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

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 twofluids 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.

Global existence and temporal decay for the 3D compressible Hall-magnetohydrodynamic system

“…Mathematical derivations of Hall-MHD equations from either two-fluids or kinetic models can be found in [1] and in this paper, the first existence result of global weak solutions is given. Comparing with the usual MHD equations, the Hall-MHD equations have the Hall term ∇ ×…”

“…When ρ = const, system (1.1) becomes the incompressible Hall-MHD system, which has received many studies, see [1,4,5,6,13,14,17,36,37,38]. When divu = 0, system (1.1) becomes the density-dependent Hall-MHD system, which has been investigated by many authors, and for more details, see [12,18].…”

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $${\mathbb{R}^{3}}$$ R 3

“…The above system has been studied a lot by physicists and mathematicians because of its physical importance, complexity, rich phenomena and mathematical challenges; see [1][2][3][4]6] and the references cited therein.…”

On global existence, energy decay and blow-up criteria for the Hall-MHD system

“…(1.1) was derived in [1] from kinetic models. Local classical solutions for (1.1) with − B (with or without − u) and global weak solutions for that with − u and − B were obtained in [2].…”