The project focuses on further development of constructive and rigorous asymptotic analysis of linear and nonlinear partial differential equations that arise in a number of important physical applications. The applications include (a) evolution of small disturbances superposed on an oscillatory flow in a channel or pipe, of interest in transition to turbulence, (b) existence and stability of self-sustaining states in channel and pipe flows, believed to be omega-limits in intermediate Reynolds number turbulent flow, (c) nonlinear stability of steadily translating fingers and bubbles in Hele-Shaw flow for small nonzero surface tension, of interest in pattern formation, (d) ionization of Hydrogen atoms in time-periodic dipole fields of arbitrary frequency and amplitude.
Our understanding of the solutions of equations arising in fluid motion and quantum mechanics is still very limited. The present project develops much needed new methods to study these problems. Among the immediate applications outside mathematics we note for instance that understanding turbulent fluid flow can lead to a more energy-efficient transportation of fluids through pipes. Also, better understanding of ionization processes has many applications, including optimal propagation of very intense laser beams through the atmosphere. Training of students --supported in the project-- will help maintain the high quality of research in the United States for the coming generation.