Long Time Behavior for Partial Differential Equations in Random Media

该项目对不同种类和随机环境中的物理、生物和社会动力学系统进行数学研究。这样的系统在自然界中无处不在，包括各种例子，如生态、经济增长和流体湍流。这类问题的数学建模涉及具有高度振荡系数的偏微分方程组。通常，这样的问题涉及大量的时间和空间尺度，解决方案的微观细节的数值模拟即使是现代计算机也达不到。典型的传播距离可以是数百或数千个波长的数量级，以及随机波动的多个相关长度。这就需要在实践中使用各种近似的宏观有效模型。该项目的首要目标是更好地理解这些宏观模型的有效性。该项目第一部分的目标是开发新的工具，更好地理解抛物型方程的有效极限，从随机热方程开始，然后是流体动力学和反应扩散模型中出现的非线性问题，重点是非常长的时间尺度，当解的波动开始以超出经典中心极限定理时间尺度的方式积累时。该项目的第二部分研究了宏观经济学中出现的积分-微分方程组的解的定性行为。这样的方程模拟了从知识传播和GDP增长到国际贸易的现象。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.