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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

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 & 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.  Google Scholar

[2]

S. I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Vol. 1 (ed. M. A. Leontovitch), 1965. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Duvaut and J.-L. Lions, inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  Google Scholar

[10]

C. Evans, "Partial Differential Equations,'' 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, 2009. Google Scholar

[11]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[17]

J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

S. I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Vol. 1 (ed. M. A. Leontovitch), 1965. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Duvaut and J.-L. Lions, inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  Google Scholar

[10]

C. Evans, "Partial Differential Equations,'' 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, 2009. Google Scholar

[11]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15-36. doi: 10.1080/03091929108229123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[17]

J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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