Analysis of nonlocal effects in nonlinear parabolic partial differential equations

理解微观相互作用如何影响宏观层面的物理现象是一个基本的科学和数学问题。 这个项目调查了这个问题的一类数学模型的系统，出现在几个科学学科。应用范围从博弈论（包括决策策略、投资者行为和社会经济科学）到天体物理学和等离子体物理学。该项目的一个基本组成部分是对数学教育和STEM领域本科生和研究生培训的贡献。 主要研究者将研究一类偏微分方程，其中微观效应由某些非局部算子描述。这项研究包含两个不同的项目。第一个项目是研究非局部动力学方程的整体适定性，特别是朗道方程。朗道方程是一个众所周知的方程，用于模拟等离子体与主要的掠射型碰撞。关于简并势的朗道方程的适定性问题，目前仍是一个未解决的问题。第二个项目是关于一类自由边界问题。对这些和相关项目的理论理解提出了分析挑战，需要来自分析，概率和几何的数学技术。通过参与这项研究项目开发的技能将由学生进入劳动力市场。

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.