On weighted estimates for the stream function of axially symmetric solutions to the Navier–Stokes equations in a bounded cylinder

Higher-order estimates in weighted Sobolev spaces for solutions to a singular elliptic equation for the stream function in an axially symmetric cylinder are provided. These estimates are essential for the proof of the global existence of regular axially symmetric solutions to incompressible Navier–Stokes equations in axially symmetric cylinders. In order to derive the estimates, the technique of weighted Sobolev spaces developed by Kondrat’ev is applied. The weight is a power function of the distance to the axis of symmetry.