Reactive Processes, Mixing, and Fluid Dynamics

在自然界、科学和工程中，诸如森林火灾、恒星中的核反应或内燃机燃烧等反应过程是普遍存在的。由于液体或气体介质的运动而产生的混合，在其中发生反应过程，通常是其动力学中的一个重要组成部分，也与其他过程有关，例如可靠地制造玻璃和合金。这一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。这个项目的中心目标是通过对这些物理过程的数学模型进行分析研究，更好地了解这些物理过程的长期行为，这些数学模型以偏微分方程组的形式表示。讨论的主要问题将是反应过程的传播速度与其发生的可燃介质的性质和结构的相关性的问题；潜在的混合过程如何能够提高这一速度以及哪种类型的混合最有效地实现这一速度的问题；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。其目标是获得数学上严格的结果，这些结果还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组给出，特别是由反应扩散方程、输运方程和流体动力学方程给出。主要的兴趣在于它们解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在一维和几维非均匀介质中传播的反应过程的长期动力学，包括行波前沿的存在，它们的一般解的渐近收敛，以及随机介质中解的齐次化。该项目混合部分的目标是研究气流的混合效率，并寻找最擅长混合受其影响的物质的气流。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的形成。这项研究的另一个目标是研究主动燃烧，由于通过浮力对流体运动的反应的直接反馈，所有这三个过程结合在一起。包含这种反馈的模型涉及耦合到流体动力学方程的反应扩散方程，重点将放在行波锋面的存在和稳定性以及重力诱导的混合上。为了解决这些问题，这项研究将利用研究人员和合作者最近开发的技术，以及能够进一步提高对反应过程动力学行为的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.