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: $u_{0}=u_{0}^{r} e_{r}+u^{\theta}_{0} e_{\theta}+u_{0}^{z} e_{z}$ , $b_{0}=b_{0}^{\theta}e_{\theta}$ . We prove that, this system is globally well-posed provided the scaling-invariant norms $\|ru^{\theta}_{0}\|_{L^{\infty}}$ , $\|r^{-1}b^{\theta}_{0}\| _{L^{\frac{3}{2}}}$ are sufficiently small.