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.

奇异积分算子是现代分析学的一个庞大的分支，吸引了许多有实力的分析家。 在过去的几年里，与奇异正则关系相关的平均算子和傅里叶积分算子的发展尤为激烈。Andrew Comech在这一领域中得到了几个一般性的结果，特别是Sobolev空间中奇异正则关系的分类和相关Fourier积分算子的正则性。 该方法是基于将某些曲率假设纳入调和分析的现代工具。 这些结果的应用包括Melrose-Taylor的Radon变换的正则性和双曲型方程解的限制的光滑性。 在这两种情况下，安德鲁的方法导致最佳的结论。 未来的研究主要是针对非线性双曲型方程解的性质，如非线性双曲型方程解的长时间行为和孤子渐近性。 从调和分析的基础，这是有关拟议的研究，包括规则性的平均值的低维品种（最重要的是，平均算子的曲线）。 Andrew Comech将继续他在调和分析和双曲方程理论这两个基础数学领域的研究。 它的研究深入到数学的现代发展中，与非线性微分方程、吸引子、孤子等密切相关。 另一方面，这项研究深深植根于自然科学和技术。安德鲁·科米奇的特别兴趣是与相对论量子物理学的非线性波动方程的性质有关，这些方程现在在物理学和数学中都得到了广泛的研究。 同样有趣的是开放的机会，由于应用到层析成像，以及涉及傅立叶分析的数值计算方面。 这些都是当今科学研究的活跃和有前途的领域。