  The asymptotic behavior of the vorticity for the steady incompressible
Navier-Stokes equations in a two-dimensional exterior domain is described in
the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero.
It is well known that the asymptotic behavior of the velocity field is given by
the fundamental solution of the Oseen system which is the linearization of the
Navier-Stokes equation around $\boldsymbol{u}_{\infty}$. The vorticity has the
property of decaying algebraically inside a parabolic region called the wake
and exponentially outside. The previously proven asymptotic expansions of the
vorticity are relevant only inside the wake because everywhere else the
remainder is larger than the asymptotic term. Here we present an asymptotic
expansion that removes this weakness. Surprisingly, the found asymptotic term
is not given by the Oseen linearization and has a power of decay that depends
on the data. This strange behavior is specific to the two dimensional problem
and is not present in three dimensions.
