On the global existence to Hall-MHD system

This paper investigates the well-posedness of Hall-magnetohydrodynamics system. By using a new current function \begin{document}$ J = \nabla\times B $\end{document} as an additional unknown. The mild solution of Hall-MHD exists globally in the nonhomogeneous Lei-Lin space setting provided that the initial data satisfies \begin{document}$ \|u_{0}\|_{\mathcal{X}^{-1} }+\|B_{0}\|_{ \mathcal{X}^{-1}}+\|J_{0}\|_{ \mathcal{X}^{-1}}<\min \{\frac{\mu }{2}, \frac{\nu }{2}\}. $\end{document}