On regularity criterion to the 3D axisymmetric incompressible MHD equations

Abstract

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u D u r e r C u  e  C u z e z and b D b  e  , we prove that if jru.x, t/j Ä C holds for 1 Ä t < 0, then .u, b/ is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier-Stokes equations.

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

“…Liu et al. [34] established some regularity criteria for a class of axisymmetric solutions to the 3D incompressible MHD equations with magnetic field of particular form, similar research made for the axisymmetric MHD equations can be found in [24, 45, 47, 54] and references therein.…”