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.

我计划针对以下具体问题开展工作。 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其应用范围。 我还想证明拟线性波动方程的几乎全局存在性，并研究在非捕获障碍物之外非线性波动方程是否存在全局存在性。 我想利用这些项目开发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，我想继续研究几何情况下出现的调和分析问题。上述问题自然地产生于数学与物理学领域（包括广义相对论、量子力学和量子混沌）之间的相互作用。 所采用的技术包括固定相和奇点传播研究。 各大大学的量子物理组中有一群非常活跃的研究人员正在研究高能本征态，我特别有兴趣在这一领域做出进一步的贡献。