New asymptotic methods and applications to physical problems

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The project focuses on further development of constructive andrigorous asymptotic analysis of linear and nonlinear partialdifferential equations that arise in a number of important physicalapplications. The applications include (a) evolution of smalldisturbances superposed on an oscillatory flow in a channel or pipe,of interest in transition to turbulence, (b) existence and stabilityof self-sustaining states in channel and pipe flows, believed to beomega-limits in intermediate Reynolds number turbulent flow, (c)nonlinear stability of steadily translating fingers and bubbles inHele-Shaw flow for small nonzero surface tension, of interest inpattern formation, (d) ionization of Hydrogen atoms in time-periodicdipole fields of arbitrary frequency and amplitude. Our understanding of the solutions of equations arising in fluidmotion and quantum mechanics is still very limited. The presentproject develops much needed new methods to study these problems.Among the immediate applications outside mathematics we note forinstance that understanding turbulent fluid flow can lead to a moreenergy-efficient transportation of fluids through pipes. Also, betterunderstanding of ionization processes has many applications, includingoptimal propagation of very intense laser beams through theatmosphere. Training of students --supported in the project-- willhelp maintain the high quality of research in the United States forthe coming generation.