Steady symmetric vortex pairs and rearrangements

Source abstract

Synopsis We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equation in a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori . The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.