On the regularity of magneto-vorticity field and the global existence for the Hall magnetohydrodynamic equations

TL;DR

This paper studies the regularity of the magneto-vorticity field in Hall MHD equations, establishing bounds and conditions for global solutions, especially in 2D cases with small initial current density, and analyzes decay rates.

Contribution

It provides new bounds on the magneto-vorticity field and proves global well-posedness of Hall MHD with 2D variables under small initial current conditions.

Findings

Bounds on magneto-vorticity in 3D under velocity conditions

Global existence of solutions in 2D with small initial current

Decay rates of magneto-vorticity over time

Abstract

In this paper, we investigate the incompressible viscous and resistive Hall magnetohydrodynamic equations (Hall MHD in short). We first study the regularity of the magneto-vorticity field $B+\omega$. In three dimensions, we derive some bounds of $B+\omega$ under a condition of the velocity field $u$. Moreover, if we consider the Hall MHD with 2D variables, the uniform-in-time bounds of $B+\omega$ come from the three dimensional case. The regularity of $B+\omega$ gives us a crucial clue of blow-up scenario and provides conditions of the existence of global-in-time solutions. In particular, we prove the global well-posedness of the Hall MHD (also the electron MHD) with 2D variables when the third component of the initial current density $J_0=\nabla\times B_0$ is sufficiently small. We also derive temporal decay rate of $B+\omega$.