Analysis of Nonlinear Partial Differential Equations in Free Boundary Fluid Dynamics, Mathematical Biology, and Kinetic Theory

该项目旨在开发新的方法，以提高目前的科学知识水平，在非线性偏微分方程（PDE）的数学分析的两个不同领域的各种公认的问题，并开发新的数学方法来研究这些系统。该项目的第一部分涉及数学生物学中偏微分方程分析的基本问题。本计画的第二部分是研究在物理动力学边界条件下，从动力学理论出发的偏微分方程。该项目将涉及来自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。该项目还将通过宾夕法尼亚大学本科生研究奖学金计划中心与本科生进行外联。主要研究者（PI）完全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，以进一步发展多元化和具有全球竞争力的STEM劳动力的目标，并改善大学一级的STEM教育。PI正在参与对传统上在数学方面代表性不足的群体的外联活动，这些活动将在本项目期间继续进行。PI还不断努力通过提供国家和国际研究报告来增加社区的科学知识。该项目的结果将通过在期刊文章中发表进一步传播，并将在主要参与者的网站上公布。 在科学和工程中，弹性结构与周围流体相互作用的问题，称为流体-结构相互作用问题（FSI）是大量存在的。这些问题在物理学、生物学和医学中有许多应用。这类FSI问题包括鸟类飞行、鱼类游泳以及血液流过心脏和血管的数学模型。流固耦合模型已被广泛研究，使用计算方法。许多数值算法已经开发了这样的问题，和FSI问题的科学计算仍然是一个非常活跃的研究领域。尽管这些计算方法很重要，但从分析的角度来看，人们对它们的理解却很少。数值分析的一个主要障碍是缺乏对基本非线性偏微分方程的分析理解。一个更好的理论理解这些偏微分方程的分析方面，应导致改进的计算算法FSI问题。PI旨在解决这些问题，专注于一组典型的分析FSI问题的斯托克斯流。这个项目的第二部分是动力学理论。含库仑势的朗道方程和含长程相互作用势的无截止Boltzmann方程是描述非平衡稀薄气体和稀热等离子体动力学的两个基本碰撞动力学模型。等离子体出现在天体物理学、核聚变和托卡马克的基本物理问题中。玻尔兹曼方程已被广泛用作数学模型，例如在高层大气空气动力学中，空气是非常稀薄的气体，流体方程可能是不够的。此外，这些动力学方程的边界效应的研究在物理上是非常重要的，因为它们描述了气体或等离子体与固体之间的相互作用，以阻力和传热的形式。本研究的目的是充分了解大初始数据的局部时间适定性，以及在接近平衡设置的全局时间适定性，对于几个不同的基本物理模型在非线性偏微分方程。我们期望作为该项目的一部分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 develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in two different areas of the mathematical analysis of nonlinear partial differential equations (PDE) and to develop new mathematical methods to study these systems. The first part of the project concerns basic questions in the analysis of partial differential equations in Mathematical Biology. The second part of this project concerns the study of partial differential equations from kinetic theory in the presence of the physical kinetic boundary conditions. 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 University of Pennsylvania Center for Undergraduate Research & Fellowships program. The principal investigator (PI) is fully committed to facilitating 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 underrepresented in mathematics, and these activities will continue over the course of this project. The PI is further consistently working to increase the scientific knowledge of the community by giving national and international research presentations. The results of this project will be further disseminated through publication in journal articles and they will be posted on the PI's website. Problems in which an elastic structure interacts with the surrounding fluid, called Fluid-Structure Interaction (FSI) problems, are plentiful in science and engineering. These problems have many applications in Physics, Biology, and the Medical Sciences. Such FSI problems include the mathematical modeling of the flying of birds, the swimming of fish, and blood flow through the heart and blood vessels. FSI models have been intensively studied using computational methods. Many numerical algorithms have been developed for such problems, and the scientific computing of FSI problems continues to be a very active area of research. Despite their importance, these computational methods are poorly understood from an analytical standpoint. A major impediment for numerical analysis has been the lack of analytical understanding of the underlying nonlinear partial differential equations. A better theoretical understanding of the analytical aspects of these PDE should lead to improved computational algorithms for FSI problems. The PI seeks to address these issues by focusing on a set of canonical analytical FSI problems in the Stokes flow. The second part of this project is in kinetic theory. The Landau equation with Coulomb potential and the non-cutoff Boltzmann equation for the long-range interaction potentials are two fundamental mathematical models in collisional kinetic theory which describe the dynamics of a non-equilibrium rarefied gas and a dilute hot plasma. Plasmas appear in fundamental physical problems from Astrophysics, Nuclear fusion, and Tokamaks. The Boltzmann equation has been used as a mathematical model in a wide variety of places, for instance in high atmosphere aerodynamics, where the air is a very rarefied gas and fluid equations are probably not sufficient. Further, the study of boundary effects for these kinetic equations is physically very important because they describe the interaction, in the form of drag and heat transfer, between gas or plasma and a solid body. The objective of this research in both parts of the project is to fully understand the local-in-time well-posedness for large initial data, and the global-in-time well-posedness in a close to equilibrium setting, for several different fundamental physical models in nonlinear PDE. We expect that the techniques developed as part of this project 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.