Waves, Particle Transport 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是现代计算机所无法达到的：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。这就需要在实际中使用各种近似的宏观有效模型。从模拟特定物理系统的随机微观方程到大尺度模型的过渡本身就是一个非常重要的问题。该项目的第一部分的目标是开发新的工具，更好地理解这种有效的限制，特别是在政权时，随机介质具有长程相关性，导致多种时间和空间尺度的各种物理现象。该项目的第二部分研究反应扩散方程解的定性行为，特别是那些在随机粒子系统中出现的解，如分支布朗运动和其他Brunet-Derrida系统。 另一组问题涉及生物学中出现的反应扩散方程，如趋化模型，以及涉及异质环境中种群动力学的生态学。该项目进行物理系统的数学研究，以及波在杂乱环境中的传播和反应扩散过程。 这些研究与科学的几个分支有关，从生物医学成像问题到生物物理学，流体动力学和生物学。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和生物反应的混合效应的数学描述。湍流在许多反应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。

AbstractNumerical simulation of the microscopic details 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 effective models in practice. The passage from the stochastic microscopic equations modeling a particular physical system to the large-scale model is a highly non-trivial problem in itself. The goal of the first part of the project is to develop new tools and better understanding of such effective limits, especially in the regime when random media have long range correlations that lead to multiple temporal and spatial scales for various physical phenomena. The second part of the project investigates the qualitative behavior of solutions of reaction-diffusion equations, in particular, those that arise in stochastic particle systems, such as branching Brownian motion, and other Brunet-Derrida systems. Another set of problems involves reaction-diffusion equations that arise in biology, such as chemotactic models, and ecology involving population dynamics in heterogenous environments. This project carries out mathematical studies of physical systems, and wave propagation in cluttered environments and of reaction-diffusion processes. These studies are relevant to several branches of science, ranging from biomedical imaging questions to geophysics, fluid dynamics, and biology. 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 mixing effects of a fluid flow on chemical and biological 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 project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.