On the global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field

In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: $u_0=u_0^r e_r+\ut_0 e_\theta+u_0^z e_z, ~b_0=b_0^\theta e_\theta.$ We prove that, this system is global well-posed provided the scaling-invariant norms $\|r\ut_0\|_{L^\infty},~\|r^{-1} b^\theta_0\|_{L^{\frac32}}$ are sufficiently small.