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

作为一个更大项目的一部分，首席研究员计划开发他几年前在NSF资助下引入的h -测度工具的变体。一个问题涉及到能够观察到大范围频率的变量，就像在湍流中发生的那样。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体。另一个问题是在研究一般变系数双曲系统的振荡传播和集中效应时考虑到振荡初始数据，特别强调与地震问题有关的弹性问题。另一个问题涉及系数的有限规则性或与边界、反射、折射、掠射射线的相互作用的影响。项目负责人还计划研究在混合材料具有有限传播速度性质的前提下，哪些弹性本构关系是允许的。在美国国家科学基金会的资助下，首席研究员开发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，这对应于解释波如何表现得像粒子一样。现在拥有这样一个新的严谨的数学框架的影响将在许多不同的领域是巨大的，但目前的发展仍然是非常技术性的，除了与地震问题相关的弹性问题外，首席研究员在未来一年的工作将主要是理论性的，为未来的研究和应用开辟新的方向。

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.