On Global special regular solutions to Navier-Stokes equations in axially symmetric domain

In this seminar we discuss existence and uniqueness of global regular solutions to the Navier-Stokes equations in cylindrical and axially symmetric domains. We assume the slip boundary conditions. We prove existence of such solutions which are close to the axially symmetric solutions. The existence will be shown in two steps. First we show how to prove existence of local solutions with large existence time which magnitude increases if initial data are more closed to the initial data for the axially symmetric solution. Next assuming some decay estimates for the external force we will be able to show that the local solution can be prolonged step by step up to infinity