  We consider variational principles related to V. I. Arnold's stability
criteria for steady-state solutions of the two-dimensional incompressible Euler
equation. Our goal is to investigate under which conditions the quadratic forms
defined by the second variation of the associated functionals can be used in
the stability analysis, both for the Euler evolution and for the the
Navier-Stokes equation at low viscosity. In particular, we revisit the
classical example of Oseen's vortex, providing a new stability proof with
stronger geometric flavor. Our analysis involves a fairly detailed
functional-analytic study of the inviscid case, which may be of independent
interest, and a careful investigation of the influence of the viscous term in
the particular example of the Gaussian vortex.
