Regularity via one vorticity component for the 3D axisymmetric MHD equations

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood-Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that w theta e theta is an element of L1(0,T;B?infinity,infinity 0)$w_{\theta }e_{\theta }\in L<^>{1}\big (0,T;\dot{B}_{\infty ,\infty }<^>{0}\big )$, which partially gives a positive answer to the marginal case for the regularity of MHD equations.

Automated summary

This paper investigates the regularity conditions of axisymmetric weak solutions to the three-dimensional incompressible magnetohydrodynamics equations with nonzero swirl component. By utilizing the Littlewood-Paley decomposition techniques, it is shown that weak solutions become regular if the swirl component of vorticity satisfies certain conditions. This result provides a positive answer to the marginal case for the regularity of MHD equations.