Abstract
A team of researchers, affiliated with UNIST has made a significant breakthrough by mathematically proving that a special type of vortex pair, called the Sadovskii vortex patch, can exist within ideal fluid flows. This marks the first time such a solution has been rigorously demonstrated, more than 50 years after the model was first proposed.
Professor Kyudong Choi from the Department of Mathematical Sciences at UNIST, along with his graduate student Young-Jin Sim and Professor In-Jee Jeong from Seoul National University, announced that they have confirmed the existence of the Sadovskii vortex patch as a solution to the Euler equations.
The Sadovskii vortex patch is a unique shape where two vortices of equal strength rotate in opposite directions while remaining in contact along a symmetry axis. Although they resemble the vortices behind airplane wings or ships, these idealized structures are based on assumptions of inviscid flow, allowing them to maintain their shape and move in a straight line indefinitely.
The idea was first introduced by Russian mathematician V. S. Sadovskii through numerical simulations in 1971. However, proving that such a vortex pair could actually exist as a real solution has been a long-standing challenge in fluid dynamics.
To do this, the researchers used a mathematical approach, called variational analysis, which involves finding the configuration that maximizes or minimizes a specific quantity-in this case, the energy of the flow-under certain constraints. They set the distance between the vortices to be small and imposed an upper limit on their circulation strength, then identified the vortex pair with the highest possible energy.
Figure 1. Schematic illustration of the variational method-based proof approach used by the research team.
Their analysis showed that this energy-maximizing configuration matches the shape and structure of the Sadovskii patch originally proposed. Beyond establishing the existence of vortex pair, the team also demonstrated that such a structure could be physically stable, meaning it could persist in an ideal fluid environment-an important step toward understanding real-world vortex phenomena.
Professor Choi explained, "We have been competing with Professor Huang-Tong's team from Peking University to mathematically prove the Sadovskii vortex's existence." He added, "What sets our work apart is that we have also shown these vortex pairs could be physically stable, which makes them more than just mathematical curiosities."
This discovery has broad implications for fluid dynamics, including turbulence research, wake analysis behind aircraft and ships, and the study of vortex interactions in atmospheric and oceanic systems. For example, understanding the dynamics of vortex pairs could help explain phenomena, like the Fujiwara effect, where multiple cyclones interact and influence each other-first observed by Japanese meteorologist Fujiwara Sakuhé in 1921.
The findings of this research have been published in the December 2025 issue of Annals of PDE, one of the leading journals in the field. This research was supported by the National Research Foundation (NRF) of Korea through the Ministry of Science and ICT (MSIT).
Journal Reference
Kyudong Choi, In-Jee Jeong, and Young-Jin Sim, "On Existence of Sadovskii Vortex Patch: A Touching Pair of Symmetric Counter-Rotating Uniform Vortices," Ann. PDE., (2025).
