Global Axisymmetric Solutions to the 3D MHD Equations with Nonzero Swirl

This paper studies sufficient conditions under which axisymmetric solutions with nonzero swirl components to the Cauchy problem of the 3D incompressible magnetohydrodynamic (MHD) equations are globally well-posed. We first establish a Serrin-type regularity criterion via the swirl component of velocity for the MHD equations without magnetic diffusion. Some new estimates were introduced to overcome the difficulty caused by the absence of magnetic diffusion. Moreover, we prove the global existence of axisymmetric solutions in the presence of magnetic diffusion provided that the scaling-invariant smallness condition was prescribed only on the swirl component of initial velocity while the initial magnetic field can be arbitrarily large.