On the Large Data Global Well-Posedness of Inviscid Axially Symmetric MHD-Boussinesq System

The global well-posedness of the 3D inviscid MHD-Boussinesq system, with large axisymmetric initial data, in the Sobolev space \(H^{m}\) is given. To overcome difficulties that arise in the time-uniform \(H^{1}\) estimate, a reformulated system of good unknowns is discovered and an intermediate estimate is shown. Based on the reformulated system, a Beale-Kato-Majda-type criterion of the inviscid MHD-Boussinesq system is verified. Then higher-order estimates are concluded by the classical energy method and estimates of commutators. At last, we show the \(H^{m}\) norm of the global-in-time solution temporally grows no faster than a four times exponential function \((\forall m\in \mathbb{N})\).