On Existence of Sadovskii Vortex Patch: A Touching Pair of Symmetric Counter-Rotating Uniform Vortices

Abstract

The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in inviscid limit of planar flows via Prandtl-Batchelor theory and as the asymptotic state for vortex ring dynamics. In this work, we prove the existence of a Sadovskii vortex patch, by solving the energy maximization problem under the exact impulse condition and an upper bound on the circulation.