American Institute of Mathematical Sciences

Advanced
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

Masahiro Suzuki 1,

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.
Keywords: large-time behavior, multicomponent system, convergence rate., Sheath, generalized Bohm criterion, stationary solutions.
Mathematics Subject Classification: Primary: 35L04, 35J65, 35B40, 82D1.
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

[1]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[2]

Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583
[3]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
[4]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[5]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[6]

Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821
[7]

Ken Shirakawa, Hiroshi Watanabe. Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint. Conference Publications, 2015, 2015 (special) : 1009-1018. doi: 10.3934/proc.2015.1009
[8]

Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure & Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73
[9]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096
[10]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[11]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[12]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
[13]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031
[14]

M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827
[15]

Eduard Feireisl, Hana Petzeltová, Konstantina Trivisa. Multicomponent reactive flows: Global-in-time existence for large data. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1017-1047. doi: 10.3934/cpaa.2008.7.1017
[16]

Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077
[17]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[18]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[19]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[20]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences