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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.
|
[2] |
S. I. Braginskii, Transport processes in a plasma,, in, (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.
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.
doi: 10.1016/S0021-9991(03)00193-1. |
[5] |
P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures,, in, (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, (2010), 10. 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.
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.
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.
|
[10] |
C. Evans, "Partial Differential Equations,'', 2nd edition, 19 (2009). Google Scholar |
[11] |
T. G. Forbes, Magnetic reconnection in solar flares,, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15.
doi: 10.1080/03091929108229123. |
[12] |
H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems,, Physica D, 208 (2005), 59.
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.
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.
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.
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, (1983), 10. Google Scholar |
[17] |
J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'', Dunod, (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.
|
[19] |
S. M. Mahajan and V. Krishan, Exact solution of the incompressible Hall magnetohydrodynamics,, Mon. Not. R. Astron. Soc., 359 (2005).
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, 16 (1999), 221.
|
[21] |
A. N. Simakov and L. Chacón, Quantitative, comprehensive, analytical model for magnetic reconnection in Hall magnetohydrodynamics,, Phys. Rev. Lett., 101 (2008).
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.
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.
|
[2] |
S. I. Braginskii, Transport processes in a plasma,, in, (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.
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.
doi: 10.1016/S0021-9991(03)00193-1. |
[5] |
P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures,, in, (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, (2010), 10. 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.
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.
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.
|
[10] |
C. Evans, "Partial Differential Equations,'', 2nd edition, 19 (2009). Google Scholar |
[11] |
T. G. Forbes, Magnetic reconnection in solar flares,, Geophysical and Astrophysical Fluid Dynamics, 62 (1991), 15.
doi: 10.1080/03091929108229123. |
[12] |
H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems,, Physica D, 208 (2005), 59.
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.
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.
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.
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, (1983), 10. Google Scholar |
[17] |
J.-L.Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'', Dunod, (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.
|
[19] |
S. M. Mahajan and V. Krishan, Exact solution of the incompressible Hall magnetohydrodynamics,, Mon. Not. R. Astron. Soc., 359 (2005).
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, 16 (1999), 221.
|
[21] |
A. N. Simakov and L. Chacón, Quantitative, comprehensive, analytical model for magnetic reconnection in Hall magnetohydrodynamics,, Phys. Rev. Lett., 101 (2008).
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.
doi: 10.1016/j.jcp.2007.01.001. |
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