Mathematical Sciences: "Mathematical Topics Related to Fluid Instabilities

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9622563 Friedlander S.Friedlander in collaboration with M.M.Vishik will continue her investigation into mathematical problems that arise in fluid motion. It is proposed to study the spectrum of small oscillations of an ideal fluid about a given steady flow: this leads to the study of the spectrum of a degenerate non-elliptic differential operator. The exact location of the unstable continuous spectrum will be determined purely in terms of dynamical systems quantities. Examples of fluid instability due to the discrete part of the spectrum will be studied analytically and numerically. It was recently proved that,under certain assumptions, linear instability of a steady inviscid flow implies nonlinear (Lyapunov) instability. It is proposed to extend the range of applicability of this result.Another line of research is the study of instabilities for the augmented system of fluid equations that govern magnetohydrodynamics. A sufficient condition for instability has been derived in terms of a system of local PDE. It is proposed to apply this criterion to demonstrate instability of certain astrophysical models. %%% All fluid motions are continually subject to small disturbances ( eg, think of a tank of water in a laboratory that is "disturbed" by a truck driving by outside ). A natural question arises as to whether the the effect of the disturbance dies away leaving the fluid in the same state as before --this is called stable--or the effect of the disturbance is to change the configuration of the fluid--this is called unstable. The question of fluid stability/instability is a classical one that has received much attention in the scientific literature for more than a century.It is fundamental to studies in meteorology, oceanography, geophysics and astrophysics ; in particular, instabilities at the air/sea interface are significant to any study of the global change of the environment. The mathematics of fluid instabilities is governed by a system of partial differ ential equations that are remarkably challenging and interesting. Despite a century of study there are many open mathematical questions connected with these equations. This proposal will continue to address some of these open problems. ***