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In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of [D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645–671].

REFINED A PRIORI ESTIMATES FOR THE AXISYMMETRIC NAVIER-STOKES EQUATIONS