Remarks on Regularity Criteria for Axially Symmetric Weak Solutions to   the Navier-Stokes Equations, II
  We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin condition and in addition angular component satisfies some condition, then the solution is regular. 
Adam Kubica
math.AP math-ph math.MP