  The Navier-Stokes equations in a two-dimensional exterior domain are
considered. The asymptotic stability of stationary solutions satisfying a
general hypothesis is proven under any $L^2$-perturbation. In particular the
general hypothesis is valid if the steady solution is the sum of the critically
decaying flux carrier with flux $|\Phi| < 2\pi$ and a small subcritically
decaying term. Under the central symmetry assumption, the general hypothesis is
also proven for any critically decaying steady solutions under a suitable
smallness condition.
