A minimization formulation of a bi-kinetic sheath

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December 2016, 9(4): 621-656. doi: 10.3934/krm.2016010

A minimization formulation of a bi-kinetic sheath

Mehdi Badsi ^1, , Martin Campos Pinto ^1, and Bruno Després ^1,

1.	UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05, France, France, France

Received October 2014 Revised July 2015 Published September 2016

Abstract
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The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampère boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.

Keywords: Vlasov-Poisson system., plasma sheath, Bohm criterion, ambipolarity, Kinetic equation.

Mathematics Subject Classification: 35B35, 35L60, 82D10, 35B3.

Citation: Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic and Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010

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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