Prandtl-Batchelor flows with a point vortex on a disk

We study steady two-dimensional incompressible Navier-Stokes flow in the unit disk in a high Reynolds number regime, driven by a localized central forcing and a nearly rigid rotating boundary. The forcing models a compact source of circulation that sustains a vortex core in a viscous flow. For sufficiently small viscosity and boundary perturbation, we construct a very weak steady solution on the whole disk and identify its inviscid limit: a point vortex embedded in a constant vorticity background and joined to the boundary motion by a thin boundary layer. The background vorticity is selected by the Batchelor-Wood formula. The singular core introduces non-coercive interactions in the linearized problem that are absent for regular Prandtl-Batchelor flows. To overcome this difficulty, we map the punctured disk to a semi-infinite cylinder by a logarithmic radial transformation and develop a Fourier-mode analysis that isolates the delicate first Fourier mode, yielding uniform stability and vorticity estimates in the presence of the point-vortex singularity.