Remarks on the Liouville type theorems for the 3D stationary MHD equations

Abstract

In this paper we study the stationary incompressible MHD equations on $ {\mathbb{R}}^3 $. We show that the velocity field $ u $ and the magnetic field $ b $, which vanish at infinity, must be identically zero, provided that $ \nabla u $ and $ \nabla b $ belong to $ L^q ({{\Bbb R}}^3) $ for some $ q \in (\frac{3}{2}, 2] $, and $ u $ belongs to $ \operatorname{BMO}^{-1}({{\Bbb R}}^3) $. Moreover, motivated by the recent work [19], we also obtain the Liouville type theorem by assuming that $ (u, b) $ is axisymmetric and $ u $ satisfies some decay property.