Kyoto and Oxford

We consider the initial-boundary value problem of the Navier-Stokes equations for axisymmetric initial data with swirl in the exterior of an infinite cylinder, subject to the slip boundary condition. We construct global solutions and give an upper bound for azimuthal component of vorticity in terms of the size of cylinder. The proof is based on the Boussinesq system. We show that the system is globally well-posed for axisymmetric data without swirl.