Long Time Behavior for Partial Differential Equations in Random Media

随机介质中偏微分方程的长时间行为

基本信息

批准号：
1910023
负责人：
Leonid Ryzhik
金额：
$ 36万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1910023&HistoricalAwards=false
关键词：
Long Time Behavior Partial Differential

项目摘要

This project carries out mathematical studies of physical, biological and social dynamics systems in heterogeneous and random environments. Such systems are ubiquitous in nature, and include examples as diverse as ecology, economic growth, and fluid turbulence. The mathematical modeling of such problems involves partial differential equations with highly oscillatory coefficients. Typically, such problems involve a multitude of temporal and spatial scales, and numerical simulation of the microscopic details of the solutions is beyond reach even of the modern computers. A typical propagation distance may be of the order of hundreds or thousands of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic effective models in practice. The overarching goal of the project is to develop a better understanding of the validity of such macroscopic models. The goal of the first part of the project is to develop new tools and better understanding of the effective limits for equations of the parabolic type, starting with the random heat equation, and then for nonlinear problems arising in the fluid dynamics and reaction-diffusion modeling, with the focus on very long time scales, when fluctuations in the solutions start building up in ways beyond the classical central limit theorem time scales. The second part of the project investigates the qualitative behavior of the solutions of systems of integro-differential equations that arise in macroeconomics. Such equations model phenomena from the diffusion of knowledge and GDP growth to the international trade.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目对异质和随机环境中的物理，生物学和社会动态系统进行了数学研究。这种系统本质上是普遍存在的，包括生态学，经济增长和流体湍流等多样化的例子。此类问题的数学建模涉及具有高振荡系数的部分微分方程。通常，此类问题涉及多种时间和空间尺度，并且对解决方案的显微镜细节的数值模拟甚至是现代计算机的范围。典型的繁殖距离可能是数百或数千个波长的顺序以及随机波动的许多相关长度。这需要在实践中使用各种近似宏观有效模型。该项目的总体目标是更好地了解此类宏观模型的有效性。该项目第一部分的目的是开发新工具，并更好地理解抛物面类型方程的有效限制，从随机热方程开始，然后是在流体动力学和反应 - 扩散建模中引起的非线性问题，重点关注很长的时间尺度，而解决方案的范围超出了整体范围的范围，则在整体中心限制中范围内开始构建较长的范围。该项目的第二部分调查了宏观经济学中出现的不差异方程式系统解决方案的定性行为。此类方程式模型现象从知识和GDP增长的扩散到国际贸易。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。