Nonlinear hyperbolic differential equations and Fourier analysis

非线性双曲微分方程和傅里叶分析

• 批准号: 0099642
• 负责人: Christopher Sogge
• 金额: $ 26.9万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099642&HistoricalAwards=false
• 关键词: Nonlinear; hyperbolic; differential; equations; Fourier

I plan on working on the following specific problems. First I would like to prove new Strichartz estimates without using explicit parametrices. This could potentially greatly widen the scope of their applications. I would also like to prove almost global existence for quasilinear wave equations and investigate whether there is global existence for nonlinear wave equations outside of nontrapping obstacles. I would like to use the techniques developed on these projects to study problems in relativity theory, including the stability of the Schwarzschild solution. Finally, I would like to continue my work on problems in harmonic analysis that arise in geometrical situations.The above problems arise naturally from interactions between mathematics and areas in physics that include general relativity, quantum mechanics, and quantum chaos. The techniques employed include stationary phase and the study of propagation of singularities. There is a very active group of researchers in quantum physics groups at major universities studying high-energy eigenstates, and I am especially interested in making further contributions to this area.

我计划解决以下具体问题。首先，我想在不使用显式参数的情况下证明新的Strichartz估计。这可能会极大地扩大它们的应用范围。我还想证明拟线性波动方程的几乎全局存在性，并研究非线性波动方程在非捕获障碍外是否存在全局存在性。我想用在这些项目中发展起来的技术来研究相对论中的问题，包括史瓦西解的稳定性。最后，我想继续我在几何情况下谐波分析问题上的工作。上述问题自然产生于数学与物理学领域的相互作用，包括广义相对论、量子力学和量子混沌。所采用的技术包括固定相位和奇点传播的研究。在主要大学的量子物理小组中，有一群非常活跃的研究人员在研究高能特征态，我对在这个领域做出进一步的贡献特别感兴趣。