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.

该项目的重点是了解物理和工程中常见的某些类型的偏微分方程（PDE），例如控制弹性和电导率的偏微分方程。当我们研究材料在应力下如何变形或导电时，我们经常使用方程来描述这些现象。然而，有些方程表现得并不顺利，特别是在处理高对比度材料或复杂形状时，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。流体动力学。理解这些问题对于实际应用至关重要设计飞机或预测天气模式等应用，并且还激发了数学和统计学方面的新想法。最后，首席研究员 (PI) 对动力学方程感兴趣，该方程通过研究描述粒子如何在核聚变实验等系统中移动和相互作用。通过这些方程，科学家们希望提高我们对极端条件下等离子体行为的理解，例如在托卡马克内部。该项目为研究生提供研究培训机会，作为该项目的一部分，首席研究员将开展与上述密切相关的研究。主题并将尝试解决这些领域的一些开放问题将集中在几个项目上，这些项目可分为三个主要主题领域：首先，该项目将开发研究复合材料中出现的椭圆方程的新方法（例如弹性问题、电导率问题）。 ). PI 对包含 Lipschitz 的域中的偏微分方程解的爆炸行为特别感兴趣，涉及 p-拉普拉斯方程，以及 Lamé 系统的绝缘问题。涉及渗透多孔介质的不可压缩流体的自由边界问题，通常称为单相 Muskat 问题，重点是研究整个空间中二维和三维单相 Muskat 问题的解的规律性。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。相对论性 Vlasov-Maxwell-Landau 系统和一般领域中的空间非齐次玻尔兹曼方程。该项目由数学科学部分析计划 (DMS) 和刺激竞争性研究既定计划 (EPSCoR) 联合资助。该奖项反映了通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。