Characterization and Regularity for Axisymmetric Solenoidal Vector Fields with Application to Navier–Stokes Equation

We consider the vorticity-stream formulation of axisymmetric incompressible flows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector field in terms of the swirling components, an extra set of pole conditions is necessary to give a full description of the regularity. In addition, smooth solutions up to the axis of rotation give rise to smooth solutions of primitive formulation in the case of the Navier–Stokes equation, but not the Euler equation. We also establish a proper weak formulation and show its equivalence to Leray's formulation.