Stability of Hill's spherical vortex

TL;DR

This paper investigates the stability of Hill's spherical vortex, an explicit solution to the 3D Euler equations, using variational methods and a concentration compactness approach to establish stability up to translation.

Contribution

It applies a variational framework and uniqueness results to prove the stability of Hill's spherical vortex, a classical explicit solution.

Findings

Stability of Hill's vortex up to translation established.

Variational maximizers correspond to Hill's vortex.

Stability proof uses concentration compactness method.

Abstract

We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981), which produced a maximizer of the kinetic energy under constraints on vortex strength, impulse, and circulation. We match the set of maximizers with the Hill's vortex via the uniqueness result of C. Amick and L. Fraenkel (Arch. Rational Mech. Anal., 1986). The matching process is done by an approximation near exceptional points (so-called metrical boundary points) of the vortex core. As a consequence, the stability up to a translation is obtained by using a concentrated compactness method.