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

On the asymptotic stability of steady flows with nonzero flux in two-dimensional exterior domains