  In this paper, we consider steady Euler flows in two-dimensional bounded
annuli, as well as in exterior circular domains, in punctured disks and in the
punctured plane. We always assume rigid wall boundary conditions. We prove
that, if the flow does not have any stagnation point, and if it satisfies
further conditions at infinity in the case of an exterior domain or at the
center in the case of a punctured disk or the punctured plane, then the flow is
circular, namely the streamlines are concentric circles. In other words, the
flow then inherits the radial symmetry of the domain. The proofs are based on
the study of the trajectories of the flow and the orthogonal trajectories of
the gradient of the stream function, which is shown to satisfy a semilinear
elliptic equation in the whole domain. In exterior or punctured domains, the
method of moving planes is applied to some almost circular domains located
between some streamlines of the flow, and the radial symmetry of the stream
function is shown by a limiting argument. The paper also contains two
Serrin-type results in simply or doubly connected bounded domains with free
boundaries. Here, the flows are further assumed to have constant norm on each
connected component of the boundary and the domains are then proved to be disks
or annuli.
