Topics in Fluid dynamics with free boundaries, and Kinetic theory

本研究提出开发新的方法，以提高当前在非线性偏微分方程数学分析的几个不同领域的各种公认问题的科学知识水平。该项目的第一部分涉及有关流体动力学的问题，这些问题的解决方案有望增加我们对水波、海啸、飓风和其他流体现象的理解。该项目的第二部分从数学角度研究等离子体动力学，我们预计这些研究将增加我们对太阳风、银河星云和范艾伦辐射带等物理现象的理解。该项目的第三部分着重于相对论动力学理论的研究，预计这项研究将增加我们对天体物理学中各种各样的物理理解，例如在高大气空气动力学中，空气是一种非常稀薄的气体，流体方程可能不够。该项目将涉及宾夕法尼亚大学及其他大学的博士后研究人员、研究生和本科生的研究培训，并有代表性不足的群体参与。PI正在努力开发创新的主动学习微积分课程，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。本文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解的全局时间存在性和解的奇点形成。本研究的一部分是研究具有自由边界的流体动力学问题，如Muskat问题和曲面准地转方程。这项工作的另一部分着眼于与相对论性弗拉索夫-麦克斯韦系统有关的问题，该系统是等离子体物理学的基本模型。本提案的第三部分将研究相对论玻尔兹曼方程的问题，它是相对论运动理论的中心模型。PI建议在加深对这些不同方程的理解的过程中，开发几种新的偏微分方程分析方法。预期所开发的技术对未来数学和物理的发展是有用的。

This research proposes to develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in a few different areas of the mathematical analysis of non-linear partial differential equations. The first part of this project involves questions regarding the dynamics of fluids, and solutions to these questions are expected to increase our understanding of water waves, tsunamis, hurricanes, and other fluid phenomena. The second part of this project studies the dynamics of plasmas from a mathematical point of view, and we anticipate that these studies will increase our understanding of the physical phenomena such as the solar wind, galactic nebulae, and the Van Allen radiations belts. The third part of this project focuses on the study the relativistic Kinetic theory and it is expected that the research will increase our physical understanding in a wide variety of places in astrophysics, for instance in high atmosphere aerodynamics where the air is a very rarefied gas and fluid equations are probably not sufficient. This project will involve training in research of postdoctoral researchers, graduate students and undergraduate students from the University of Pennsylvania and beyond, with participation of under-represented groups. The PI is working to develop innovative Active Learning Calculus courses 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 objective of the proposed research is to fully understand both global in time existence of solutions and singularity formation of solutions when this occurs for several different fundamental physical models in non-linear partial differential equations. One part of this research is to study fluid dynamics problems with free boundaries such as the Muskat problem and the Surface Quasi-Geostrophic equations. Another part of this work looks at problems related to the relativistic Vlasov-Maxwell system which is a fundamental model of plasma physics. And a third part of this proposal will study problems on the the relativistic Boltzmann equation which is the central model in relativistic Kinetic theory. The PI proposes to develop several new methods in the Analysis of partial differential equations in the course of developing deeper understanding of these different equations. It is expected that the techniques developed to be useful for future mathematical and physical developments.