Reaction-Diffusion Front Speeds in Chaotic and Stochastic Flows

流体流动中的反应扩散锋传播存在于许多科学领域，如对流中的颗粒传输、湍流燃烧和风中的野火传播等。我们的目标是对原型方程进行数学分析和计算，以获得对流动对运输的非常重要的影响的基本理解。近年来，当流线(流线)结构良好(规则运动)或完全随机(遍历运动)时，在这个方向上已经完成了大量的渐近和数值工作。经常遇到的较少研究的情况是流线既包括规则运动又包括随机运动，而两者都不占据整个相空间。阿诺德-贝尔特米-奇尔德里斯(ABC)流就是一个例子，它是一类具有混沌流线的三维不可压缩平均零周期流。本研究的目的是研究具有混沌和随机流线的各种ABC流中反应-扩散-平流和哈密顿-雅可比方程的大时间前沿速度。流线中无序量的量度是由庞加莱截面上的点占据的相空间体积给出的。该项目将分析和计算方法结合起来，研究前沿速度对混沌程度、非线性以及与有效扩散的相关性，有效扩散是一种相关的传输特性。利用前沿速度变分公式和校正方程，将无界空间域上的非线性动力学问题归结为有限空间域上的主特征值或Lyapunov指数(增长率)问题。相关的教育活动、博士后辅导和数据管理也在计划中。开展这些工作将增进我们对自然界中无序流动中物质运输的理解，并对污染物输送、森林火灾蔓延、动力效率高的内燃机和低废气排放等科学和工程领域产生广泛的影响。该项目开发的数学和数值方法是解决复杂现实问题的潜在估计和咨询工具。它们可以帮助决策者及时采取行动，将火灾危险和污染物造成的损害降至最低，并帮助制造商在针对绿色环境的发动机设计中提高能效。该项目产生的结果和数据还将使课程开发和课程提供方面的教育工作者受益，这反过来又会刺激更多的美国学生在科学、技术、工程和数学学科方面攻读更高的学位。

Reaction-diffusion front propagation in fluid flows appears in many scientific areas such as particle transport in convection, turbulent combustion and wild fire spread in winds. We aim to carry out mathematical analysis and computation on prototype equations to gain fundamental understanding of the highly nontrivial effects of flow on transport. Significant amount of asymptotic and numerical work in this direction has been accomplished in recent years when the flow lines (streamlines) are either well-structured (regular motion) or fully random (ergodic motion). The often encountered yet less studied case is when the streamlines consist of both regular and stochastic motions, while neither one takes up the entire phase space. An example is the Arnold-Beltrami-Childress (ABC) flow, a class of three dimensional incompressible mean zero periodic flow with chaotic flow lines. The research program is to study large time front speeds of reaction-diffusion-advection and Hamilton-Jacobi equations in various ABC flows with chaotic and stochastic streamlines in channel domains orin the entire space. A measure of the amount of disorder in the streamlines is given by the phase space volume occupied by the points on the Poincare sections. The project combines analytical and computational approaches to study the dependence of front speeds on the degree of chaos, nonlinearities, and correlation with effective diffusion, a related transport property. Front speed variational formulas and corrector equations are used to reduce the original nonlinear dynamical problems on unbounded spatial domains to a principal eigenvalue or Lyapunov exponent (growth rate) problem on a finite spatial domain. Related educational activities, postdoc mentoring, and data management are also planned.Carrying out the proposed work will advance our understanding of material transport in disordered flows arising in nature, and generate broad impact to the science and engineering of pollutant transport, forest fire spreading, internal combustion engine with power efficiency and low waste gas emission to name a few. The mathematical and numerical methods developed in the project are potentially estimation and consulting tools for resolving complex real-world problems.They may aid decision makers to act timely to minimize damage from fire hazards and pollutants, and help manufacturers improve energy efficiency in engine design for green environment. The results and data generated in the project will also benefit educators in curriculum development and course offerings, which in turn stimulates more US students to pursue higher degrees in science, technological, engineering and mathematical disciplines.