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

自由边界流体动力学和运动理论中非线性偏微分方程的分析

基本信息

批准号：
1764177
负责人：
Robert Strain
金额：
$ 18万
依托单位：
University of Pennsylvania
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-15 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764177&HistoricalAwards=false
关键词：
Analysis Non Linear Partial Differential

项目摘要

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.

该项目旨在在研究非线性部分微分方程的研究中促进科学理解并开发新方法，这些方程可预测流体流动流动边界的未来行为，并在动力学理论中预测粒子动力学。该项目的第一部分考虑了石油工程中的基本数学模型，该模型是由石油工程师M. Muskat在1934年制定的，将水的混合物描述为油砂。该项目的第二部分考虑了用于稀热等离子体的高度精确数学模型的研究。这些类型的等离子体经常出现在天体物理学，核融合和Tokamaks的基本物理问题中。在该项目的第三部分中，PI将研究相对论粒子动力学的基本部分微分方程，并且可以预期，这项研究将增加我们在天体物理学各个地方的身体理解，例如，在高气层空气动力学中，空气是一种非常稀有的气体和流体方程式不再是合适的数学模型。该项目将涉及宾夕法尼亚大学和其他大学的博士后研究人员，研究生和本科生的研究和教学培训。该项目还将通过UPENN本科研究与奖学金计划与本科生的宣传。 PI完全致力于通过教学课程，定期直接指导以及定期进行研究研讨会来促进这些学生的培训和教育。 PI正在积极努力在宾夕法尼亚大学开发新的创新数学课程，以进一步促进发展多样化且具有全球竞争性的STEM劳动力并在大学水平上改善STEM教育的目标。 PI正在向传统上代表数学不足的团体进行外展活动，这些活动将在该项目的整个过程中继续进行。这项研究的目的是充分了解非线性偏微分方程中几种不同基本物理模型的全球存在和奇异性形成。这项工作的第一部分研究流体动力学问题，即自由边界，例如表面准斑系方程和麝香式问题。这项工作的另一部分着眼于血浆物理学的相对论兰道方程的存在和独特问题。该项目的第三部分考虑了与动力学理论的相对论鲍尔茨曼方程中的数学问题，并具有物理上相关的不可积分粒子相互作用。 PI提议开发新方法，以在非线性部分偏微分方程的数学分析中，在这些不同领域的各种公认问题集合中提高当前的科学知识水平。预计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NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。