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

By Marion Arichetogaray, Pierre Degond, A. Frouvelle, Jian‐Guo Liu. 2011

264 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/8a09038fa2fa52c51398aa93d67661265f002206

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Singularity formation for the incompressible Hall-MHD equations without resistivity (2013, unknown): “for the existence of global weak solutions [9, 18], regularity criterion [11, 12] and global smooth small solutions [17, 21]. The Hall-MHD has received little attention from mathematicians. The paper [1] provided a derivation ofHall-MHD system froma two-ﬂuids Euler-Maxwell system forelectrons and io”
Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations (2015, methodology): “A derivation of Hall-MHD system from a two-fluids Euler-Maxwell system fo r electrons and ions was presented in [1], through a set of scaling limits.”
Well-posedness for Hall-magnetohydrodynamics (2012, methodology): “The details of the functional analysis are the same as in [1].”
On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics (2013, background): “In [1], Acheritogaray, Degond, Frouvelle and Liu derived the Hall-MHD equations from either two fluids model or kinetic models in a mathematically rigorous way.”