A constrained variational problem for an existence theorem of a steady vortex pair in two-phase shear flow

We prove an existence theorem for a steady vortex pair in two-phase shear flow in a planar domain. The method used is a variational principle in which the kinetic energy is maximised subject to the vorticity belonging to the weak closure of the set of rearrangements of a prescribed function, and subject to another functional representing the “generalised impulse” having a prescribed value. We prove also that when the prescribed value of the “generalised impulse” is large enough, the constrained maximiser of the kinetic energy is in fact a rearrangement of the prescribed function.