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

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December 2011, 4(4): 901-918. doi: 10.3934/krm.2011.4.901

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

Marion Acheritogaray ^1, , Pierre Degond ^1, , Amic Frouvelle ^1, and Jian-Guo Liu ^2,

1.	1-Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathmatiques de Toulouse, F-31062 Toulouse, France, France, France
2.	Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, United States

Received July 2011 Revised August 2011 Published November 2011

Abstract
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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.

Keywords: incompressible viscous flow, Hall-MHD, global weak solutions, KMC waves., entropy dissipation, kinetic formulation, generalized Ohm's law, resistivity.

Mathematics Subject Classification: Primary: 35L60; Secondary: 35K55, 35Q8.

Citation: Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic and Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901

References:

[1]	L. Arnold, J. Dreher and R. Grauer, A semi-implicit Hall-MHD solver using whistler wave preconditioning, Comput. Phys. Comm., 178 (2008), 553-557.
[2]	S. I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Vol. 1 (ed. M. A. Leontovitch), 1965.
[3]	B. Cassany and P. Grua, Analysis of the operating regimes of microsecond-conduction-time plasma opening switches, J. Appl. Phys., 78 (1995), 67-76. doi: 10.1063/1.360583.
[4]	L. Chacòn and D. A. Knoll, A 2D high-$\beta$ Hall MHD implicit nonlinear solver, J. Comput. Phys., 188 (2003), 573-592. doi: 10.1016/S0021-9991(03)00193-1.
[5]	P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures, in "Material Substructures in Complex Bodies: From Atomic Level to Continuum" (eds. G. Capriz and P.-M. Mariano), Elsevier, 2007. doi: 10.1016/B978-008044535-9/50002-9.
[6]	P. Degond, F. Deluzet, G. Dimarco, G. Gallice, P. Santagati and C. Tessieras, Simulation of non-equilibrium plasmas with a numerical noise-reduced particle-in-cell method, in "Proceedings of the 27th International Symposium on Rarefied Gas Dynamics," July 10-15, Pacific Grove, California, 2010.
[7]	P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys., 25 (1996), 595-633. doi: 10.1080/00411459608222915.
[8]	J. Dreher, V. Runban and R. Grauer, Axisymmetric flows in Hall-MHD: A tendency towards finite-time singularity formation, Physica Scripta, 72 (2005), 451-455. doi: 10.1088/0031-8949/72/6/004.
[9]	G. Duvaut and J.-L. Lions, inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
[10]	C. Evans, "Partial Differential Equations,'' 2^nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, 2009.
[11]	T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123.
[12]	H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Physica D, 208 (2005), 59-72. doi: 10.1016/j.physd.2005.06.003.
[13]	D. S. Harned and Z. Mikić, Accurate semi-implicit treatment of the Hall effect in magnetohydrodynamic computations, J. Comput. Phys., 83 (1989), 1-15. doi: 10.1016/0021-9991(89)90220-9.
[14]	J. D. Huba and L. I. Rudakov, Hall magnetohydrodynamics of reversed field current layers, Physica Scripta, T107 (2004), 20-26. doi: 10.1238/Physica.Topical.107a00020.
[15]	F. Kazeminezhad, J. N. Leboeuf, F. Brunel and J. M. Dawson, A discrete model for MHD incorporating the Hall term, J. Comput. Phys., 104 (1993), 398-417. doi: 10.1006/jcph.1993.1039.
[16]	A. S. Kingsep, Yu. V. Mokhov and Y. V. Chukbar, Nonlinear skin phenomenas in plasmas, Nonlinear and Turbulent Processes in Physics, in "Proceedings of the Second International Workshop held 10-25 October, 1983" (ed. R. Z. Sagdeev), Harwood Academic Publishers, 1984.
[17]	J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.
[18]	J.-G. Liu and W.-C. Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal., 41 (2009), 1825-1850.
[19]	S. M. Mahajan and V. Krishan, Exact solution of the incompressible Hall magnetohydrodynamics, Mon. Not. R. Astron. Soc., 359 (2005), L27-L29. doi: 10.1111/j.1745-3933.2005.00028.x.
[20]	F. Méhats and J.-M. Roquejoffre, A nonlinear oblique derivative boundary value problem for the heat equation. Part 1: Basic results, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 16 (1999), 221-253.
[21]	A. N. Simakov and L. Chacón, Quantitative, comprehensive, analytical model for magnetic reconnection in Hall magnetohydrodynamics, Phys. Rev. Lett., 101 (2008), 105003. doi: 10.1103/PhysRevLett.101.105003.
[22]	F. Valentini, P. Tràvníček, F. Califano, P. Hellinger and A. Mangeney, A hybrid-Vlasov model based on the current advance method for the simulation of collisionless magnetized plasma, J. Comput. Phys., 225 (2007), 753-770. doi: 10.1016/j.jcp.2007.01.001.

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References:

[1]	L. Arnold, J. Dreher and R. Grauer, A semi-implicit Hall-MHD solver using whistler wave preconditioning, Comput. Phys. Comm., 178 (2008), 553-557.
[2]	S. I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Vol. 1 (ed. M. A. Leontovitch), 1965.
[3]	B. Cassany and P. Grua, Analysis of the operating regimes of microsecond-conduction-time plasma opening switches, J. Appl. Phys., 78 (1995), 67-76. doi: 10.1063/1.360583.
[4]	L. Chacòn and D. A. Knoll, A 2D high-$\beta$ Hall MHD implicit nonlinear solver, J. Comput. Phys., 188 (2003), 573-592. doi: 10.1016/S0021-9991(03)00193-1.
[5]	P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures, in "Material Substructures in Complex Bodies: From Atomic Level to Continuum" (eds. G. Capriz and P.-M. Mariano), Elsevier, 2007. doi: 10.1016/B978-008044535-9/50002-9.
[6]	P. Degond, F. Deluzet, G. Dimarco, G. Gallice, P. Santagati and C. Tessieras, Simulation of non-equilibrium plasmas with a numerical noise-reduced particle-in-cell method, in "Proceedings of the 27th International Symposium on Rarefied Gas Dynamics," July 10-15, Pacific Grove, California, 2010.
[7]	P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys., 25 (1996), 595-633. doi: 10.1080/00411459608222915.
[8]	J. Dreher, V. Runban and R. Grauer, Axisymmetric flows in Hall-MHD: A tendency towards finite-time singularity formation, Physica Scripta, 72 (2005), 451-455. doi: 10.1088/0031-8949/72/6/004.
[9]	G. Duvaut and J.-L. Lions, inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
[10]	C. Evans, "Partial Differential Equations,'' 2^nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, 2009.
[11]	T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123.
[12]	H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Physica D, 208 (2005), 59-72. doi: 10.1016/j.physd.2005.06.003.
[13]	D. S. Harned and Z. Mikić, Accurate semi-implicit treatment of the Hall effect in magnetohydrodynamic computations, J. Comput. Phys., 83 (1989), 1-15. doi: 10.1016/0021-9991(89)90220-9.
[14]	J. D. Huba and L. I. Rudakov, Hall magnetohydrodynamics of reversed field current layers, Physica Scripta, T107 (2004), 20-26. doi: 10.1238/Physica.Topical.107a00020.
[15]	F. Kazeminezhad, J. N. Leboeuf, F. Brunel and J. M. Dawson, A discrete model for MHD incorporating the Hall term, J. Comput. Phys., 104 (1993), 398-417. doi: 10.1006/jcph.1993.1039.
[16]	A. S. Kingsep, Yu. V. Mokhov and Y. V. Chukbar, Nonlinear skin phenomenas in plasmas, Nonlinear and Turbulent Processes in Physics, in "Proceedings of the Second International Workshop held 10-25 October, 1983" (ed. R. Z. Sagdeev), Harwood Academic Publishers, 1984.
[17]	J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.
[18]	J.-G. Liu and W.-C. Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal., 41 (2009), 1825-1850.
[19]	S. M. Mahajan and V. Krishan, Exact solution of the incompressible Hall magnetohydrodynamics, Mon. Not. R. Astron. Soc., 359 (2005), L27-L29. doi: 10.1111/j.1745-3933.2005.00028.x.
[20]	F. Méhats and J.-M. Roquejoffre, A nonlinear oblique derivative boundary value problem for the heat equation. Part 1: Basic results, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 16 (1999), 221-253.
[21]	A. N. Simakov and L. Chacón, Quantitative, comprehensive, analytical model for magnetic reconnection in Hall magnetohydrodynamics, Phys. Rev. Lett., 101 (2008), 105003. doi: 10.1103/PhysRevLett.101.105003.
[22]	F. Valentini, P. Tràvníček, F. Califano, P. Hellinger and A. Mangeney, A hybrid-Vlasov model based on the current advance method for the simulation of collisionless magnetized plasma, J. Comput. Phys., 225 (2007), 753-770. doi: 10.1016/j.jcp.2007.01.001.

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