Global Small Solutions to the Inviscid Hall-MHD System

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu (Ann. Inst. H. Poincaré Anal. Non Linéaire 31:555–565, 2014). However, as far as we know, how to construct the global small solutions to the inviscid Hall-MHD system is still an open problem. In the present paper, we give a positive answer in T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {T}^3$$\end{document} when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.