Remarks on Liouville type theorems for the steady MHD and Hall-MHD equations

In this note we investigate Liouville type theorems for the steady three dimensional MHD and Hall-MHD equations, and show that the velocity field $u$ and the magnetic field $B$ are vanishing provided that $B\in L^{6,\infty}($\mathbb{R}$^3)$and$u\in BMO^{-1}($\mathbb{R}$^3)$, which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three dimensional MHD equations.