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.
In a remarkable breakthrough in the field of Mathematical Science, Professor Kyudong Choi from the Department of Mathematical Sciences at UNIST has provided an irrefutable proof that certain spherical vortices exist in a stable state. This groundbreaking discovery holds significant implications for predicting weather anomalies and advancing weather prediction technologies.
A vortex is a rotating region of fluid, such as air or water, characterized by intense rotation. Common examples include typhoons and tornadoes frequently observed in news reports. Professor Choi’s mathematical proof establishes the stability of specific types of vortex structures that can be encountered in real-world fluid flows.
The study builds upon the foundational Euler equation formulated by Leonhard Euler in 1757 to describe the flow of eddy currents. In 1894, British mathematician M. Hill mathematically demonstrated that a ball-shaped vortex could maintain its shape indefinitely while moving along its axis.
Professor Choi’s research confirms that Hill’s spherical vortex maximizes kinetic energy under certain conditions through the application of variational methods. By incorporating functional analysis and partial differential equation theory from mathematical analysis, this study extends previous investigations on two-dimensional fluid flows to encompass three-dimensional fluid dynamics with axial symmetry conditions.
One notable feature identified by Hill is the presence of strong upward airflow at the front of the spherical vortex—an attribute often observed in phenomena like typhoons and tornadoes. Professor Choi’s findings serve as a starting point for further studies involving measurements related to residual time associated with these ascending air currents.
“Research on vortex stability has gained international attention,” stated Professor Choi. “[A]nd it holds long-term potential for advancements in today’s weather forecasting technology.”
Supported by funding from Korea Research Foundation under the Ministry of Science and ICT as well as UNIST, this study was published ahead of official release on July 24th via the online edition of Communications on Pure and Applied Mathematics.
Kyudong Choi, “Stability of Hill’s spherical vortex,” Commun. Pure Appl. Math., (2023).