Reaction-Diffusion Front Speeds in Chaotic and Stochastic Flows

流体流动中的反应扩散锋面传播在对流中的粒子输移、湍流燃烧和风中的野火蔓延等许多科学领域都有出现。我们的目标是对原型方程进行数学分析和计算，以获得对流动对输运的高度非平凡影响的基本理解。近年来，当流线（流线）是结构良好的（规则运动）或完全随机的（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。经常遇到但研究较少的情况是流线由规则运动和随机运动组成，而两者都不占整个相空间。以Arnold-Beltrami-Childress （ABC）流为例，该流是一类具有混沌流线的三维不可压缩平均零周期流。本课题主要研究通道域或整个空间中具有混沌和随机流线的各种ABC流的反应-扩散-平流和Hamilton-Jacobi方程的大时间前速度。流线中无序量的量度是由庞加莱截面上的点所占据的相空间体积给出的。该项目结合了分析和计算方法来研究前方速度与混沌程度、非线性以及与有效扩散（相关的输运性质）的相关性的关系。采用前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简化为有限空间域上的主特征值或李雅普诺夫指数（增长率）问题。相关的教育活动、博士后指导和数据管理也在计划之中。这项工作的开展将促进我们对自然界中无序流动中物质运移的认识，并对污染物运移、森林火灾蔓延、节能内燃机和低废气排放等科学与工程产生广泛影响。项目中开发的数学和数值方法是解决复杂现实问题的潜在评估和咨询工具。它们可以帮助决策者及时采取行动，最大限度地减少火灾危险和污染物的损害，并帮助制造商提高绿色环境发动机设计的能源效率。该项目产生的结果和数据也将使课程开发和课程设置的教育工作者受益，这反过来又刺激更多的美国学生在科学、技术、工程和数学学科攻读更高的学位。

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.