On existence of Sadovskii vortex patch: A touching pair of symmetric counter-rotating uniform vortex

TL;DR

This paper proves the existence of a Sadovskii vortex patch, a symmetric pair of touching vortex patches, by solving an energy maximization problem under specific physical constraints.

Contribution

It provides a rigorous mathematical proof of the Sadovskii vortex patch's existence, confirming long-standing numerical and theoretical conjectures.

Findings

Existence of Sadovskii vortex patch established mathematically.

Solution involves energy maximization under impulse and circulation bounds.

Supports relevance of Sadovskii vortex in inviscid flow limits and vortex dynamics.

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.