Analysis of nonlocal effects in nonlinear parabolic partial differential equations

非线性抛物型偏微分方程中的非局部效应分析

• 批准号: 1412748
• 负责人: Maria Pia Gualdani
• 金额: $ 17万
• 依托单位: George Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1412748&HistoricalAwards=false
• 关键词: Analysis; nonlocal; effects; nonlinear; parabolic

It is a fundamental scientific and mathematical problem to understand how microscopic interactions influence physical phenomena at a macroscopic level. This project investigates this question for a class of mathematical models of systems that arise in several scientific disciplines. Applications range from game theory, which includes decision strategies, investors' behavior, and socioeconomic sciences, to astrophysics and plasma physics. A fundamental component of this project concerns the contribution to mathematical education and training of undergraduate and graduate students in STEM fields. The principal investigator will study a class of partial differential equations where microscopic effects are described by certain non-local operators. This research contains two different projects. The first project concerns the study of global well-posedness of non-local kinetic equations, with particular attention to the Landau equation. The Landau equation is a well-known equation that is used to model plasma with predominant grazing-type collisions. At present the well-posedness of the Landau equation for degenerate potentials is still an open problem and has given rise to several unresolved conjectures. The second project is related to a class of free boundary problems. The theoretical understanding of these and related projects presents analytical challenges and requires mathematical techniques that come from analysis, probability, and geometry. The skills developed through participation in this research project will be taken by students into the workforce.

了解微观相互作用如何在宏观层面影响物理现象是一个基本的科学和数学问题。 该项目研究了在几个科学学科中出现的一类数学模型的问题。应用程序范围从游戏理论（包括决策策略，投资者的行为和社会经济科学）到天体物理学和血浆物理学。该项目的一个基本组成部分涉及对STEM领域的本科生和研究生的数学教育和培训的贡献。 主要研究者将研究一类部分微分方程，其中某些非本地算子描述了微观效应。这项研究包含两个不同的项目。第一个项目涉及研究非本地动力学方程的全球适应性，并特别关注Landau方程。 Landau方程是一种众所周知的方程式，用于与主要放牧型碰撞建模等离子体。目前，降级电位的兰道方程的适合性仍然是一个开放的问题，已经引起了几个未解决的猜想。第二个项目与一类自由边界问题有关。对这些及相关项目的理论理解提出了分析挑战，并且需要分析，概率和几何形状带来的数学技术。通过参与该研究项目发展的技能将由学生带入劳动力。