Well-posedness for Hall-magnetohydrodynamics

TL;DR

This paper establishes local and global existence of smooth solutions for the Hall-MHD equations under various conditions and proves a Liouville theorem for stationary solutions, advancing understanding of this complex fluid model.

Contribution

It provides the first rigorous proofs of local and global well-posedness for the Hall-MHD equations with large and small data, respectively, and introduces a Liouville theorem for stationary solutions.

Findings

Proved local existence of smooth solutions for large initial data.

Established global smooth solutions for small initial data.

Proved a Liouville theorem for stationary solutions.

Abstract

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.