Analysis of nonlocal effects in nonlinear parabolic partial differential equations

理解微观相互作用如何在宏观水平上影响物理现象是一个基本的科学和数学问题。这个项目研究了在几个科学学科中出现的一类系统数学模型的这个问题。应用范围从博弈论（包括决策策略、投资者行为、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STEM领域的本科生和研究生的数学教育和培训的贡献。首席研究员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。第一个项目涉及非局部动力学方程的全局适定性研究，特别关注朗道方程。朗道方程是一个著名的方程，用于模拟以掠食型碰撞为主的等离子体。目前，简并势朗道方程的适定性仍然是一个未解的问题，并引起了一些未解的猜想。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而发展的技能将被学生带到工作岗位上。

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.