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

平均场博弈中非线性奇异源扩散方程分析

• 批准号: 1109682
• 负责人: Maria Pia Gualdani
• 金额: $ 20.84万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109682&HistoricalAwards=false
• 关键词: Analysis; Diffusion; Equations; Nonlinear; Singular

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.

本项目的目标是研究一类在生命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）分析一维和二维空间中自由边值问题解的存在性和长时间行为：（2）研究与Fokker-Planck扩散方程耦合的Hamilton-Jacobi-Bellman方程解的正则性;（3）具有非线性奇异源的抛物型方程的长时间性态分析。这些方程提出了一些困难，由于高微分阶以及潜在的非线性结构，和新的分析工具需要开发：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的方法相结合。特别感兴趣的是研究模型中的稳定性和有限时间爆破现象：在稳定的情况下，将研究大时间的解的行为（可能的分叉或收敛到平衡状态）。在爆破的情况下，将分析奇点的形成与相关量的爆破之间的关系。 社会学、经济学和生命科学中的复杂的现实系统通常包含大量相互作用并可以发展集体行为的个体;这些正是本项目从中引出主题的科学领域，数学可以提供巨大的见解和贡献。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。该项目的重点是研究一类平均场和动力学模型，可以很好地描述代理，集体行为和平均过程之间的相互作用。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），生物学（肿瘤生长，群集）和神经科学。 该项目还将通过研究向本科生和研究生提供教育和培训。