The Kinetic Theory of Waves and Reactive-Diffusive Fronts

波和反应扩散前沿的动力学理论

• 批准号: 0604687
• 负责人: Leonid Ryzhik
• 金额: $ 24.65万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604687&HistoricalAwards=false
• 关键词: Kinetic; Theory; Waves; Reactive; Diffusive

Numerical simulation of the microscopic description of wave propagation in random media is still beyond reach of modern computers: a typical propagation distance may be of 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 two-fold: on one hand, to develop new tools and better understanding of kinetic limits, and second, to consider the applications of kinetic methods to the inverse problems of wave propagation, finding sources and scatterers in a cluttered environment. 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. We will investigate the interaction of the mixing, dynamic, and geometric properties of the underlying flow and the effects of diffusion and reaction. 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 the reactants in a Boussinesq reactive system.brbrThis project carries out mathematical studies of wave propagation in complex media and of reaction-diffusion equations. The mathematical models are relevant to several branches of science, ranging from biomedical imaging questions 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. An objective of this project is to develop imaging methods that are less sensitive to unpredictable fluctuations of the clutter. We will strive to understand the universality and the limits of applicability of macroscopic models and develop inversion algorithms that arise from the macroscopic rather than detailed microscopic models and that are therefore inherently more stable with respect to fluctuations of the environment. 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 reaction phenomena: it may drastically enhance the rate of reaction, leading to higher efficiency, or, in some situations, extinguish the chemical process. The mathematical theory is far from complete, due to the inherent complexity and richness of the phenomena. The project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.

对随机介质中波传播的显微镜描述的数值模拟仍然是现代计算机的范围：典型的传播距离可能是数百个波长和随机波动的许多相关长度的顺序。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。 该项目的第一部分的目的是两个方面：一方面，开发新的工具并更好地理解动力学限制，其次，考虑动力学方法在波浪传播的反问题上的应用，在杂乱的环境中找到来源和散射器。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。我们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Boussinesq反应系统中能量，动量和反应物的运输的定量研究。BrbrBrthisProject在复杂介质和反应扩散方程中进行了波传播的数学研究。 数学模型与科学的几个分支有关，从生物医学成像问题到地球物理学，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成像方法对杂物的不可预测的波动不太敏感。 我们将努力理解宏观模型的普遍性和适用性的局限性，并开发由宏观而非详细的显微镜模型引起的反转算法，因此，对于环境波动而言，它们天生更稳定。 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。在许多反应现象中，湍流流动起着重要作用：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。 由于现象的固有复杂性和丰富性，数学理论远非完整。 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。