Parallel and circular flows for the two-dimensional Euler equations *

International audienceWe consider steady flows of ideal incompressible fluids in two-dimensional domains. These flows solve the Euler equations with tangential boundary conditions. If such a flow has no stagnation point in the domain or at infinity, in the sense that the infimum of its norm over the domain is positive, then it inherits the geometric properties of the domain, for some simple classes of domains. Namely, if the domain is a strip or a half-plane, then such a flow turns out to be parallel to the boundary of the domain. If the domain is the plane, the flow is then a parallel flow, that is, its trajectories are parallel lines. If the domain is an annulus, then the flow is circular, that is, the streamlines are concentric circles. The results are based on qualitative properties and classification results for some semilinear elliptic equations satisfied by the stream function