In the short article we study the ideal incompressible magnetohydrodynamic equations in three dimensions in which the Faraday law is inviscid. We prove the global well-posedness of classical solutions for a family of special axisymmetric initial data whose swirl components of the velocity field and magnetic vorticity field are trivial.
Lei, Z. (2015). On axially symmetric incompressible magnetohydrodynamics in three dimensions. Journal of Differential Equations, 259(7), 3202–3215. https://doi.org/10.1016/j.jde.2015.04.017