The Kinetic Theory of Waves and Reactive-Diffusive Fronts

对波在随机介质中传播的微观描述的数值模拟仍然是现代计算机无法达到的：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。该项目第一部分的目标有两个方面：一方面，开发新的工具和更好地理解动力学极限，其次，考虑动力学方法在波传播逆问题中的应用，在混乱的环境中寻找源和散射体。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。我们将研究混合、动力和底层流动的几何特性以及扩散和反应的影响之间的相互作用。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。该项目解决了Boussinesq反应系统中能量、动量和反应物传输的定量研究。这个项目进行波在复杂介质中的传播和反应扩散方程的数学研究。数学模型与科学的几个分支相关，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。该项目的目标是开发对杂波不可预测波动不太敏感的成像方法。我们将努力理解宏观模型的普遍性和适用性的局限性，并开发由宏观而不是详细的微观模型产生的反演算法，因此，相对于环境的波动，这些算法本质上更稳定。这个项目的另一个领域涉及流体流动对化学反应的影响的数学描述。紊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度，从而提高效率，或者在某些情况下，使化学过程熄灭。由于现象本身的复杂性和丰富性，数学理论还远远不够完善。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.