Waves and fronts in heterogeneous media

该项目对异构环境中的物理系统进行数学研究。这样的系统在自然界中是普遍存在的，并且包括诸如通过大气的光传播、地震后地球中的地震波、股票市场波动和医学成像问题等各种各样的示例。这类问题的数学建模涉及具有高振荡系数的偏微分方程。通常，这样的问题涉及大量的时间和空间尺度：典型的传播距离可以是数百或数千个波长的量级，以及随机波动的许多相关长度。即使是最强大的现代计算机，也无法对解的微观细节进行数值模拟。为了预测和理解这些多尺度系统的行为，必须使用各种近似的宏观有效模型，这些模型保留了过程的显着特征，而无需跟踪所有微观细节。该项目的总体目标是更好地理解这种宏观模型的有效性。该项目第一部分的目标是开发新的工具，更好地了解波传播的有效极限，重点是具有长程相关性的随机介质，这些介质导致各种物理现象的多个时间和空间尺度。第二部分研究了具有随机扰动的反应扩散方程解的定性行为。 随机涨落和非均匀性对波前传播性质有重要影响，导致新的解类型和渐近行为。该项目的一个目标是了解随机反应扩散问题与最近的正则结构理论之间的联系。

This project carries out mathematical studies of physical systems in heterogeneous environments. Such systems are ubiquitous in nature, and include examples as diverse as light propagation through the atmosphere, seismic waves in the Earth following an earthquake, stock market fluctuations, and medical imaging problems. 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: a typical propagation distance may be of the order of hundreds or thousands of wavelengths and as many correlation lengths of random fluctuations. Numerical simulation of the microscopic details of the solutions is beyond the reach even for the most powerful modern computers. For the prediction and understanding of behavior of these multi-scale system, it is imperative to use various approximate macroscopic effective models that preserve the salient features of the processes without keeping track of all microscopic details. 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 in wave propagation, with the focus on random media with long-range correlations that lead to multiple temporal and spatial scales for various physical phenomena. The second part of the proposal investigates the qualitative behavior of the solutions of reaction-diffusion equations with stochastic perturbations. Stochastic fluctuations and heterogeneities play a significant effect on front propagation properties, leading to new types of solutions and asymptotic behavior. One goal of the project will be to understand the connection between the stochastic reaction-diffusion problems and the recent theory of the regularity structures.