Point vortex in a viscous incompressible fluid

A plane steady problem of a point vortex in a domain filled by a viscous incompressible fluid and bounded by a solid wall is considered. The existence of the solution of Navier–Stokes equations, which describe such a flow, is proved in the case where the vortex circulation $\Gamma$ and viscosity $\nu$ satisfy the condition $|\Gamma|<2\pi\nu$. The velocity field of the resultant solution has an infinite Dirichlet integral. It is shown that this solution can be approximated by the solution of the problem of rotation of a disk of radius $\gamma$ with an angular velocity $\omega$ under the condition $2\pi\gamma^2\omega\to\Gamma$, as $\gamma\to0$ and $\omega\to\infty$.