Harmonic Analysis and Homogenization of Elliptic Equations in Perforated Domains

穿孔域椭圆方程的调和分析与齐次化

• 批准号: 2153585
• 负责人: Zhongwei Shen
• 金额: $ 21.31万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153585&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Homogenization; Elliptic; Equations

Partial differential equations with rapidly oscillating coefficients are used to describe various processes in materials such as composite materials and porous media. The theory of homogenization, whose goal is to describe macroscopic properties of microscopically inhomogeneous or heterogeneous materials, shows that such strongly inhomogeneous material, whose characteristics change sharply with respect to space variables, may be approximately described via a homogenized or effective homogeneous material. As a result, the theory of homogenization of partial differential equations with rapidly oscillating coefficients has many important applications in physics, mechanics, and materials science. The long-term goal of this project is to establish optimal quantitative results in the homogenization theory for a large class of partial differential equations in various settings, arising in applications. The proposed research focuses on several challenging problems in the modelling of fluid flows and acoustic propagation in porous media and inclusions in composite materials. The new approaches and techniques to be developed will provide theoretical foundation and guidance for numerical simulations of diffusion processes in highly heterogenous media. Funding for this project provides support for training of Ph.D. students. The main focus of this project on quantitative homogenization of partial differential equations is on large-scale regularity properties and convergence rates for second-order elliptic equations and systems in perforated domains. More specifically, the problems investigated as part of the project include (1) uniform regularity estimates for Darcy’s law; (2) large-scale regularity estimates for Brinkman’s law; (3) elliptic equations and systems with periodic and high-contrast coefficients; and (4) boundary value problems in perforated domains. The resolution of these problems will provide a deeper understanding of some fundamental issues in periodic homogenization. This project lies at the interface of harmonic analysis and partial differential equations. Existing and new techniques from harmonic analysis are expected to play a significant role in the work of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

具有快速振荡系数的偏微分方程用于描述复合材料和多孔介质等材料的各种过程。均质化理论的目标是描述微观上不均匀或非均质材料的宏观性质，表明这种特性随空间变量急剧变化的强非均质材料可以通过均质化或有效均质材料来近似描述。因此，具有快速振荡系数的偏微分方程的均匀化理论在物理、力学和材料科学中有许多重要的应用。本项目的长期目标是为应用中出现的各种设置下的大类偏微分方程建立均匀化理论的最佳定量结果。提出的研究重点是多孔介质和复合材料包裹体中流体流动和声传播建模中的几个具有挑战性的问题。所开发的新方法和新技术将为高度非均质介质中扩散过程的数值模拟提供理论基础和指导。本项目的经费用于培养博士研究生。本项目在偏微分方程的定量均匀化方面的主要重点是研究二阶椭圆方程和系统在穿孔区域的大规模正则性和收敛速率。更具体地说，作为项目的一部分，研究的问题包括(1)达西定律的统一规则估计；(2) Brinkman定律的大尺度正则性估计；(3)具有周期系数和高对比系数的椭圆方程和系统；(4)穿孔区域的边值问题。这些问题的解决将使我们对周期均匀化的一些基本问题有更深入的了解。本课题是调和分析与偏微分方程的结合。现有的和新的谐波分析技术有望在该项目的工作中发挥重要作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。