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

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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.