Problems in Regularity Theory of Partial Differential Equations

偏微分方程正则论中的问题

• 批准号: 2350129
• 负责人: Hongjie Dong
• 金额: $ 35.12万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350129&HistoricalAwards=false
• 关键词: Problems; Regularity; Theory; Partial; Differential

This project focuses on understanding certain types of partial differential equations (PDE) commonly encountered in physics and engineering, such as those governing elasticity and conductivity. When we study how materials deform under stress or conduct electricity, we often use equations to describe these phenomena. However, some equations don't behave smoothly, especially when dealing with high contrast materials or complex shapes. These situations can lead to equations that are much harder to analyze, and traditional methods may not work. Another area of study is equations from fluid dynamics. Understanding these questions is crucial for practical applications like designing airplanes or predicting weather patterns, and it also inspires new ideas in mathematics and statistics. Finally, the Principal Investigator (PI) is interested in kinetic equations, which describe how particles move and interact in systems like nuclear fusion experiments. By studying these equations, scientists hope to improve our understanding of how plasmas behave in extreme conditions, such as inside a tokamak. The project provides research training opportunities for graduate students. As part of this project, the PI will carry out research closely related to the aforementioned topics and will attempt to address some of the open problems in these areas. The focus will be on several projects that can be gathered into three main topical areas. First, the project will develop new methods to study elliptic equations arising in composite materials (e.g., elasticity problems, conductivity problems). The PI is particularly interested in the blowup behaviors of solutions to PDE in domains with Lipschitz inclusions, equations involving the p-Laplacian, and the insulated problem for the Lamé system. Second, the project will explore the free boundary problem involving an incompressible fluid permeating a porous medium, often referred to as the one-phase Muskat problem. The focus will be on investigating the regularity of solutions to the two- and three-dimensional one-phase Muskat problem in the whole space, as well as on exploring the short-term and long-term smoothing effects of these solutions. Finally, the project will investigate boundary regularity of linear kinetic equations as well as the stability and global well-posedness of nonlinear kinetic equations, including the relativistic Vlasov-Maxwell-Landau system and the spatially inhomogeneous Boltzmann equations in general domains.This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences (DMS) and the Established Program to Stimulate Competitive Research (EPSCoR).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.

本课题侧重于理解物理和工程中常见的某些类型的偏微分方程组(PDE)，例如控制弹性和电导率的偏微分方程组。当我们研究材料在应力或导电作用下如何变形时，我们经常用方程来描述这些现象。但是，有些方程的运行并不顺畅，尤其是在处理高对比度材质或复杂形状时。这些情况可能会导致方程更难分析，而传统方法可能不起作用。另一个研究领域是流体动力学方程。理解这些问题对于设计飞机或预测天气模式等实际应用至关重要，它还启发了数学和统计学的新想法。最后，首席调查员(PI)对动力学方程感兴趣，它描述了粒子在核聚变实验等系统中如何运动和相互作用。通过研究这些方程，科学家们希望提高我们对等离子体在极端条件下的行为的理解，比如在托卡马克内部。该项目为研究生提供了研究培训机会。作为该项目的一部分，国际和平研究所将开展与上述专题密切相关的研究，并将试图解决这些领域的一些公开问题。重点将放在几个项目上，这些项目可以集中到三个主要的专题领域。首先，该项目将开发新的方法来研究复合材料中出现的椭圆型方程(例如，弹性问题、传导性问题)。PI特别关注具有Lipschitz包含的区域中的偏微分方程解的爆破行为，涉及p-Laplace算子的方程，以及Lamé系统的绝缘问题。其次，该项目将探索涉及渗透在多孔介质中的不可压缩流体的自由边界问题，通常被称为单相马斯卡特问题。重点将研究二维和三维单相马斯卡特问题在整个空间的解的规律性，以及探索这些解的短期和长期平滑效果。最后，该项目将研究线性动力学方程的边界正则性以及非线性动力学方程的稳定性和全局适定性，包括相对论的Vlasov-Maxwell-Landau系统和一般领域的空间非齐次Boltzmann方程。该项目由数学科学部门(DMS)的分析计划和已建立的刺激竞争研究计划(EPSCoR)联合资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。