Global small solutions to the inviscid Hall-MHD system

TL;DR

This paper proves the existence of global small solutions to the inviscid Hall-MHD system on a 3D torus, given initial magnetic fields close to a background field satisfying a Diophantine condition.

Contribution

It establishes the first known global small solutions for the inviscid Hall-MHD system under specific initial magnetic field conditions.

Findings

Global small solutions exist on ^3 for initial magnetic fields near a Diophantine background.

The result extends understanding of the inviscid Hall-MHD system's long-term behavior.

Provides a mathematical framework for future studies on inviscid MHD systems.

Abstract

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu [Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {31} (2014), 555--565]. However, as we known, 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 $ \mathbb{T}^3$ when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.