Collaborative Research: Waves and Fronts in Heterogeneous Media

波在随机介质中传播的微观细节的数值模拟仍然超出了现代计算机的能力范围：典型的传播距离可能是数百个波长的数量级和随机波动的相关长度。这就需要使用各种近似的宏观模型，其中动力学方程构成了一个重要的类。然而，从微观波动方程到大尺度动力学的过渡本身就是一个复杂的问题。 该项目第一部分的目标是开发新的工具，更好地了解动力学极限，特别是在随机介质具有长程相关性的情况下，导致各种波现象的多个时间和空间尺度。该项目的第二部分研究反应-扩散-对流方程的解的定性行为，主要关注流体流动的影响。将研究底层流动的混合、动力学和几何性质的相互作用以及扩散和反应的影响。 在反应过程对流体流动的反馈不能被忽略的情况下，问题变得特别复杂。该项目致力于对Boussinesq反应系统中的能量、动量和反应物的传输、其他具有反馈的反应扩散系统以及随机介质中的火焰前沿传播进行定量研究。该项目对杂乱环境中的波传播以及流体中的反应扩散过程进行数学研究。这些研究与科学的几个分支有关，从生物医学成像到生物物理学，流体动力学和天体物理学。在杂乱的环境中成像，无论是人体、地球内部还是树叶，由于媒体的复杂性，都是固有的不稳定。这个项目的一个目标是开发成像方法，是不太敏感的不可预知的波动的杂波。 该项目的另一个领域涉及流体流动对化学反应影响的数学描述。湍流在许多化学现象中起着重要的作用：它可以大大提高反应速率，从而提高效率，或者在某些情况下，熄灭化学过程。该项目将在更简单的数学模型中解决这些问题，以阐明整个问题中存在的机制。

Numerical simulation of the microscopic details of wave propagation in random media is still beyond the reach of modern computers: a typical propagation distance may be on the order of hundreds of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic models, of which kinetic equations constitute an important class. However, the passage from microscopic wave equations to large-scale kinetics is a complicated problem in itself. The goal of the first part of the project is to develop new tools and better understanding of kinetic limits, especially in the regime when random media have long range correlations that lead to multiple temporal and spatial scales for various wave phenomena. The second part of the project investigates the qualitative behavior of solutions of reaction-diffusion-advection equations, with the main focus on the effect of a fluid flow. The interaction of the mixing, dynamic, and geometric properties of the underlying flow and the effects of diffusion and reaction will be investigated. The problem becomes especially complex in the situations where the feedback from the reaction process on the fluid flow cannot be ignored. The project addresses the quantitative study of the transport of the energy, momentum, and reactants in a Boussinesq reactive system, other reaction-diffusion systems with feedback, and flame-front propagation in random media.This project carries out mathematical studies of wave propagation in cluttered environments and of reaction-diffusion processes in fluids. These studies are relevant to several branches of science, ranging from biomedical imaging to geophysics, fluid dynamics, and astrophysics. Imaging in a cluttered environment, whether it is a human body, earth interior, or foliage, is inherently unstable because of media complexity. One objective of this project is to develop imaging methods that are less sensitive to unpredictable fluctuations of the clutter. Another area of this project concerns the mathematical description of the effect of a fluid flow on chemical reactions. Turbulent fluid flow plays an important role in many chemical phenomena: it may drastically enhance the rate of reaction, leading to higher efficiency, or, in some situations, extinguish the chemical process. The project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.