Mathematical Sciences: Mathematical Techniques for Unstable Physical Processes

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This project will study the general area of instability in various physical systems using advanced mathematical concepts from perturbation theory, dynamical systems theory and Hamiltonian mechanics. Among the topics to be considered are the stability of dispersive wave solutions of the Ginzburg-Landau equation and the Zakharov equation, the advective mixing of fluid trajectories at the onset of chaos in viscous flows and the escape of particle trajectories of rotating Hamiltonian systems from KAM regions. The phenomenon of turbulence is perhaps the most important and the most difficult problem in fluid dynamics today. In this project the principal investigator will study simple models of turbulent flows using techniques from perturbation theory and nonlinear mechanics in order to understand better how instabilities in flows are magnified to the point where the flows go completely unstable and become turbulent. This too is an important problem, predicting where and when a flow first becomes turbulent, since very often one wants to delay the onset of turbulence for as long as possible.