

微局域分析和非线性波

• 批准号: 9706489
• 负责人: Mark Williams
• 金额: $ 6.89万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706489&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Nonlinear; Waves

9706489 Williams The proposed research aims to apply and further develop microlocal methods (localization in phase space; Fourier analysis; pseudodifferential, Fourier integral, and paradifferential operators) for the study of several types of nonlinear wave problems: rigorous nonlinear geometric optics for continuous and discontinuous (shock) solutions of quasilinear boundary problems, new mechanisms for blowup, and semilinear diffraction of generic singularities by smooth obstacles. The work on shocks has the closest connection to real-world applications. Shocks arise in many contexts involving the flow of some type of "fluid", which usually means a gas like air or a liquid like water. They form in connection with flights of jets, reentry of space shuttles, and as a result of combustion or explosion, for example. Geometric optics is a mathematical method that can be used to obtain qualitative information and make predictions about shocks, which is based on constructing and analyzing "approximate shocks". In the past there has been uncertainty about whether the approximate shocks really behave like the exact physical shocks. Our work aims to understand precisely when the approximate shocks are and are not close to exact shocks.

9706489 威廉姆斯 拟议的研究旨在应用和进一步发展微局部方法（相空间局部化;傅立叶分析;伪微分，傅立叶积分和paradiomatic算子），用于研究几种类型的非线性波问题：严格 非线性几何光学的连续和不连续（冲击）解决方案的准线性边界问题，新的机制爆破，和半线性衍射的一般奇点光滑的障碍。 关于冲击的工作与现实世界的应用有着最密切的联系。 在许多情况下，冲击会出现在涉及某种类型的“流体”的流动中，这通常意味着像空气这样的气体或像水这样的液体。它们的形成与喷气式飞机的飞行、航天飞机的重返以及燃烧或爆炸等有关。 几何光学是一种数学方法，可用于获得定性信息，并作出预测的冲击，这是基于构建和分析“近似冲击”。 在过去，人们一直不确定近似冲击是否真的表现得像确切的物理冲击。 我们的工作旨在精确地理解近似冲击何时接近和不接近精确冲击。