In this project the principal investigator intends to carry out a rigorous investigation of the existence and nonlinear stability (or instability) of certain important multidimensional structures arising in the mathematical study of compressible fluids. These structures include boundary layers, shock waves, detonation fronts, and vortex sheets. There is a vast applied literature on these topics, but until relatively recently they have resisted careful mathematical analysis, particularly in the multidimensional case. In recent work by the principal investigator, his collaborators, and others, new tools have become available, coming from PDE, harmonic analysis, and dynamical systems, that permit a rigorous study of the highly singular perturbation problems that arise in investigating the stability of such structures.
Much of the earlier work done by physicists and engineers on problems of the type under consideration in this project has been of a formal or heuristic nature. Approaching the same problems from a mathematically rigorous point of view not only puts previous work on a firm foundation but can also deepen understanding, uncover new phenomena, and bring errors to light. The proposed work on boundary layers is related to problems in engineering concerned with the stabilization of high speed aircraft and may be of interest to engineers working on such problems. The results of the research would be disseminated by publication in high quality journals and by lectures given at conferences and other universities. The grant will also foster international research collaboration of the principal investigator with colleagues in France and China. The research carried out under this grant is expected to have an impact on the work of other mathematicians, on graduate course preparation and seminars, and on the training of graduate students and postdocs.
