We revisit Arnold's variational approach to the stability of
steady-state solutions of the two-dimensional Euler equations. In the
case of planar vortices, we study in detail the quadratic form that
represents the second variation of the energy on the isovortical
surface. We show in particular that, for a large class of radially
symmetric vortices with strictly decreasing profile, the second
variation is negative definite for all perturbations that preserve
the total circulation. We use that property to give a new stability
proof for the Oseen vortex as a self-similar solution of the 2D
Navier-Stokes equations, and to investigate the vanishing viscosity
limit of axisymmetric vortex rings. This talk is based on joint work
with V. Sverak.
