Dynamical systems and singular perturbation theory for multi-scale reaction-diffusion phenomena

本研究项目涉及一系列跨学科和基础的数学方法，用于多长度和时间尺度的反应扩散问题。第一个项目涉及Oya-Vallochi生物修复模型，这是一个由平流、反应、扩散方程组成的系统，用于降解底物、营养物和活性细菌的浓度。将进行数值模拟和数学分析，以确定稳定行波存在的参数域，以及在这些域的边界上发展的不稳定性。第二个项目涉及生物化学、化学工程、燃烧和空气污染工程中出现的化学反应和扩散问题的还原方法。这些反应通常涉及30-100种物质，50-400种反应，时间尺度从纳秒到秒不等。此外，这些反应通常发生在反应流中，在扩展的区域上，因此必须对区域的每个部分的化学进行建模。约简方法对于分析大量数据是必不可少的，因为它们确定了长期动态发挥作用的关键进展变量和低维流形。第三个项目涉及活化剂-抑制剂反应-扩散方程中的模式形成，特别是对斑点模式、自我复制斑点、新发现的控制脉冲和斑点散射的“分离器”解决方案的竞争不稳定性的分析。此外，将使用PI和合作者开发的一种新的重整化群方法建立强相互作用脉冲的非线性稳定性。这个应用数学研究项目涉及工程和科学中多尺度反应扩散系统关键应用的数学方法。这些项目的目标是解释最近的实验，改进计算方法，并发展对下一代工程技术至关重要的新数学理论。第一个项目涉及生物修复，即诱导土壤中的微生物降解对环境有害的有机化合物的过程。将确定以定期和最佳方式进行补救的操作条件。第二个项目涉及还原方法，这对于理解生物化学、化学工程、燃烧和空气污染工程中的复杂化学反应至关重要。迄今为止，PI开发的数学理论有助于改进工程师目前使用的还原方法，而提出的新理论将有助于开发下一代更快、更准确的还原方法。第三个项目的重点是涉及斑点和相互作用的相干结构的化学模式的动力学和稳定性。将使用来自动力系统理论、奇异摄动和微分方程的数学方法，并将开发新的数学技术。这些项目还涉及博士生和博士后，其中很大一部分是女性，以及在阿贡国家实验室和外国大学的合作。

This research project concerns a series of interdisciplinary and fundamental mathematical methods for reaction-diffusion problems with multiple length and time scales. The first project concerns the Oya-Vallochi model of bioremediation, a system of advection, reaction, diffusion equations for the concentrations of the substrate to be degraded, the nutrient, and the active bacteria. Numerical simulations and mathematical analysis will be carried out to determine parameter domains in which stable traveling waves exist, as well as which instabilities develop on the boundaries of these domains. The second project involves reduction methods for chemical reaction and diffusion problems arising in biochemistry, chemical engineering, combustion, and air pollution engineering. These reactions typically involve 30-100 species, 50-400 reactions, and time scales ranging from nanoseconds to seconds. Moreover, these reactions typically occur in reactive flows, over extended domains, so that one must model the chemistry in each part of the domain. Reduction methods are indispensable to the analysis of this large amount of data, because they identify key progress variables and low-dimensional manifolds on which the long-term dynamics play out. The third project concerns pattern formation in activator-inhibitor reaction-diffusion equations, specifically the analysis of competing instabilities of spot patterns, self-replicating spots, newly-discovered `separator' solutions that govern the scattering of pulses and spots. Also, nonlinear stability of strongly interacting pulses will be established using a novel renormalization group approach the PI and collaborators have developed.brbrThis applied mathematics research project concerns mathematical methods for key applications of multiple-scale reaction diffusion systems in engineering and science. The goals of these projects are to explain recent experiments, to improve computational methods, and to develop new mathematical theory that will be essential for the next-generation of engineering techniques. The first project concerns bioremediation, a process in which microorganisms in soil are induced to degrade environmentally-harmful organic compounds. Operating conditions in which remediation takes place in a regular and optimal fashion will be identified. The second project concerns reduction methods that are essential for understanding complex chemical reactions in biochemistry, chemical engineering, combustion, and air pollution engineering. The mathematical theory the PI has developed to date has helped to improve reduction methods currently used by engineers, and the proposed new theory will help the development of the next generation of faster and more accurate reduction methods. The third project centers on the dynamics and stability of chemical patterns involving spots and interacting coherent structures. Mathematical methods from dynamical systems theory, singular perturbations, and differential equations will be used, and new mathematical techniques will also be developed. The projects also involve PhD students and postdoctoral fellows, a significant percentage of whom are women, as well as collaborations at Argonne National Laboratory and foreign universities.