Collaborative Research: Waves and Fronts in Heterogeneous Media

合作研究：异构媒体中的波与前沿

• 批准号: 1016106
• 负责人: Leonid Ryzhik
• 金额: $ 41.2万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-11-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016106&HistoricalAwards=false
• 关键词: Collaborative; Research; Waves; Fronts; Heterogeneous

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.

随机介质中波传播的显微镜细节的数值模拟仍然超出了现代计算机的范围：典型的传播距离可能是数百个波长的顺序，并且随机波动的许多相关长度。这需要使用各种近似宏观模型，其中动力学方程构成了重要类。但是，从微观波方程到大规模动力学的通道本身就是一个复杂的问题。 该项目第一部分的目的是开发新工具并更好地理解动力学限制，尤其是在随机媒体具有长距离相关性的情况下，导致各种波浪现象的多个时间和空间尺度。该项目的第二部分研究了反应 - 扩散 - 辅助方程解决方案的定性行为，主要重点是流体流的效果。将研究基础流的混合，动态和几何特性的相互作用以及扩散和反应的影响。 在不忽略反应过程中反应过程的反馈过程中，问题变得特别复杂。该项目介绍了在BoussinesQ反应系统中的能量，动量和反应物，其他反应反馈的其他反应扩散系统以及随机介质中的火焰 - 前传播的定量研究。该项目在混乱环境中进行了波传播的数学研究，并在液体中进行反应降低过程。这些研究与科学的几个分支有关，从生物医学成像到地球物理，流体动力学和天体物理学。在混乱的环境中进行成像，无论是人体，地球内部还是树叶，由于媒体的复杂性而来都是不稳定的。该项目的一个目的是开发对杂物不可预测的波动不太敏感的成像方法。 该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。湍流流动在许多化学现象中起着重要作用：它可能会大大提高反应速率，导致效率更高，或者在某些情况下，消除了化学过程。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。