Title: A Liouville theorem for the Euler equations in the plane,
Abstract: This paper is concerned with qualitative properties of bounded steady flows
of an ideal incompressible fluid with no stagnation point in the
two-dimensional plane R^2. We show that any such flow is a shear flow, that is,
it is parallel to some constant vector. The proof of this Liouville-type result
is firstly based on the study of the geometric properties of the level curves
of the stream function and secondly on the derivation of some estimates on the
at most logarithmic growth of the argument of the flow. These estimates lead to
the conclusion that the streamlines of the flow are all parallel lines.