Instabilites in Fluid Motion

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DMS-9500466 Friedlander Recent results of Friedlander and Vishik give a sufficient condition for linear instability of a steady Euler flow. It is proposed to extend this work in several directions. Friedlander is showing that for classes of PDE, including the Euler equations, linear instability under certain spectral conditions implies nonlinear instability. Using the theory of oscillatory integrals it is proposed to extend the instability criteria to the more complicated system of magnetohydrodynamics. Another line of research is the study of the completeness of a system of Floquet root vectors for time periodic problems of fluid dynamics. Aspects of this work will be undertaken in collaboration with M.M. Vishik, W. Strauss and V.I. Yudovich. The question of stability/instability of fluid motion is a classical problem that has received much attention in the past 100 years. It is fundamental to studies in meteorology, oceanography, geophysics, astrophysics and plasma physics where physical phenomena are governed by the underlying fluid instabilities. Despite its long history, a number of important questions remain open. It is proposed to increase our understanding of the fundamental behaviour of fluids by studying the mathematical partial differential equations that describe fluid flow .