Point-vortex limit for a massive or massless particle in a two-dimensional perfect fluid

Let $h(t)$ be the position of a particle immersed in a two-dimensional incompressible perfect fluid, let $m$ be its mass, and let $\gamma$ be the circulation around it. Denote by $u_\text{bd}$ the background fluid velocity, by $\delta_h$ the Dirac mass at $h$, and by $K_{\mathbb{R}^2}$ the full-plane Biot–Savart law. Define the desingularized drift velocity by

$$u_\text{d}(h)=\left(u_\text{bd}-K_{\mathbb{R}^2}[\gamma\delta_h]\right)(h).$$

Point-vortex limit conjecture. A massive particle moves according to

$$m h”=\gamma\bigl(h'-u_\text{d}(h)\bigr)^\perp,$$

whereas a massless particle with nonzero circulation moves as a point vortex according to

$$h'=u_\text{d}(h).$$

In both cases, the vortex strength is the circulation $\gamma$, while the genuine fluid vorticity $\omega$ is transported by the background velocity $u_\text{bd}$. This describes the expected zero-radius dynamics, with the self-induced velocity removed from the particle velocity; the source presents it as a belief in a very general setting, and no resolution is supplied here.