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

Abstract

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 $u\in BM{O}^{-1}({\mathbb{R}}^3)$, 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.

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