American Institute of Mathematical Sciences

Advanced
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

[1]

Ning Duan, Yasuhide Fukumoto, Xiaopeng Zhao. Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in $ \mathbb{R}^3 $. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3035-3057. doi: 10.3934/cpaa.2019136
[2]

Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
[3]

Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084
[4]

Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867
[5]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[6]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337
[7]

Zhong Tan, Huaqiao Wang, Yucong Wang. Time-splitting methods to solve the Hall-MHD systems with Lévy noises. Kinetic & Related Models, 2019, 12 (1) : 243-267. doi: 10.3934/krm.2019011
[8]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[9]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[10]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
[11]

Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 965-987. doi: 10.3934/dcdsb.2018341
[12]

Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
[13]

Augusto Visintin. Ohm-Hall conduction in hysteresis-free ferromagnetic processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 551-563. doi: 10.3934/dcdsb.2013.18.551
[14]

Mimi Dai. Phenomenologies of intermittent Hall MHD turbulence. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021285
[15]

Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure & Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963
[16]

Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032
[17]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[18]

Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090
[19]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643
[20]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (159)
  • HTML views (0)
  • Cited by (105)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy