The Kinetic Theory of Waves and Reactive-Diffusive Fronts

波在随机介质中传播的微观描述的数值模拟仍然不是现代计算机所能达到的：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的宏观模型，其中动力学方程构成了一类重要的模型。然而，从微观波动方程到大尺度动力学本身就是一个复杂的问题。该项目第一部分的目标有两个：一方面，开发新的工具和更好地了解动能极限，其次，考虑将动力学方法应用于波传播的反问题，在杂乱的环境中寻找源和散射体。项目的第二部分研究反应-扩散-平流方程解的定性行为，主要集中在流体流动的影响上。我们将研究底层流动的混合、动力学和几何性质之间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视的情况下，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进行定量研究。brbr该项目对波在复杂介质中的传播和反应扩散方程进行数学研究。这些数学模型与几个科学分支有关，从生物医学成像问题到地球物理、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，本质上是不稳定的。该项目的一个目标是开发对杂波的不可预测波动不那么敏感的成像方法。我们将努力了解宏观模型的普遍性和适用范围，并开发基于宏观而不是详细的微观模型的反演算法，因此相对于环境的波动，这种算法本质上更稳定。这个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。湍流在许多反应现象中扮演着重要的角色：它可能会大大加快反应速度，导致更高的效率，或者在某些情况下，扑灭化学过程。由于现象的内在复杂性和丰富性，数学理论远远不完整。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。

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.