On steady vortex flow in two dimensions, 1

The purpose of this paper (and its sequel) is to give a mathematically rigorous - as well as physically natural - discussion of certain steady solutions of the Euler dynamical equations for an ideal, two-dimensional fluid. The flows considered have a prescribed distribution of vorticity omega = curl u (u devotes the velocity field), and are separated into regions where omega = 0 and omega > 0. The shape and position of the vortex core (region where omega > 0) for a flow satisfying the dynamical requirements is then determined by the geometry of the fluid domain (assumed bounded here). Solutions of the fluid dynamical equations are most conveniently characterized by a variational principle which involves finding an extreme value for the kinetic energy of the flow subject to certain natural constraints. This approach permits a precise analysis of the properties of solutions to be carried out in a unified manner. In this respect, special emphasis is placed upon deriving the (classical) point vortex as the limit of solutions with concentrated vorticity.