Isolated yet highly unstable 2D Euler equilibria

Abstract

We construct steady solutions of the two-dimensional Euler equations on smooth annular domains that are isolated, modulo scaling, in the $L^{\infty }$ vorticity topology. They are highly unstable: any Yudovich trajectory remaining sufficiently close to such an equilibrium on an open time interval is a stationary multiple of it and, moreover, at fixed energy, enstrophy and circulations, distinct nearby trajectories cannot return even weakly to the equilibrium. Further, on any non-circular annulus, we show that this kind of isolated, highly unstable equilibria occurs arbitrarily close in the Yudovich norm to smooth equilibria which are themselves non-isolated.