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∈L6,∞(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\in L^{6,\infty }(\mathbb {R}^3)$$\end{document} and u∈BMO-1(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in BMO^{-1}(\mathbb {R}^3)$$\end{document}, 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.