On Existence of Relative Equilibria of Two--Dimensional Vortex Sheets

In this study we consider the existence of relative equilibria of two--dimensional vortex sheets. We focus on open sheets and derive conditions which must be satisfied by equilibrium configurations of such sheets. It is shown that, in contrast to the time--dependent case, such sheets must be everywhere orthogonal to the velocity field of the coordinate system in which they are stationary. Finally, we provide a rigorous demonstration that for vortex sheets arising from desingularization of translating (counter--rotating) and corotaing pairs of point vortices such equilibrium configuration do not in fact exist. The argument is based on classical results concerning existence of solutions of singular integral equations.