Liouville type theorem for the stationary equations of magneto-hydrodynamics

We show that any smooth solution $(\mathbf{u},\mathbf{H})$ to the stationary equations of magneto-hydrodynamics (MHD) belonging to both spaces $L^6 ($\mathbb{R}$^3)$and$BMO^{-1}($\mathbb{R}$^3)$ must be identically zero. This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption $\nabla $\mathbf{u}$, \nabla $\mathbf{H}$ \in L^2 ($\mathbb{R}$^3) $, i.e., finite Dirichlet integral.