Reactive Processes, Mixing, and Fluid Dynamics

反应过程，如森林火灾、恒星中的核反应或内燃机燃烧，在自然界、科学和工程中无处不在。由于发生反应过程的液体或气体介质的运动而引起的混合经常是其动力学的重要组成部分，并且也与其他过程有关，例如玻璃和合金的可靠制造。这种运动可能会受到流体湍流的影响，其影响在物理和工程的许多领域都是至关重要的。该项目的中心目标是通过对其数学模型的分析研究，更好地理解这些物理过程的长期行为，这些数学模型以偏微分方程的形式表示。讨论的主要问题将是反应过程的扩散速度取决于发生反应的燃烧介质的性质和结构的问题；底层混合过程如何提高这一速度以及哪种混合方式最有效地实现这一目标的问题；以及湍流的自发发展和流体运动中意外的奇异行为的问题。目标是获得数学上严谨的结果，这也可以进一步阐明正在建模的实际物理过程的动力学行为。本研究项目研究几个重要物理过程的数学模型，包括反应过程、流体动力学和混合。模型由线性和非线性偏微分方程给出，特别是由反应扩散方程、输运方程和流体动力学方程给出。主要兴趣在于其解决方案的长期动态以及奇点的形成。该项目的反应扩散部分的目标是理解和描述反应过程在一个和几个维度上通过非均匀介质传播的长期动力学，包括行进锋的存在性，它们的一般解的渐近收敛性，以及随机介质中解的均质化。该项目的混合部分的目标是研究流动的混合效率，并寻找最适合混合它们平流的物质的流动。该项目流体动力学部分的目标是研究湍流，特别是在二维流体和大气运动模型中创建小尺度和有限时间奇点形成。研究的另一个目标是研究主动燃烧，由于浮力对流体运动的反应直接反馈，所有这三个过程都结合在一起。结合这种反馈的模型包括耦合到流体动力学方程的反应扩散方程，重点将放在行进锋的存在性和稳定性以及重力诱导混合上。为了解决这些问题，该研究将利用研究者和合作者最近开发的技术，以及能够进一步推进对反应过程动力学行为理解的新方法的开发。

Reactive processes such as forest fires, nuclear reactions in stars, or burning in internal combustion engines are ubiquitous in nature, science, and engineering. Mixing due to motion of a liquid or gaseous medium in which reactive processes occur is frequently an important component in their dynamics, and is also relevant to other processes, such as reliable manufacturing of glasses and alloys. This motion may be subject to fluid turbulence, the effects of which are of paramount importance in many areas of physics and engineering. The central aim of this project is a better understanding of the long term behavior of these physical processes through the analytical study of their mathematical models, which are expressed in the form of partial differential equations. The main questions addressed will be the question of dependence of the speed of spreading of reactive processes on the properties and structure of combustive media in which they occur; the question of how an underlying mixing process can enhance this speed and which types of mixing are most efficient at achieving this; and the question of spontaneous development of turbulence and unexpected singular behaviors in the motion of fluids. The goal is to obtain mathematically rigorous results which can also shed further light on the dynamical behavior of the actual physical processes being modeled. This research project studies mathematical models of several important physical processes, which include reactive processes, fluid dynamics, and mixing. The models are given by linear and nonlinear partial differential equations, in particular, by reaction-diffusion equations, transport equations, and equations of fluid dynamics. The main interest is in the long term dynamics of their solutions as well as in the formation of singularities. The goal of the reaction-diffusion portion of the project is the understanding and description of long term dynamics of reactive processes spreading through inhomogeneous media in one and several dimensions, including existence of traveling fronts, asymptotic convergence of general solutions to them, and homogenization of solutions in random media. The goal of the mixing portion of the project is the study of mixing efficiency of flows and the search for those which are best at mixing substances advected by them. The goal of the fluid dynamics portion of the project is the study of turbulence, particularly creation of small scales and finite time singularity formation in models of fluid and atmospheric motion in two dimensions. Another goal of the research is the study of active combustion, where all three of these processes come together due to a direct feedback of the reaction on fluid motion via the buoyancy force. Models incorporating such feedback involve reaction-diffusion equations coupled to fluid dynamics equations, and the focus will be on existence and stability of traveling fronts and on gravity-induced mixing. To address these questions, the research will make use of techniques recently developed by the investigator and collaborators, as well as the development of new methods capable of further advancing understanding of the dynamical behavior of reactive processes.