Analysis of Diffusion Equations with Nonlinear Singular Sources in Mean Field Games

本课题的目标是研究一类在生活科学和社会科学中作为数学模型出现的非线性扩散方程的演化动力学：(1)分析一、二维空间中自由边界问题解的存在性和长时间行为；(2)耦合于Fokker-Planck扩散方程的Hamilton-Jacobi-Bellman方程解的正则性研究；(3)非线性奇异源抛物型方程的长时间行为分析。这些方程由于其高阶微分和潜在的非线性结构而存在一些困难，需要开发新的分析工具：一些关键技术涉及经典偏微分方程技术与统计力学、最优控制和动力学理论方法的结合。特别感兴趣的是对模型的稳定性和有限时间爆破现象的研究：在稳定的情况下，将研究大时间解的行为（可能的分岔或收敛到平衡状态）。在爆破情况下，分析奇点的形成与相关量的爆破之间的关系。在社会学、经济学和生命科学中，复杂的、现实生活中的系统通常包含大量相互作用并能形成集体行为的个体；这些正是本项目所涉及的科学领域，数学可以在这些领域提供深刻的见解和贡献。对扩散过程和集体行为的理解在许多科学领域都是必不可少的。该项目侧重于研究一类平均场和动力学模型，这些模型可以很好地描述代理之间的相互作用、集体行为和平均过程。这种研究的应用范围从博弈论（决策策略、投资者行为）、社会经济（意见形成、人口动态）到生物学（肿瘤生长、群集）和神经科学。该项目还将通过研究为本科生和研究生提供教育和培训。

The goal of this project is to investigate the evolution dynamics of a class of nonlinear diffusion equations that appear as mathematical models in life and social science: (1) Analysis of existence and long time behavior of solutions of a free boundary problem in one and two dimensional space;(2) Study of the regularity of solutions to a Hamilton-Jacobi-Bellman equation coupled to a Fokker-Planck diffusion equation;(3) Analysis of long-time behavior of parabolic equations with nonlinear singular sources. These equations present several difficulties due to high differential order as well as the underlying nonlinear structure, and new analytical tools need to be developed: some key techniques involve combination of classical partial differential equations technique with methods from statistical mechanics, optimal control and kinetic theory. Of particular interest is the study of stability and finite time blow-up phenomena in the model: in case of stability, behavior of solution for large time will be investigated (possible bifurcations or convergence towards equilibrium state). In case of blow-up, relation between formation of singularities and blow-up of the related quantities will be analyzed. Complex, real-life systems in sociology, economics, and life sciences often contain a large number of individuals that interact and can develop a collective behavior; these are exactly the areas of sciences where this project draws the topics from and where mathematics could offer great insights and contributions. The understanding of diffusion processes and collective behavior is essential in many areas of science. The project focuses on investigating a class of mean field and kinetic models that can describe very well interactions among agents, collective behavior and averaging processes. Applications of such a study range from game theory (decisional strategies, behavior of investors), socio-economic (opinion formation, population dynamics), to biology (tumor growth, flocking) and neuroscience. The project will also provide education and training through research to undergraduate and graduate students.