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A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror
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A minimization formulation of a bi-kinetic sheath
1. | UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05, France, France, France |
References:
[1] |
J. Apell, The superposition operator in function spaces. A survey, Expositiones Mathematicae, 1988. |
[2] |
A. Ambroso, X. Fleury, B. Lucquin-Desreux and P. A. Raviart, Some remarks on a stationary Vlasov-Poisson system with source term arising in ion beam neutralization, Transport Theory Statist. Phys., 30 (2001), 587-616.
doi: 10.1081/TT-100107418. |
[3] |
S. Baalrud and C. Hegna, Kinetic theory of the presheath and the Bohm criterion, Plasma Sources Science and Technology, 20 (2011). |
[4] |
H. Beresticky and T. Lachant-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Math. Nachr., 266 (2004), 3-19.
doi: 10.1002/mana.200310139. |
[5] |
D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, McGraw Hill, New York, 1949. |
[6] |
R. Chalise and R. Khanal, A Kinetic Trajectory Simulation for Magnetized Plasma Sheath, Institute of Physics Publishing, 2012. |
[7] |
F. Chen, Introduction to Plasma Physics, Plenum Press, 1974. |
[8] |
R. J. Diperna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[9] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010.
doi: 10.1090/gsm/019. |
[10] |
M. Feldman, S. Y. HA and M. Slemrod, A Geometric level-set formulation of a plasma sheath interface, Arch. Rat. Mech. Anal., 178 (2005), 81-123.
doi: 10.1007/s00205-005-0368-3. |
[11] |
Y. Güçlü, The plasma sheath for 1D-1V Vlasov-Poisson solvers, Invited talk at the Numkin 2013 Conference, Garching, Germany, http://www.ipp.mpg.de/1525371/Gueclue.pdf. |
[12] |
Y. Guo, C.-H. Shu and T. Zie, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635 (electronic).
doi: 10.1137/S0036141003421133. |
[13] |
O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques, Springer-Verlage, 1993. |
[14] |
H. Kohno, J. R. Myra and D. A. D'Ippolito, Radio-frequency sheath-plasma interactions with magnetic field tangency points along the sheath surface, Physics of Plasmas, 20 (2013), 082514. |
[15] |
J. G. Laframboise, Theory of spherical and cylindrical Langmuir probes in a collision less, Maxwellian plasma at rest, Institute for Aerospace Studies, University of Toronto, Report No. 100, 1966. |
[16] |
P.-H. Maire, Établissement et Comparaison de Modèles Fluides Pour un Plasma Faiblement Ionisé Quasi-neutre. Détermination des Conditions aux Limites à la Paroi, Thèse de Doctorat (in French) de l'Université Paris 6, 1996. |
[17] |
G. Manfredi and S. Devaux, Magnetized Plasma-Wall Transition. Consequences for Wall Sputtering and Erosion, Institute of Physics Publishing, 2008. |
[18] |
R. E. Marshak, The variational method for asymptotic neutron densities, Physical Review, 71 (1947), 688-693. |
[19] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, National Institute of Standards and Technology, Cambridge University Press, 2010. |
[20] |
P.-A. Raviart and C. Greengard, A Boundary-Value problem for the stationary Vlasov-Poisson equations: The Plane Diode, Comm. Pure Appl. Math., 43 (1990), 473-507.
doi: 10.1002/cpa.3160430404. |
[21] |
K.-U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 493.
doi: 10.1088/0022-3727/24/4/001. |
[22] |
T. E. Sheridan, Solution of the Plasma-Sheath Equation with a Cool Maxwellian Ion Source, AIP Publishing, 2001. |
[23] |
G. Stampachia and D. Kinderlehrer, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980. |
[24] |
P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. |
[25] |
F. Valsaque and G. Manfredi, Numerical study of plasma wall transition in an oblique magnetic field, Journal of Nuclear Materials, 290-293 (2001), 763-767.
doi: 10.1016/S0022-3115(00)00454-2. |
show all references
References:
[1] |
J. Apell, The superposition operator in function spaces. A survey, Expositiones Mathematicae, 1988. |
[2] |
A. Ambroso, X. Fleury, B. Lucquin-Desreux and P. A. Raviart, Some remarks on a stationary Vlasov-Poisson system with source term arising in ion beam neutralization, Transport Theory Statist. Phys., 30 (2001), 587-616.
doi: 10.1081/TT-100107418. |
[3] |
S. Baalrud and C. Hegna, Kinetic theory of the presheath and the Bohm criterion, Plasma Sources Science and Technology, 20 (2011). |
[4] |
H. Beresticky and T. Lachant-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Math. Nachr., 266 (2004), 3-19.
doi: 10.1002/mana.200310139. |
[5] |
D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, McGraw Hill, New York, 1949. |
[6] |
R. Chalise and R. Khanal, A Kinetic Trajectory Simulation for Magnetized Plasma Sheath, Institute of Physics Publishing, 2012. |
[7] |
F. Chen, Introduction to Plasma Physics, Plenum Press, 1974. |
[8] |
R. J. Diperna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[9] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010.
doi: 10.1090/gsm/019. |
[10] |
M. Feldman, S. Y. HA and M. Slemrod, A Geometric level-set formulation of a plasma sheath interface, Arch. Rat. Mech. Anal., 178 (2005), 81-123.
doi: 10.1007/s00205-005-0368-3. |
[11] |
Y. Güçlü, The plasma sheath for 1D-1V Vlasov-Poisson solvers, Invited talk at the Numkin 2013 Conference, Garching, Germany, http://www.ipp.mpg.de/1525371/Gueclue.pdf. |
[12] |
Y. Guo, C.-H. Shu and T. Zie, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635 (electronic).
doi: 10.1137/S0036141003421133. |
[13] |
O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques, Springer-Verlage, 1993. |
[14] |
H. Kohno, J. R. Myra and D. A. D'Ippolito, Radio-frequency sheath-plasma interactions with magnetic field tangency points along the sheath surface, Physics of Plasmas, 20 (2013), 082514. |
[15] |
J. G. Laframboise, Theory of spherical and cylindrical Langmuir probes in a collision less, Maxwellian plasma at rest, Institute for Aerospace Studies, University of Toronto, Report No. 100, 1966. |
[16] |
P.-H. Maire, Établissement et Comparaison de Modèles Fluides Pour un Plasma Faiblement Ionisé Quasi-neutre. Détermination des Conditions aux Limites à la Paroi, Thèse de Doctorat (in French) de l'Université Paris 6, 1996. |
[17] |
G. Manfredi and S. Devaux, Magnetized Plasma-Wall Transition. Consequences for Wall Sputtering and Erosion, Institute of Physics Publishing, 2008. |
[18] |
R. E. Marshak, The variational method for asymptotic neutron densities, Physical Review, 71 (1947), 688-693. |
[19] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, National Institute of Standards and Technology, Cambridge University Press, 2010. |
[20] |
P.-A. Raviart and C. Greengard, A Boundary-Value problem for the stationary Vlasov-Poisson equations: The Plane Diode, Comm. Pure Appl. Math., 43 (1990), 473-507.
doi: 10.1002/cpa.3160430404. |
[21] |
K.-U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 493.
doi: 10.1088/0022-3727/24/4/001. |
[22] |
T. E. Sheridan, Solution of the Plasma-Sheath Equation with a Cool Maxwellian Ion Source, AIP Publishing, 2001. |
[23] |
G. Stampachia and D. Kinderlehrer, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980. |
[24] |
P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. |
[25] |
F. Valsaque and G. Manfredi, Numerical study of plasma wall transition in an oblique magnetic field, Journal of Nuclear Materials, 290-293 (2001), 763-767.
doi: 10.1016/S0022-3115(00)00454-2. |
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