Problems in Regularity Theory of Partial Differential Equations

该项目侧重于理解物理和工程中常见的某些类型的偏微分方程（PDE），例如那些管理弹性和导电性的偏微分方程。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，有些方程的表现并不平滑，尤其是在处理高对比度材料或复杂形状时。这些情况可能导致方程更难分析，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，它也激发了数学和统计学的新思想。最后，首席研究员（PI）对动力学方程感兴趣，动力学方程描述了粒子如何在核聚变实验等系统中移动和相互作用。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（如托卡马克内部）行为的理解。该项目为研究生提供研究培训机会。 作为该项目的一部分，PI将开展与上述主题密切相关的研究，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，这些项目可归纳为三个主要专题领域。首先，该项目将开发新的方法来研究复合材料中产生的椭圆方程（例如，弹性问题、导电性问题）。PI特别感兴趣的是解的爆破行为，以偏微分方程在域与Lipschitz包含，方程涉及p-Laplacian，和绝缘问题的Lamé系统。其次，该项目将探讨涉及渗透多孔介质的不可压缩流体的自由边界问题，通常被称为单相Muskat问题。重点将是调查在整个空间的二维和三维的单相Muskat问题的解决方案的规律性，以及探索这些解决方案的短期和长期的平滑效果。最后，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Vlasov-Maxwell-朗道系统和一般域上的空间非齐次Boltzmann方程。本项目由数学科学部分析计划和刺激竞争研究计划联合资助。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。

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.