Topics in Fluid dynamics with free boundaries, and Kinetic theory

自由边界流体动力学和动力学理论主题

基本信息

批准号：
1500916
负责人：
Robert Strain
金额：
$ 27万
依托单位：
University of Pennsylvania
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-06-01 至 2019-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500916&HistoricalAwards=false
关键词：
Topics Fluid dynamics free boundaries

项目摘要

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.

这项研究建议开发新的方法，以在非线性部分偏微分方程的数学分析的几个不同领域中促进当前的科学知识水平。该项目的第一部分涉及有关流体动力学的问题，预计这些问题的解决方案将增加我们对水浪，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的角度研究了等离子体的动力学，我们预计这些研究将增加我们对诸如太阳风，银河系和范·艾伦辐射带等物理现象的理解。该项目的第三部分侧重于研究相对论动力学理论，预计该研究将增加我们在天体物理学各种各样的各种位置的身体理解，例如，在高气氛空气动力学中，空气是一种非常稀有的气体和流体方程，可能不足。该项目将涉及培训宾夕法尼亚大学及其他地区的博士后研究人员，研究生和本科生的研究，并参与了代表性不足的群体。 PI正在努力开发创新的积极学习演算课程，以进一步发展发展多样化且具有全球竞争性的STEM劳动力并在大学一级改善STEM教育的目标。拟议的研究的目的是在非线性偏微分方程中的几种不同的基本物理模型发生时，完全了解解决方案的全局存在和解决方案的奇异性形成。这项研究的一部分是研究具有自由边界的流体动力学问题，例如Muskat问题和表面准地斑方程。这项工作的另一部分着眼于与相对论弗拉索夫·马克斯韦（Vlasov-Maxwell System）有关的问题，该系统是血浆物理学的基本模型。该提案的第三部分将研究相对论鲍尔茨曼方程的问题，该方程是相对论动力学理论的中心模型。 PI建议在对这些不同方程式的更深入了解过程中开发几种新方法，以分析部分微分方程。可以预期，这些技术将对未来的数学和物理发展有用。