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Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system. (English) Zbl 1251.35076
Summary: 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
35D30 Weak solutions to PDEs
35L60 First-order nonlinear hyperbolic equations
76W05 Magnetohydrodynamics and electrohydrodynamics
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