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

这个项目提出了在非线性偏微分方程组的数学分析中开发新的方法。其目的是更深入地理解具有三个不同类别的初始数据的动力学方程的全局时间存在唯一性定理。第一类是近真空初始数据。第二类包含具有无大小限制的初始数据的空间齐次方程。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，解开始时足够接近合适的Banach空间中的某个大的齐次解)。该项目的另一个目标是确定这些解的大时间收敛到均衡。这项研究将加深对偏微分方程的理解，进而引入新的技术，有望对未来的数学和物理发展有用。本项目从运动学理论和相关领域寻求一种多方面的方法来解决非线性偏微分方程中出现的物理问题。研究愿景包括开发新的工具来研究关于基本数学模型的分析问题；这些工具有望推广到应用数学中的一系列问题。该项目的成果将通过发表在期刊文章中传播，将在首席调查员的网站上公布，并将在国际会议上介绍。首席调查员计划召集与该提案有关的领域的专家。该项目将涉及宾夕法尼亚大学和其他大学的研究生和本科生。首席调查员致力于通过各种手段，包括与该项目有关的指导研究和研讨会，帮助培养这些学生。他将积极寻找来自不同背景的学生参与该项目。

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.