In this talk, we investigate the confinement properties of two-point and three-point vortex systems in a planar domain. When the initial point vortices are placed near a stable stationary configuration, a natural question is how long this configuration persists, and how the confinement time depends on the initial radius of closeness.

For the two-point vortex system, we show that this concentration persists indefinitely for almost every initial condition sufficiently close to the stationary point. However, when all initial conditions are considered, the confinement time follows a power-law behavior. For the three-point vortex system, we prove that this power-law behavior holds for all initial conditions.

Our approach to obtaining these power-law estimates does not rely on the integrability of the systems in the whole plane. Instead, we employ a non-standard Birkhoff normal form, which suggests strong potential for extending these results to the general N-point vortex system.

This is joint work with S. Ibrahim (Victoria), R. Qin (Ningbo), and S. Chen (Ningbo).