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.
Categories: math.AP