Analysis of Non-Linear Partial Differential Equations in Free Boundary Fluid Dynamics and Kinetic Theory

该项目旨在促进科学理解和开发新的方法，研究预测未来运动边界流体流动行为的非线性偏微分方程，以及动力学理论中的粒子动力学。项目的第一部分考虑了石油工程中的一个基本数学模型，该模型是由石油工程师M. Muskat在1934年提出的，用于描述油砂中水的混合物。本课题的第二部分考虑了稀热等离子体的高精度数学模型的研究。这些类型的等离子体经常出现在天体物理学、核聚变和托卡马克等基本物理问题中。在这个项目的第三部分，PI将研究相对论粒子动力学的基本偏微分方程，预计这项研究将增加我们对天体物理学各个领域的物理理解，例如在高大气空气动力学中，空气是一种非常稀薄的气体，流体方程不再是合适的数学模型。本项目将涉及宾夕法尼亚大学和其他大学的博士后、研究生和本科生的科研和教学培训。该项目还包括通过宾夕法尼亚大学本科生研究中心和奖学金项目向本科生进行拓展。学院致力于通过教学课程、定期直接指导和定期举办研究研讨会，促进这些学生的培训和教育。PI正积极致力于在宾夕法尼亚大学开发新的创新数学课程，以进一步实现培养多样化和具有全球竞争力的STEM劳动力的目标，并改善大学水平的STEM教育。PI正在向传统上在数学方面代表性不足的群体开展外展活动，这些活动将在本项目的过程中继续进行。本研究的目的是充分理解非线性偏微分方程中几种不同基本物理模型的整体存在性和奇点形成。本文的第一部分研究了具有自由边界的流体动力学问题，如曲面准地转方程和Muskat问题。本工作的另一部分着眼于等离子体物理学相对论朗道方程的存在性和唯一性问题。本项目的第三部分考虑了与物理相关的不可积粒子相互作用的动力学相对论玻尔兹曼方程中的数学问题。PI建议开发新的方法，以提高当前在非线性偏微分方程的数学分析这些不同领域的各种公认问题的科学知识水平。预计所开发的技术将对未来的数学和物理发展有用。该奖项反映了美国国家科学基金会的法定使命，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 aims to advance scientific understanding and develop new methods in the study of non-linear partial differential equations that predict the future behavior of fluid flow with moving boundaries, and particle dynamics in Kinetic Theory. The first part of the project considers a basic mathematical model in petroleum engineering, which was formulated by the petroleum engineer M. Muskat in 1934 to describes the mixture of water into an oil sand. The second part of this project considers the study of a highly accurate mathematical model for a dilute hot plasma. These types of plasmas appear regularly in fundamental physical problems from astrophysics, nuclear fusion, and tokamaks. In the third part of this project the PI will study fundamental partial differential equations from relativistic particle dynamics and it is expected that this research will increase our physical understanding in a variety of places in astrophysics, for instance in high atmosphere aerodynamics where the air is a very rarefied gas and fluid equations are no longer a suitable mathematical model. This project will involve training in research and teaching of postdoctoral researchers, graduate students and undergraduate students from the University of Pennsylvania and other universities. The project will also involve outreach to undergraduate students through the UPenn Center for Undergraduate Research & Fellowships program. The PI is fully committed to facilitate the training and education of these students through teaching courses, regular direct mentoring, and running regular research seminars. The PI is actively working to develop new innovative mathematics courses at the University of Pennsylvania in order to further the goal of developing a diverse and globally competitive STEM workforce and to improve STEM education at the collegiate level. The PI is engaging in outreach activities to groups that are traditionally under-represented in mathematics, and these activities will continue over the course of this project. The objective of this research is to fully understand both global existence and singularity formation for several different fundamental physical models in non-linear partial differential equations. The first part of this work studies fluid dynamics problems with free boundaries, such as the Surface Quasi-Geostrophic equations and the Muskat problem. Another part of this work looks at existence and uniqueness problems for the relativistic Landau equation from plasma physics. And a third part of this project considers mathematical problems in the the relativistic Boltzmann equation from Kinetic theory with physically relevant non-integrable particle interactions. The PI proposes to develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in these different areas in the mathematical analysis of non-linear partial differential equations. It is expected that the techniques developed will be useful for future mathematical and physical developments.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.