# American Institute of Mathematical Sciences

September  2016, 9(3): 587-603. doi: 10.3934/krm.2016008

## Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

 1 Department of Computer Science and Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan

Received  April 2015 Revised  February 2016 Published  May 2016

The main concern of this paper is to analyze a boundary layer called a sheath that occurs on the surface of materials when in contact with a multicomponent plasma. For the formation of a sheath, the generalized Bohm criterion demands that ions enter the sheath region with a high velocity. The motion of a multicomponent plasma is governed by the Euler--Poisson equations, and a sheath is understood as a monotone stationary solution to those equations. In this paper, we prove the unique existence of the monotone stationary solution by assuming the generalized Bohm criterion. Moreover, it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in weighted Sobolev space. We also obtain the convergence rate, which is subject to the decay rate of the initial perturbation, of the time global solution toward the stationary solution.
Citation: Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008
##### References:
 [1] A. Ambroso, Stability for solutions of a stationary Euler-Poisson problem,, Math. Models Methods Appl. Sci., 16 (2006), 1817.  doi: 10.1142/S0218202506001728.  Google Scholar [2] A. Ambroso, F. Méhats and P.-A. Raviart, On singular perturbation problems for the nonlinear Poisson equation,, Asympt. Anal., 25 (2001), 39.   Google Scholar [3] D. Bohm, Minimum ionic kinetic energy for a stable sheath,, in The characteristics of electrical discharges in magnetic fields (eds. A. Guthrie and R.K.Wakerling), (1949), 77.   Google Scholar [4] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion,, $2^{nd}$ edition, (1984).   Google Scholar [5] S.-H. Ha and M. Slemrod, Global existence of plasma ion-sheaths and their dynamics,, Comm. Math. Phys., 238 (2003), 149.  doi: 10.1007/s00220-003-0871-z.  Google Scholar [6] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar [7] I. Langmuir, The interaction of electron and positive ion space charges in cathode sheaths,, Phys. Rev., 33 (1929), 954.  doi: 10.1103/PhysRev.33.954.  Google Scholar [8] M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing,, $2^{nd}$ edition, (2005).  doi: 10.1002/0471724254.  Google Scholar [9] S. Nishibata, M. Ohnawa and M. Suzuki, Asymptotic stability of boundary layers to the Euler-Poisson equations arising in plasma physics,, SIAM J. Math. Anal., 44 (2012), 761.  doi: 10.1137/110835657.  Google Scholar [10] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar [11] K.-U. Riemann, The Bohm criterion and sheath formation. Initial value problems,, J. Phys. D: Appl. Phys., 24 (1991), 493.   Google Scholar [12] K.-U. Riemann, The Bohm criterion and boundary conditions for a multicomponent system,, IEEE Trans. Plasma Sci., 23 (1995), 709.  doi: 10.1109/27.467993.  Google Scholar [13] M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics,, Kinet. Relat. Models, 4 (2011), 569.  doi: 10.3934/krm.2011.4.569.  Google Scholar

show all references

##### References:
 [1] A. Ambroso, Stability for solutions of a stationary Euler-Poisson problem,, Math. Models Methods Appl. Sci., 16 (2006), 1817.  doi: 10.1142/S0218202506001728.  Google Scholar [2] A. Ambroso, F. Méhats and P.-A. Raviart, On singular perturbation problems for the nonlinear Poisson equation,, Asympt. Anal., 25 (2001), 39.   Google Scholar [3] D. Bohm, Minimum ionic kinetic energy for a stable sheath,, in The characteristics of electrical discharges in magnetic fields (eds. A. Guthrie and R.K.Wakerling), (1949), 77.   Google Scholar [4] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion,, $2^{nd}$ edition, (1984).   Google Scholar [5] S.-H. Ha and M. Slemrod, Global existence of plasma ion-sheaths and their dynamics,, Comm. Math. Phys., 238 (2003), 149.  doi: 10.1007/s00220-003-0871-z.  Google Scholar [6] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar [7] I. Langmuir, The interaction of electron and positive ion space charges in cathode sheaths,, Phys. Rev., 33 (1929), 954.  doi: 10.1103/PhysRev.33.954.  Google Scholar [8] M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing,, $2^{nd}$ edition, (2005).  doi: 10.1002/0471724254.  Google Scholar [9] S. Nishibata, M. Ohnawa and M. Suzuki, Asymptotic stability of boundary layers to the Euler-Poisson equations arising in plasma physics,, SIAM J. Math. Anal., 44 (2012), 761.  doi: 10.1137/110835657.  Google Scholar [10] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar [11] K.-U. Riemann, The Bohm criterion and sheath formation. Initial value problems,, J. Phys. D: Appl. Phys., 24 (1991), 493.   Google Scholar [12] K.-U. Riemann, The Bohm criterion and boundary conditions for a multicomponent system,, IEEE Trans. Plasma Sci., 23 (1995), 709.  doi: 10.1109/27.467993.  Google Scholar [13] M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics,, Kinet. Relat. Models, 4 (2011), 569.  doi: 10.3934/krm.2011.4.569.  Google Scholar
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