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.

该项目的重点是理解物理和工程中通常遇到的某些类型的部分微分方程（PDE），例如那些管理弹性和电导率的部分。当我们研究材料在压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。但是，某些方程式的行为并不顺利，尤其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式更难分析，并且传统方法可能行不通。研究的另一个领域是流体动力学的方程式。理解这些问题对于实用应用至关重要，例如设计飞机或预测天气模式，并且还激发了数学和统计学方面的新想法。最后，主要研究者（PI）在动力学方程中很有趣，该方程描述了粒子在核融合实验等系统中的移动和相互作用。通过研究这些方程式，科学家希望提高我们对等离子体在极端条件（例如Tokamak）中的行为的理解。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，PI将进行与近似元素主题密切相关的研究，并将试图解决这些领域的一些开放问题。重点将放在几个可以收集到三个主要主题领域的项目上。首先，该项目将开发新的方法来研究复合材料（例如弹性问题，电导率问题）中产生的椭圆方程。 PI对具有Lipschitz夹杂物的域中PDE解决方案的爆炸行为特别感兴趣，涉及P-Laplacian的方程以及Lamé系统的绝缘问题。其次，该项目将探索涉及多孔培养基的不可压缩流体的自由边界问题，通常称为单相麝香问题。重点将是研究整个空间中二维和三维单相的麝香问题的规律性，以及探索这些解决方案的短期和长期平滑效应。最后，该项目将研究线性动力学方程的边界规则性，以及非线性动力学方程的稳定性和全球范围，包括相对论Vlasov-Maxwell-landau系统以及空间上的不均匀性Boltzmann方程在一般领域中的竞争性计划（Mathem Interiality in Mathem Interials）在Mathem Inalital Insportive（Matherm of Science）中进行了竞争。 （EPSCOR）。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子优点和更广泛的影响评估标准评估被认为是珍贵的支持。