The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids

Abstract

We give a new description of the coadjoint operator \(\mathrm {Ad}^{*}_{\eta ^{-1}(t)}\) along a geodesic \(\eta (t)\) of the \(L^{2}\) metric in the group of volume-preserving diffeomorphisms, important in hydrodynamics. When the underlying manifold is two dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations modulo a compact operator; when the manifold is three dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations plus a bounded operator. We give two applications of this result when the underlying manifold is two dimensional: conjugate points along geodesics of the \(L^{2}\) metric are characterized in terms of the coadjoint operator and thus determining the conjugate locus is a purely algebraic question. We also prove that Eulerian and Lagrangian stability of the 2D Euler equations are equivalent and that instabilities in the 2D Euler equations are contained and small.