Dynamical systems theory and singular perturbation analysis for patterns, bubbles, and chemical reduction methods

点击翻译按钮获取中文摘要

Kaper0306523 This project concerns systems with multiple length and timescales, with the goals of analyzing recent experiments, ofimproving computational methods, and of establishing newmathematical theory for such systems. First, in the area ofchemical patterns with multiple length scales, therecently-discovered phenomena of self-replicating spots andpulses has posed new challenges for modeling and for stabilityanalysis of solutions of partial differential equations. Theinvestigator and collaborators build on their analysis of thedynamics, time scales, and mechanisms responsible forself-replication to study the underlying bifurcation hierarchiesthat organize the self-replication regime, to further examine thezero-pole cancellation phenomenon in the nonlocal eigenvalueproblem stability analysis, to extend the renormalization grouptechnique to establish the fully-nonlinear stability of pulses,and to develop extensions to systems with more than two lengthscales. Second, in the increasingly-important area of reductionmethods for large systems of chemical reactions with multipletime scales, the validity and accuracy of certain methods areanalyzed, with special focus on the computational singularperturbation method of Lam and Goussis. Third, the investigatoranalyzes the Oya-Vallochi model of subsurface bioremediation.Bioremediation is a process in which microorganisms, in thepresence of electron acceptors, degrade environmentally-harmfulorganic compounds. The investigator studies traveling waves ofbiomass activity and advection versus dispersion. Fourth, heconducts fundamental studies of nonspherical deformations of gasbubbles in Newtonian fluids. Finally, a challenging open problemconcerning the existence of self-similar, blow-up solutions ofthe nonlinear Schroedinger equation in spatial dimensions betweentwo and four is attempted. This project concerns mathematics for problems ofsignificant current interest in biology, chemistry, engineering,and physics, which exhibit both fast and slow dynamicalprocesses. First, with collaborators and a doctoral student, theinvestigator analyzes computational methods used to simulatelarge, complex systems of reactions in biochemistry, combustion,and air pollution engineering. These processes, such as theproduction of certain proteins, the burning of natural gas, andthe formation of nitrous oxides in the atmosphere, typicallyinvolve a few hundred species, each of which participates inseveral reactions, with the reaction times ranging fromnanoseconds to milliseconds, even to minutes. Methods that reducethe system complexity, while retaining a desired accuracy, arecritical for modeling these processes. The investigator aims toshow that there is a highly accurate method that can be used toimprove the accuracy of other widely-used methods, which areembedded in major computer codes. Second, the investigator and adoctoral student study mathematical models of bioremediation, inwhich microorganisms are used to degrade environmentally-harmfulorganic compounds. Mathematics provides an advantageous approachto determine important quantities, such as the wave speed withwhich the biologically-active zone propagates through a wet soilcolumn and how this speed depends on the many physicalparameters. Third, fundamental research is conducted on thedynamics of gas bubbles in water. Deformations of sphericalbubbles lead to oscillations on time scales much shorter thanthat on which the spherical mode itself oscillates, and the maingoal is to model the nonlinear transfer of energy between thespherical and nonspherical modes that can lead to bubblecavitation and the attendant production of underwater sound byturbine blades, for example. Finally, the investigator developsfurther theory for self-replicating chemical patterns and for aprototypical equation that governs nonlinear wave propagation.