Mathematical Sciences: H-Measures and the Study of Oscillations and Concentration Effects in Systems of PartialDifferential Equations of Continuum Mechanics

数学科学：连续介质偏微分方程组中的 H 测度以及振荡和浓度效应的研究

• 批准号: 9100834
• 负责人: Luc Tartar
• 金额: $ 15万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-01 至 1994-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9100834&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Measures; Study; Oscillations

The mathematical theory of weak convergence has been shown to have important applications to problems arising in continuum physics because it describes the relations between microscopic variables and their averages at a macroscopic level. In order to average nonlinear quantities, a mathematical tool called compensated compactness has been developed by F. Murat and the principal investigator and it has been used to obtain various relations of a macroscopic nature from corresponding microscopic relations. An improvement of this tool, based on so called H- measures that first appeared in questions of homogenization, has been developed recently by the principal investigator. The goal of this project is to further study the mathematical properties of H-measures, enlarge the class of its applications to nonlinear partial differential equations of continuum mechanics, and define more general mathematical objects that could remedy some of the already known limitations of H- measures. Particular areas to be studied include effective properties of mixtures, nonlocal effects induced by homogenization, evolution of microstructures in composite materials and damping of waves in heterogeneous materials.

弱收敛的数学理论已经被证明对连续统物理学中出现的问题有重要的应用，因为它描述了微观变量和它们在宏观水平上的平均值之间的关系。为了求非线性量的平均，F. Murat和主要研究者开发了一种称为补偿紧性的数学工具，并利用它从相应的微观关系中得到各种宏观性质的关系。该工具的改进，基于所谓的H-措施，首次出现在均质化问题，已由首席研究员最近发展。本项目的目标是进一步研究H测度的数学性质，扩大其在连续介质力学非线性偏微分方程中的应用类别，并定义更一般的数学对象，以弥补一些已知的H测度的局限性。需要研究的特定领域包括混合物的有效特性、均质化引起的非局部效应、复合材料中微结构的演化以及非均质材料中的波阻尼。