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January  2021, 14(1): 149-174. doi: 10.3934/krm.2020052

Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem

Mehdi Badsi

Université de Nantes, CNRS UMR 6629, Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssinière, 44332 Nantes

Received  April 2020 Revised  July 2020 Published  November 2020

The mathematical description of the interaction between a collisional plasma and an absorbing wall is a challenging issue. In this paper, we propose to model this interaction by considering a stationary bi-species Vlasov-Poisson-Boltzmann boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be uniquely determined from the ambipolarity of the particles flows as the unique solution of a nonlinear equation. We also prove that it is an increasing function of the electrons re-emission coefficient at the wall. Based on the Schauder fixed point theorem, our analysis establishes the existence of a solution provided, on the one hand that the incoming ions density satisfies a moment condition that generalizes the Historical Bohm criterion, and on the other hand that the collision frequency does not exceed a critical value whose definition is subordinated to the strict validity of our generalized Bohm criterion.

Keywords: Vlasov-Poisson-Boltzmann, stationary transport problems, weakly singular Volterra equations, collisional sheath, floating potential, kinetic Bohm criterion.
Mathematics Subject Classification: Primary:35Q83, 35Q20;Secondary:35B35.
Citation: Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052
References:
[1]

S. Andras, Weakly singular Volterra and Fredholm-Volterra integral equations, Studia. Univ. "Babes-Bolyai", Mathematica, 48 2003, 147-155.  Google Scholar

[2]

A. A. Arsenev, Existence in the large of a weak solution of Vlasov's system of equations, Mat. Mat. Fiz, 15 (1975), 136–147.  Google Scholar

[3]

M.Badsi, M. Campos Pinto and B. Després, A Minimization formulation of a bi-kinetic sheath, Kinetic and related models, 9 (2016), 621-656. doi: 10.3934/krm.2016010.  Google Scholar

[4]

M. Badsi, Linear electron stability for a bi-kinetic sheath model, Journal of Mathematical Analysis and Applications, 453 (2017), 954-972. doi: 10.1016/j.jmaa.2017.04.055.  Google Scholar

[5]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Annales de l'institut Henri Poincaré, 2 (1985), 101-118. doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

[6]

N. Ben Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, M2AS, 17 (1994), 451-476 doi: 10.1002/mma.1670170604.  Google Scholar

[7]

N. Ben Abdallah and J. Dolbeault, Relative entropies for kinetic equations in bounded domains, Arch. Rat. Mech. Anal, 168 (2003), 253-298. doi: 10.1007/s00205-002-0239-0.  Google Scholar

[8]

D. Bohm, The characteristics of electrical discharges in magnetic fields, New York: Mc Graw Hill, Chap 3, 1949. Google Scholar

[9]

M. Bostan, Existence and uniquness of the mild solution for the 1d Vlasov-Poisson initial-boundary value problem, SIAM J. Math.Anal., 37 (2005), 156-188. doi: 10.1137/S0036141003434649.  Google Scholar

[10]

M. Bostan, I. M. Gamba, T. Goudon and A. Vasseur, Boundary Value problems for the stationary Vlasov-Boltzmann-Poisson equation, Indiana Univ. Math., 59 (2010), 1629-1660. doi: 10.1512/iumj.2010.59.4025.  Google Scholar

[11]

H. Brunner, The numerical solution of a weakly singular Volterra integral equations by collocation on graded meshes, Mathematics of Computation, 45 (1985), 417-437. doi: 10.1090/S0025-5718-1985-0804933-3.  Google Scholar

[12]

F. F. Chen, Introduction to Plasma Physics, Plenum press, 1974. doi: 10.1007/978-1-4757-0459-4.  Google Scholar

[13]

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.  Google Scholar

[14]

D. Gérard-Varet, D. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 62 (2013), 359-402. doi: 10.1512/iumj.2013.62.4900.  Google Scholar

[15]

Y. Guo, C-W. Shu and T. Zhou, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635. doi: 10.1137/S0036141003421133.  Google Scholar

[16]

N. Jiang and X. Zhang, The Boltzmann equation with incoming boundary condition : Global solutions and Navier-Stokes limit, SIAM J. Math. A, 51 (2019), 2504-2534. doi: 10.1137/17M114697X.  Google Scholar

[17]

C. W. Jurgensen and E. S. G. Shaqfeh, Nonlocal transport models of the self-consistent potential distribution in a plasma sheath with charge transfer collisions, J. Applied Physics, 64 (1988). doi: 10.1063/1.342077.  Google Scholar

[18]

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. Google Scholar

[19]

G. Manfredi and S. Devaux, Magnetized plasma-wall transition. Consequences for wall sputtering and erosion, Institute of Physics Publishing, 2008. Google Scholar

[20]

S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys, 210 (2000), 447–466. doi: 10.1007/s002200050787.  Google Scholar

[21]

S. Mukherjee, Effect of charge exchange collisions on the static properties of a fully collisional ion sheath, IEEE Transactions on Plasma Science, 23 (1995), 816-821.  doi: 10.1109/27.473200.  Google Scholar

[22]

C. Greengard and P.-A. Raviart, A Boundary-Value problem for the stationary Vlasov-Poisson equations : The Plane Diode, Communications on Pure and Applied Mathematics, 43 (1990), 473-507.  doi: 10.1002/cpa.3160430404.  Google Scholar

[23]

K.-U. Riemann, The Bohm criterion and sheath formation, Phys. Plasmas, 24 (1991). doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[24]

K.-U. Riemann, Kinetic analysis of the collisional plasma-sheath transition, Journal of Physical D : Applied Physics, 36 (2003). doi: 10.1088/0022-3727/36/22/007.  Google Scholar

[25]

J. Schaeffer, Global existence of smooth solutions to the Vlasov Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.  Google Scholar

[26]

Terrence E. Sheridan, Solution of the plasma-sheath equation with a cool Maxwellian ion source, AIP Publishing, (2001). doi: 10.1063/1.1391448.  Google Scholar

[27]

P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. Google Scholar

[28]

L. Tonks and I. Langmuir, A general theory of the Plasma of an Arc, Physical Review, 1929. doi: 10.1103/PhysRev.34.876.  Google Scholar

[29]

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.  Google Scholar

show all references

References:
[1]

S. Andras, Weakly singular Volterra and Fredholm-Volterra integral equations, Studia. Univ. "Babes-Bolyai", Mathematica, 48 2003, 147-155.  Google Scholar

[2]

A. A. Arsenev, Existence in the large of a weak solution of Vlasov's system of equations, Mat. Mat. Fiz, 15 (1975), 136–147.  Google Scholar

[3]

M.Badsi, M. Campos Pinto and B. Després, A Minimization formulation of a bi-kinetic sheath, Kinetic and related models, 9 (2016), 621-656. doi: 10.3934/krm.2016010.  Google Scholar

[4]

M. Badsi, Linear electron stability for a bi-kinetic sheath model, Journal of Mathematical Analysis and Applications, 453 (2017), 954-972. doi: 10.1016/j.jmaa.2017.04.055.  Google Scholar

[5]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Annales de l'institut Henri Poincaré, 2 (1985), 101-118. doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

[6]

N. Ben Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, M2AS, 17 (1994), 451-476 doi: 10.1002/mma.1670170604.  Google Scholar

[7]

N. Ben Abdallah and J. Dolbeault, Relative entropies for kinetic equations in bounded domains, Arch. Rat. Mech. Anal, 168 (2003), 253-298. doi: 10.1007/s00205-002-0239-0.  Google Scholar

[8]

D. Bohm, The characteristics of electrical discharges in magnetic fields, New York: Mc Graw Hill, Chap 3, 1949. Google Scholar

[9]

M. Bostan, Existence and uniquness of the mild solution for the 1d Vlasov-Poisson initial-boundary value problem, SIAM J. Math.Anal., 37 (2005), 156-188. doi: 10.1137/S0036141003434649.  Google Scholar

[10]

M. Bostan, I. M. Gamba, T. Goudon and A. Vasseur, Boundary Value problems for the stationary Vlasov-Boltzmann-Poisson equation, Indiana Univ. Math., 59 (2010), 1629-1660. doi: 10.1512/iumj.2010.59.4025.  Google Scholar

[11]

H. Brunner, The numerical solution of a weakly singular Volterra integral equations by collocation on graded meshes, Mathematics of Computation, 45 (1985), 417-437. doi: 10.1090/S0025-5718-1985-0804933-3.  Google Scholar

[12]

F. F. Chen, Introduction to Plasma Physics, Plenum press, 1974. doi: 10.1007/978-1-4757-0459-4.  Google Scholar

[13]

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.  Google Scholar

[14]

D. Gérard-Varet, D. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 62 (2013), 359-402. doi: 10.1512/iumj.2013.62.4900.  Google Scholar

[15]

Y. Guo, C-W. Shu and T. Zhou, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635. doi: 10.1137/S0036141003421133.  Google Scholar

[16]

N. Jiang and X. Zhang, The Boltzmann equation with incoming boundary condition : Global solutions and Navier-Stokes limit, SIAM J. Math. A, 51 (2019), 2504-2534. doi: 10.1137/17M114697X.  Google Scholar

[17]

C. W. Jurgensen and E. S. G. Shaqfeh, Nonlocal transport models of the self-consistent potential distribution in a plasma sheath with charge transfer collisions, J. Applied Physics, 64 (1988). doi: 10.1063/1.342077.  Google Scholar

[18]

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. Google Scholar

[19]

G. Manfredi and S. Devaux, Magnetized plasma-wall transition. Consequences for wall sputtering and erosion, Institute of Physics Publishing, 2008. Google Scholar

[20]

S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys, 210 (2000), 447–466. doi: 10.1007/s002200050787.  Google Scholar

[21]

S. Mukherjee, Effect of charge exchange collisions on the static properties of a fully collisional ion sheath, IEEE Transactions on Plasma Science, 23 (1995), 816-821.  doi: 10.1109/27.473200.  Google Scholar

[22]

C. Greengard and P.-A. Raviart, A Boundary-Value problem for the stationary Vlasov-Poisson equations : The Plane Diode, Communications on Pure and Applied Mathematics, 43 (1990), 473-507.  doi: 10.1002/cpa.3160430404.  Google Scholar

[23]

K.-U. Riemann, The Bohm criterion and sheath formation, Phys. Plasmas, 24 (1991). doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[24]

K.-U. Riemann, Kinetic analysis of the collisional plasma-sheath transition, Journal of Physical D : Applied Physics, 36 (2003). doi: 10.1088/0022-3727/36/22/007.  Google Scholar

[25]

J. Schaeffer, Global existence of smooth solutions to the Vlasov Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.  Google Scholar

[26]

Terrence E. Sheridan, Solution of the plasma-sheath equation with a cool Maxwellian ion source, AIP Publishing, (2001). doi: 10.1063/1.1391448.  Google Scholar

[27]

P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. Google Scholar

[28]

L. Tonks and I. Langmuir, A general theory of the Plasma of an Arc, Physical Review, 1929. doi: 10.1103/PhysRev.34.876.  Google Scholar

[29]

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.  Google Scholar

Figure 1.  Schematic characteristic ions trajectories associated with a decreasing potential $ \phi $. The solid lines corresponds to characteristic curves originating from $ x = 0 $ with positive velocities, and they span $ D_{i,0} $ the lighter gray region. The dashed lines correspond to characteristic curves originating from the wall with negative velocities, and they span the darker gray region $ D_{i,1} $
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Figure 2.  Schematic characteristic electrons trajectories associated with a decreasing potential $ \phi $. The solid lines corresponds to characteristic curves originating from $ x = 0 $ with positive velocities, and they span $ D_{e,0} $ the lighter gray region. The dashed lines correspond to characteristic curves originating from the wall with negative velocities, and they span the darker gray region $ D_{e,1} $
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