Long and Short Time Asymptotics of Systems of Nonlinear Partial Differential Equations Arising in Mean-Field Theory and Fluid-Dynamics

这个研究项目是对非线性偏微分方程组解的定性行为的分析研究，而传统的分析方法并不适用。本文主要研究四个方面的问题：(1)模拟价格形成的非线性扩散系统的自由边界问题解的长期行为；(2)不可压缩欧拉方程径向对称解的可能爆破；(3)梯度流法研究量子漂移-扩散四阶非线性抛物系统；(4)等离子体物理中的经典和量子动力学模型。该项目旨在通过动力学理论、最佳质量运输方法和变分技术建立不同方法之间的联系，从而促进对这些系统的分析。本项目分析了对几个具有实际重要性的系统建模的方程，包括经济学中的价格形成、流体流动和等离子体动力学。这些系统的数学模型提出了实质性的分析挑战，这项工作旨在改进现有的方法，以便更深入地理解控制方程解的性质。这项工作的结果将有助于更好地预测这些复杂系统的行为。

This research project is an analytical study of the qualitative behavior of solutions to nonlinear systems of partial differential equations to which conventional methods of analysis do not apply. Four problem areas are under investigation: (1) long-time behavior of solutions to free-boundary problems for a nonlinear diffusion system modeling price formation in economics; (2) possible blow up for radially symmetric solutions for the incompressible Euler equation; (3) gradient flow methods for study of the quantum drift-diffusion fourth-order nonlinear parabolic system; and (4) classical and quantum kinetic models in plasma physics. The project aims to facilitate analysis of these systems by establishing links between different approaches via kinetic theory, optimal mass transportation methods, and variational techniques.This project analyzes equations that model several systems of practical importance, including price formation in economics, fluid flow, and the dynamics of plasmas. The mathematical models for these systems present substantial analytical challenges, and this work aims to improve on existing methods to enable deeper understanding of the properties of solutions to the governing equations. The results of the work will facilitate better prediction of the behavior of these complicated systems.