Waves and Particles: Nonlinear Microlocal Analysis of Microstructures in Partial Differential Equations

波与粒子：偏微分方程中微观结构的非线性微局域分析

• 批准号: 9704762
• 负责人: Luc Tartar
• 金额: $ 5万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704762&HistoricalAwards=false
• 关键词: Waves; Particles; Nonlinear; Microlocal; Analysis

9704762 TARTAR As part of a much larger project, the Principal Investigator plans to develop variants of the tool of H-measures that he has introduced a few years ago under NSF grants. One question concerns variants able to observe large domains of frequency, as is supposed to occur in turbulence. Another question concerns parabolic effects, which require a variant using different scalings in different directions. Another question concerns the taking into account of oscillating initial data in studying the propagation of oscillations and concentration effects for general hyperbolic systems with variable coefficients, a particular emphasis being put on questions of Elasticity in connection with seismic problems. Another question concerns the effects of limited regularity of coefficients or interaction with the boundaries, reflection, refraction, grazing rays. The Principal Investigator also plans to study which constitutive relations in Elasticity are admissible under the necessity of having the finite propagation speed property for mixtures. The Principal Investigator has developed under NSF grants a new approach to general questions of Continuum Mechanics and Physics, which corresponds to explaining how waves can appear to behave like particles. The repercussions of having now such a new rigourous mathematical framework will be enormous in many different areas, but at the moment the developments are still of a very technical nature, and apart from problems related to Elasticity which are connected to seismic problems, the work of the Principal Investigator during the coming year will be mostly theoretical, opening new directions for future research and applications.

9704762 TARTAR作为一个更大的项目的一部分，首席研究员计划开发他几年前在NSF赠款下推出的H测量工具的变体。一个问题涉及能够观察到大的频率域的变体，因为应该发生在湍流中。另一个问题涉及抛物线效应，这需要在不同方向上使用不同比例的变量。另一个问题涉及考虑到振荡的初始数据在研究传播的振荡和浓度效应的一般双曲型系统的变系数，特别强调的问题，弹性与地震问题。另一个问题涉及有限的规则性系数或与边界的相互作用，反射，折射，掠射射线的影响。主要研究者还计划研究在混合物具有有限传播速度属性的必要性下，弹性中的哪些本构关系是可接受的。 主要研究者在NSF的资助下赠款了一种新的方法来解决连续介质力学和物理学的一般问题，该方法对应于解释波如何表现得像粒子。 现在有这样一个新的严格的数学框架的影响将是巨大的，在许多不同的领域，但目前的发展仍然是一个非常技术性的，除了与地震问题有关的弹性问题，首席研究员在未来一年的工作将主要是理论，为未来的研究和应用开辟新的方向。