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Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach
Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
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 |
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,'' 2nd 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. |
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. |
[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,'' 2nd 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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