Collaborative Research: Waves and Fronts in Heterogeneous Media

对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：典型的传播距离可能是数百个波长的数量级，以及随机波动的许多相关长度。这就需要使用各种近似宏观模型，其中动力学方程是一类重要的模型。然而，从微观波动方程过渡到大尺度动力学本身就是一个复杂的问题。该项目的第一部分的目标是开发新的工具和更好地理解动力学极限，特别是当随机介质具有长期相关性时，导致各种波现象的多个时间和空间尺度。项目的第二部分研究了反应-扩散-平流方程解的定性行为，主要关注流体流动的影响。混合、动力和底层流动的几何性质的相互作用以及扩散和反应的影响将被研究。在反应过程对流体流动的反馈不可忽视的情况下，问题变得尤为复杂。该项目涉及能量、动量和反应物在Boussinesq反应系统、其他反馈反应扩散系统中的传输的定量研究，以及随机介质中的火焰前传播。该项目对混乱环境中的波传播和流体中的反应扩散过程进行数学研究。这些研究涉及几个科学分支，从生物医学成像到地球物理学、流体动力学和天体物理学。在一个杂乱的环境中成像，无论是人体、地球内部还是树叶，由于介质的复杂性，本质上是不稳定的。该项目的一个目标是开发对杂波不可预测波动不太敏感的成像方法。这个项目的另一个领域涉及流体流动对化学反应的影响的数学描述。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度，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.