A Uniqueness theorem for the Vortex-Wave system with non-constant vorticity near the point vortex

We study the uniqueness of the vortex wave system, i.e.\ the 2D incompressible Euler equations with vorticity consisting of a point vortex and a bounded part. We show that if the initial vorticity and point vortex location $(w_0,X_0)$ satisfy a constraint of the type $\mathrm{\mid }{w}_0(x)-w_0(X_0)\mathrm{\mid }=O(\mathrm{\mid }x-X_0{\mathrm{\mid }}^{\alpha })$ for $\alpha >2$ and the vorticity is in $L^{\mathrm{\infty }}$, then solutions are unique, globally in time, even though the velocity is not log-Lipschitz.