# American Institute of Mathematical Sciences

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

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

 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  February 2021 Early access  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.

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Citation: Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic and 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. [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. [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. [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. [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. [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. [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. [8] D. Bohm, The characteristics of electrical discharges in magnetic fields, New York: Mc Graw Hill, Chap 3, 1949. [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. [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. [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. [12] F. F. Chen, Introduction to Plasma Physics, Plenum press, 1974. doi: 10.1007/978-1-4757-0459-4. [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. [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. [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. [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. [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. [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. [19] G. Manfredi and S. Devaux, Magnetized plasma-wall transition. Consequences for wall sputtering and erosion, Institute of Physics Publishing, 2008. [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. [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. [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. [23] K.-U. Riemann, The Bohm criterion and sheath formation, Phys. Plasmas, 24 (1991). doi: 10.1088/0022-3727/24/4/001. [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. [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. [26] Terrence E. Sheridan, Solution of the plasma-sheath equation with a cool Maxwellian ion source, AIP Publishing, (2001). doi: 10.1063/1.1391448. [27] P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. [28] L. Tonks and I. Langmuir, A general theory of the Plasma of an Arc, Physical Review, 1929. doi: 10.1103/PhysRev.34.876. [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.

show all references

##### References:
 [1] S. Andras, Weakly singular Volterra and Fredholm-Volterra integral equations, Studia. Univ. "Babes-Bolyai", Mathematica, 48 2003, 147-155. [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. [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. [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. [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. [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. [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. [8] D. Bohm, The characteristics of electrical discharges in magnetic fields, New York: Mc Graw Hill, Chap 3, 1949. [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. [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. [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. [12] F. F. Chen, Introduction to Plasma Physics, Plenum press, 1974. doi: 10.1007/978-1-4757-0459-4. [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. [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. [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. [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. [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. [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. [19] G. Manfredi and S. Devaux, Magnetized plasma-wall transition. Consequences for wall sputtering and erosion, Institute of Physics Publishing, 2008. [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. [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. [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. [23] K.-U. Riemann, The Bohm criterion and sheath formation, Phys. Plasmas, 24 (1991). doi: 10.1088/0022-3727/24/4/001. [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. [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. [26] Terrence E. Sheridan, Solution of the plasma-sheath equation with a cool Maxwellian ion source, AIP Publishing, (2001). doi: 10.1063/1.1391448. [27] P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. [28] L. Tonks and I. Langmuir, A general theory of the Plasma of an Arc, Physical Review, 1929. doi: 10.1103/PhysRev.34.876. [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.
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 $D_{i,1}$
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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