Reaction, Diffusion, and Fluid Flow

该项目侧重于流体流动中发生的反应过程模型的分析研究。这些模型涉及非线性偏微分方程，如反应-扩散方程，它可能与流体动力学的Navier-Stokes方程耦合。提出的研究旨在提高我们对流体运动对燃烧的影响的理解，并有两个主要目标。第一个是继续目前在周期性介质中的被动燃烧工作，以及开发适用于一般无序介质的新技术。要解决的问题包括周期性流动的几何性质对反应传播加速的影响以及淬火现象。我们还将研究反应在无序介质中的传播，特别是行进前解的存在性和这些特殊解的任意解的渐近逼近。第二个主要目标是研究通过浮力直接反馈反应对流体运动的主动燃烧。包含这种反馈的模型，特别是与高湍流燃烧状态相关的模型，涉及到与流体动力学方程耦合的反应-扩散方程，本质上非常复杂。我们将集中精力研究行进锋的存在性和稳定性，反应传播速度的界限，以及重力诱导的混合。此外，我们打算将开发的技术应用于研究液相悬浮液的相关模型中的相变。该项目解决的问题涉及丰富而微妙的数学，但也具有跨学科的特点。诸如内燃机燃烧、恒星核反应、森林火灾和大气中臭氧的产生等反应过程在自然界、科学和工程中无处不在。下面的液体或气体介质的运动往往起着至关重要的作用，要么加速反应，要么使反应熄灭。提出的研究旨在更好地从数学上理解所产生的湍流的各种性质对反应过程的影响。它与诸如天体物理学、生物学、环境科学和化学工程等科学分支相关，并可能为现实生活中的现象提供有用的定性见解。作为芝加哥大学本科生暑期项目研究经验的一部分，首席研究员还计划教授一门本科水平的课程，以及一门专门的研究生水平的反应扩散方程课程。

The project focuses on analytical study of models of reaction processes taking place in fluid flow. These models involve nonlinear partial differential equations such as reaction-diffusion equations, which may be coupled to the Navier-Stokes equations of fluid dynamics. The proposed research aims at improving our understanding of the effects of fluid motion on combustion and has two main goals. The first is a continuation of current work on passive combustion in periodic media as well as development of new techniques applicable to general disordered media. The questions to be addressed include the effect of geometric properties of periodic flows on speed-up of propagation of reaction as well as the phenomenon of quenching. We will also investigate propagation of reaction in disordered media, in particular, the existence of traveling front solutions and asymptotic approach of arbitrary solutions to these special ones. The second main goal is the study of active combustion with direct feedback of reaction on the fluid motion via the buoyancy force. Models incorporating such feedback, particularly relevant in highly turbulent combustion regimes, involve reaction-diffusion equations coupled to fluid dynamics equations and are inherently very complex. We will focus our efforts on the existence and stability of traveling fronts, bounds on the speed of propagation of reaction, and gravity-induced mixing. In addition, we intend to apply the developed techniques to the study of phase transitions in a related model of liquid suspensions. The problems addressed by the project involve rich and subtle mathematics but also have an interdisciplinary character. Reaction processes such as burning in internal combustion engines, nuclear reactions in stars, forest fires, and production of ozone in the atmosphere are ubiquitous in nature, science, and engineering. Motion of the underlying liquid or gaseous medium often plays a crucial role by either speeding up reaction or quenching it. The proposed research aims at a better mathematical understanding of the effects of various properties of the resulting turbulent flows on reactive processes. It is relevant to branches of science such as astrophysics, biology, environmental science, and chemical engineering, and may provide useful qualitative insights in real life phenomena. The principal investigator also plans to teach an undergraduate-level course as part of the Research Experience for Undergraduates summer program at the University of Chicago, as well as a specialized graduate-level course in reaction-diffusion equations.