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We consider variational principles related to V. I. Arnold's stabilitycriteria for steady-state solutions of the two-dimensional incompressible Eulerequation. Our goal is to investigate under which conditions the quadratic formsdefined by the second variation of the associated functionals can be used inthe stability analysis, both for the Euler evolution and for the theNavier-Stokes equation at low viscosity. In particular, we revisit theclassical example of Oseen's vortex, providing a new stability proof withstronger geometric flavor. Our analysis involves a fairly detailedfunctional-analytic study of the inviscid case, which may be of independentinterest, and a careful investigation of the influence of the viscous term inthe particular example of the Gaussian vortex.

Arnold's variational principle and its application to the stability of planar vortices