Global axially symmetric solutions with large swirl to the Navier-Stokes equations

Long time existence of axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder and with boundary slip conditions is proved. The axially symmetric solutions with nonvanishing azimuthal component of velocity (swirl) are examined. The solutions are such that swirl is small in a neighbourhood close to the axis of symmetry but it is large in some positive distance from it. There is a great difference between the proofs of global axially symmetric solutions with vanishing and nonvanishing swirl. In the first case global estimate follows at once but in the second case we need a lot of considerations in weighted spaces to show it. The existence is proved by the Leray-Schauder fixed point theorem