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

平均场理论和流体动力学中非线性偏微分方程组的长时和短时渐近

基本信息

批准号：
0807636
负责人：
Maria Pia Gualdani
金额：
$ 7.8万
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：
2008
资助国家：
美国
起止时间：
2008-08-01 至 2012-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807636&HistoricalAwards=false
关键词：
Long Short Time Asymptotics Systems

项目摘要

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.

本研究项目是对传统分析方法不适用的非线性偏微分方程组的解的定性行为进行分析研究。本文研究了四个方面的问题：（1）经济学中模拟价格形成的非线性扩散系统自由边界问题解的长时间行为;（2）不可压Euler方程径向对称解的可能爆破;（3）研究量子漂移扩散四阶非线性抛物方程组的梯度流方法;等离子体物理中的经典和量子动力学模型。本课题的目的是通过动力学理论、最佳物质输运方法、变分技术等，建立不同方法之间的联系，使这些系统的分析变得容易。本课题分析了经济学中的价格形成、流体流动、等离子体动力学等几个具有实际意义的系统的模型方程。这些系统的数学模型提出了大量的分析挑战，这项工作的目的是改进现有的方法，使更深入地了解控制方程的解决方案的属性。工作的结果将有助于更好地预测这些复杂系统的行为。