Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system

TL;DR

This paper derives the Hall-MHD equations from a two-fluid model, introduces a kinetic formulation, proves global weak solutions, and discusses special solutions and regularizations, advancing theoretical understanding of Hall-MHD systems.

Contribution

It provides a new kinetic formulation for Hall-MHD and establishes the global existence of weak solutions, which was previously unresolved.

Findings

Existence of global weak solutions for incompressible viscous resistive Hall-MHD

Kinetic formulation encompassing various Hall-MHD models

Analysis of axisymmetric swirling magnetic field solutions

Abstract

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.