Homogenization of Elliptic and Parabolic Partial Differential Equations

椭圆和抛物型偏微分方程的齐次化

• 批准号: RGPIN-2018-06371
• 负责人: Lin, Jessica
• 金额: $ 1.53万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678214
• 关键词: Homogenization; Elliptic; Parabolic; Partial; Differential

The theory of stochastic homogenization identifies the average, macroscopic behavior of a phenomenon which is subject to microscopic, random effects. For example, one may be interested in determining the general properties of a porous material with randomly distributed impurities, or predicting the evolution of a population in a heterogeneous medium with random obstacles. Such phenomena are typically modeled by partial differential equations (PDEs) with random coefficients which depend on microscopic lengthscales describing the heterogeneities. The random coefficients take into account all possible realizations of a physical environment, and by imposing certain hypotheses, one may expect that asymptotically on average, almost all such solutions exhibit the same effective behavior. ******The main goal of this proposal is to further the general understanding of elliptic and parabolic PDEs through the rich source of problems based in stochastic homogenization. The study of homogenization offers contributions to both theoretical and applied mathematics: homogenization often exposes many interesting problems in the analysis of the relevant equations, and it can be directly used to model physical processes. ******I plan to focus my efforts on two main classes of elliptic and parabolic PDEs: (a) Nondivergence Form Equations, and (b) Reaction-Diffusion Equations. Such equations serve as the primary mathematical models in stochastic control theory, finance, and geometry and chemical kinetics, combustion, and biology respectively. The projects I am proposing are motivated by the following two objectives: (1) To broaden the class of PDEs for which homogenization takes place and (2) To obtain more specific information about the process of homogenization, such as error estimates and properties of the effective behavior. ******The study of homogenization combines tools from several different areas of mathematics, including analysis, PDEs, dynamical systems, and probability. I am committed to using collaborative approaches for the proposed research program; drawing inspiration and techniques from various subfields. This flexible perspective promotes a unified understanding of the physical phenomena, as well as enhancing the theory for both nondivergence form and reaction-diffusion equations. Furthermore, progress in this specific research program may influence developments in the above related areas of mathematics.******Aside from the immediate applications to other subfields of mathematics, the study of multiscale problems has been a source of interest for specialists in several outside areas including materials science, chemical engineering, and biology. Consequently, this work contributes towards strengthening the relationship between the mathematical theory of PDEs and applications to other scientific disciplines.

随机均质化理论确定了一种现象的平均宏观行为，该现象受到微观，随机效应的影响。例如，人们可能有兴趣确定具有随机分布杂质的多孔材料的一般特性，或者预测具有随机障碍物的异质培养基中种群的演变。这种现象通常是通过偏微分方程（PDE）建模的，其随机系数取决于描述异质性的显微镜长度尺度。随机系数考虑了对物理环境的所有可能实现，并且通过施加某些假设，人们可能期望平均而言，几乎所有这些解决方案都表现出相同的有效行为。 *****该提案的主要目标是通过基于随机均质化的丰富问题来源进一步了解椭圆和抛物线PDE。同质化研究为理论和应用数学提供了贡献：均质化通常在相关方程的分析中暴露了许多有趣的问题，并且可以直接用于建模物理过程。 ******我计划将精力集中在两个主要类别的椭圆和抛物线pdes上：（a）非散发形式方程，以及（b）反应扩散方程。这些方程分别用作随机控制理论，金融以及几何学和化学动力学，燃烧和生物学的主要数学模型。我提出的项目是由以下两个目标激励的：（1）扩大发生同质化的PDE类别，以及（2）获得有关均质化过程的更具体信息，例如有效行为的错误估计和特性。 ******均化研究结合了来自几个不同数学领域的工具，包括分析，PDE，动力学系统和概率。我致力于为拟议的研究计划使用协作方法；从各个子场中汲取灵感和技术。这种灵活的观点促进了对物理现象的统一理解，并增强了非散发形式和反应扩散方程的理论。此外，该特定研究计划的进展可能会影响上述数学相关领域的发展。******除了直接应用到其他数学的其他子领域外，对多尺度问题的研究还对包括材料，化学工程和生物学在内的多个外部领域的专家的兴趣来源。因此，这项工作有助于加强PDE的数学理论与其他科学学科的应用之间的关系。