Analysis of non-linear partial differential equations in Kinetic theory and related fields

本计画旨在发展非线性偏微分方程式之数学分析新方法。本课程的目标是对三种不同类型的初始数据的动力学方程的整体时间存在性和唯一性定理有更深的理解。第一类是邻近真空初始数据。第二类包括空间上均匀的方程，其初始数据没有大小限制。在此基础上，第三类初始数据将是那些大但弱不均匀的数据（即，该解开始时足够接近适当的Banach空间中的某个大齐次解）。该项目的另一个目标是确定这些解决方案的大时间收敛到平衡。本研究课题旨在通过对偏微分方程的深入理解，为今后的数学和物理学的发展引进新的技术。本研究课题从动力学理论和相关领域出发，对非线性偏微分方程中的物理问题进行多方面的研究。研究愿景包括开发新的工具来研究基本数学模型的分析问题;这些工具预计将推广到应用数学中的各种问题。该项目的结果将通过在期刊文章中发表来传播，将张贴在主要研究员的网站上，并将在国际会议上介绍。首席研究员计划召集与本提案有关领域的专家。该项目将涉及来自宾夕法尼亚大学和其他大学的研究生和本科生。首席研究员致力于通过各种方式帮助促进这些学生的培训和教育，包括与该项目相关的定向研究和研讨会。他将积极寻找来自不同背景的学生来参与这个项目。

This project proposes to develop new methods in the mathematical analysis of nonlinear partial differential equations. The objective is to gain a deeper understanding of global-in-time existence and uniqueness theorems for kinetic equations with initial data from three different categories. The first category is nearby vacuum initial data. The second category encompasses spatially homogenous equations with initial data that do not have a size restriction. Building on that, the third category of initial data will be those that are large but weakly inhomogeneous (i.e., the solution starts out sufficiently close to some large homogeneous solution in a suitable Banach space). Another goal of the project is to determine the large-time convergence to equilibrium of these solutions. This research will lead to a deeper understanding of partial differential equations, which in turn introduce novel techniques that are expected to be useful for future mathematical and physical developments.This project pursues a multifaceted approach to physical questions arising in nonlinear partial differential equations from kinetic theory and related fields. The research vision includes developing new tools to study analytical problems about fundamental mathematical models; these tools are expected to be generalizable to a wide array of problems in applied mathematics. The results of the project will be disseminated through publication in journal articles, will be posted on the principal investigator's web site, and will be presented at international conferences. The principal investigator plans to bring together experts in areas related to this proposal. The project will involve graduate and undergraduate students from the University of Pennsylvania and other universities. The principal investigator is committed to help foster the training and education of these students through a variety of means, including directed studies and seminars related to the project. He will actively seek out students from diverse backgrounds to work on the project.