Remarkably little is known about the detailed structure of the stability spectrum of the Euler equations that govern the motion of an inviscid fluid. This is the case even though it is a classical problem and one fundamental to an understanding of turbulent flow. Friedlander will continue to exploit the concept of a "fluid Lyapunov exponent" which provides an effective means for detecting high frequency instabilities in the essential part of the spectrum. Instabilities may also arise in the discrete spectrum and in this situation no general approach is, at present, possible. Analysis of the eigenvalue problem will be given for a specific 2-dimensional flow with a "cat's eye" type structure which appears to typify an interesting class of flows containing both oscillatory regions and hyperbolic stagnation points. Friedlander also proposes to extend the range of applicability of a result of Friedlander et al that proves under certain conditions that linear instability in a PDE implies nonlinear instability. Friedlander et al recently proved that in two dimensions any steady flow that is sufficiently "close" to a nondegenerate unstable flow is also unstable. The harder problem of robustness of instability for 3- dimensional flows will now be investigated. A novel type of flow on a torus with nonzero flux will be investigated using techniques from free boundary problems.
The issue of stability of liquids or gases presents an important example of a physical question that may be addressed through sophisticated mathematical techniques. The answers have direct physical interpretations: stable flows are robust under inevitable disturbances in the environment while unstable flows break up rapidly. The question of stability or instability of a fluid flow is fundamental to studies of physical phenomena including those in the oceans and the atmosphere. In the view of many scientists, waves and instabilities lie at the heart of our attempts to use mathematical models for long term weather prediction and to better understand global climate change. However there are important open questions connected with the fundamental equations that underlie all models for fluid behavior. This proposal seeks the answers to some of these questions related to fluid instabilities.
