Remarks on Liouville type theorems for the steady MHD and Hall-MHD equations

TL;DR

This paper proves Liouville type theorems for steady 3D MHD and Hall-MHD equations, showing that under certain conditions on the velocity and magnetic fields, these fields must vanish, highlighting the velocity's critical role.

Contribution

It establishes new Liouville theorems for steady MHD and Hall-MHD equations with specific function space conditions, including cases with partial viscosity or diffusivity.

Findings

Velocity and magnetic fields vanish under given conditions.

Results hold for partial viscosity or diffusivity cases.

Highlights the importance of velocity field in the equations.

Abstract

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 $B\in L^{6,\infty}(\mathbb{R}^3)$ and $u\in BMO^{-1}(\mathbb{R}^3)$, 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.