Properties of solutions of linear and non-linear hyperbolic equations, singular Fourier integral operators, averages over curves

线性和非线性双曲方程解的性质、奇异傅里叶积分算子、曲线平均值

• 批准号: 9970330
• 负责人: Andrew Comech
• 金额: $ 6.62万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970330&HistoricalAwards=false
• 关键词: Properties; solutions; linear; non; hyperbolic

Singular integral operators is a vast modern branch of Analysis, whichnow attracts many strong analysts. Particularly intense developmentfor the last several years has been in the field of averagingoperators and Fourier integral operators associated to singularcanonical relations. Andrew Comech obtained several general results inthis field; in particular, the classification of singular canonicalrelations and regularity properties of the associated Fourier integraloperators in the Sobolev spaces. The approach was based onincorporating certain curvature assumptions into the modern tools ofHarmonic Analysis. Among the applications of these results are theregularity properties of the Radon Transform of Melrose-Taylor andsmoothness of restrictions of solutions to hyperbolic equations. Inboth cases Andrew's methods lead to the optimal conclusions. Thefuture research is aimed at the properties of solutions to non-linearhyperbolic equations, such as long-time behavior and solitonasymptotics of solutions to nonlinear hyperbolic equations. Questionsfrom the very foundations of Harmonic Analysis, which are related tothe proposed research, include the regularity properties of theaverages taken over lower-dimensional varieties (most importantly,averaging operators over curves). Andrew Comech is going to continue his research on the wedge of thetwo fundamental mathematical fields: Harmonic Analysis and the Theoryof Hyperbolic Equations. The research grows far into the moderndevelopment of the mathematics, being intimately related to nonlinear differential equations, attractors, and solitons. On the other hand, the research has its roots deep in the natural sciences and technology. Particular interests of Andrew Comech are related to the properties of nonlinear wave equations of relativistic Quantum Physics, which are now widely investigated both in Physics and in Mathematics. Not less interesting are the opportunities open due to applications to Tomography, as well as aspects of numerical computations involving Fourier Analysis. These are active and promising areas of today's scientific research.

奇异积分算子是分析学的一个巨大的现代分支，现在吸引了许多强大的分析师。 过去几年中，与奇异正则关系相关的平均算子和傅里叶积分算子领域取得了特别激烈的发展。安德鲁·科梅奇（Andrew Comech）在该领域取得了一些一般性成果；特别是，索博列夫空间中相关傅里叶积分算子的奇异正则关系和正则性质的分类。 该方法基于将某些曲率假设纳入现代调和分析工具中。 这些结果的应用包括梅尔罗斯-泰勒氡变换的正则性和双曲方程解的限制的光滑性。 在这两种情况下，安德鲁的方法都会得出最佳结论。 未来的研究主要针对非线性双曲方程解的性质，如非线性双曲方程解的长期行为和孤立渐近性等。 来自调和分析基础的问题，与所提出的研究相关，包括低维变量的平均值的规律性特性（最重要的是，曲线上的平均算子）。 安德鲁·科梅奇 (Andrew Comech) 将继续他在两个基础数学领域的研究：调和分析和双曲方程理论。 该研究深入到现代数学的发展，与非线性微分方程、吸引子和孤子密切相关。 另一方面，这项研究深深植根于自然科学和技术。安德鲁·科梅奇 (Andrew Comech) 特别感兴趣的是相对论量子物理学的非线性波动方程的性质，该方程目前在物理学和数学领域得到广泛研究。 同样有趣的是由于断层扫描的应用以及涉及傅立叶分析的数值计算方面而出现的机会。 这些是当今科学研究活跃且有前途的领域。