# American Institute of Mathematical Sciences

June  2011, 4(2): 569-588. doi: 10.3934/krm.2011.4.569

## Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics

 1 Research Institute of Nonlinear Partial Differential Equations, Organization for University Research Initiatives, Waseda University, Tokyo 169-8555, Japan

Received  February 2010 Revised  November 2010 Published  April 2011

The main concern of the present paper is to analyze a sheath formed around a surface of a material with which plasma contacts. Here, for a formation of the sheath, the Bohm criterion requires the velocity of positive ions should be faster than a certain physical constant. The behavior of positive ions in plasma is governed by the Euler-Poisson equations. Mathematically, the sheath is regarded as a special stationary solution. We first show that the Bohm criterion gives a sufficient condition for an existence of the stationary solution by using the phase plane analysis. Then it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in the weighted Sobolev space. Moreover we obtain the convergence rate of the time global solution towards the stationary solution subject to the decay rate of the initial perturbation. These theorems are proved by a weighted energy method.
Citation: Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569
##### 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] F. F. Chen, "Introduction to Plasma Physics and Controlled Fusion,'', 2$^nd$ edition, (1984).   Google Scholar [4] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis of an isothermal Euler-Poisson model,, Ann. Fac. Sci. Toulouse Math., 5 (1996), 599.   Google Scholar [5] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis and jump relations for Euler-Poisson model in the quasineutral limit,, Asymptotic Anal., 11 (1995), 209.   Google Scholar [6] P. Degond and P. Markowich, On a one-dimensional steady-state hydrodynamic model,, Appl. Math. Lett., 3 (1990), 25.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar [7] S.-H. Ha and M. Slemrod, Global existence of plasma ion-sheaths and their dynamics,, Comm. Math. Phys., 238 (2003), 149.   Google Scholar [8] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions,, Arch. Ration. Mech. Anal., 179 (2006), 1.  doi: 10.1007/s00205-005-0369-2.  Google Scholar [9] T. Kato, Linear evolution equations of "hyperbolic'' type,, J. Math. Soc. Japan., 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar [10] 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 [11] S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Ration. Mech. Anal., 170 (2003), 297.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [12] M. A. Lieberman and A. J. Lichtenberg, "Principles of Plasma Discharges and Materials Processing,'', 2$^{nd}$ edition, (2005).  doi: 10.1002/0471724254.  Google Scholar [13] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differ. Equ., 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar [14] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639.   Google Scholar [15] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors,, Arch. Ration. Mech. Anal., 192 (2009), 187.  doi: 10.1007/s00205-008-0129-1.  Google Scholar [16] S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors,, J. Differ. Equ., 249 (2010), 1385.  doi: 10.1016/j.jde.2010.06.008.  Google Scholar [17] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar [18] M. Slemrod, The radio-frequency driven plasma sheath: asymptotics and analysis,, SIAM J. Appl. Math., 63 (2003), 1737.  doi: 10.1137/S0036139902411831.  Google Scholar [19] N. Sternberg and V. A. Godyak, Solving the mathematical model of the electrode sheath in symmetrically driven rf discharges,, J. Comput. Phys., 111 (1994), 347.  doi: 10.1006/jcph.1994.1068.  Google Scholar [20] Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows,, Nonlinearity, 17 (2004), 835.  doi: 10.1088/0951-7715/17/3/006.  Google Scholar [21] K.-U. Riemann, The Bohm criterion and sheath formation. Initial value problems,, J. Phys. D: Appl. Phys., 24 (1991), 493.  doi: 10.1088/0022-3727/24/4/001.  Google Scholar [22] M.-H. Vignal, A boundary layer problem for an asymptotic preserving scheme in the quasi-neutral limit for the Euler-Poisson system,, SIAM J. Appl. Math., 70 (2010), 1761.  doi: 10.1137/070703272.  Google Scholar

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##### 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] F. F. Chen, "Introduction to Plasma Physics and Controlled Fusion,'', 2$^nd$ edition, (1984).   Google Scholar [4] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis of an isothermal Euler-Poisson model,, Ann. Fac. Sci. Toulouse Math., 5 (1996), 599.   Google Scholar [5] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis and jump relations for Euler-Poisson model in the quasineutral limit,, Asymptotic Anal., 11 (1995), 209.   Google Scholar [6] P. Degond and P. Markowich, On a one-dimensional steady-state hydrodynamic model,, Appl. Math. Lett., 3 (1990), 25.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar [7] S.-H. Ha and M. Slemrod, Global existence of plasma ion-sheaths and their dynamics,, Comm. Math. Phys., 238 (2003), 149.   Google Scholar [8] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions,, Arch. Ration. Mech. Anal., 179 (2006), 1.  doi: 10.1007/s00205-005-0369-2.  Google Scholar [9] T. Kato, Linear evolution equations of "hyperbolic'' type,, J. Math. Soc. Japan., 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar [10] 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 [11] S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Ration. Mech. Anal., 170 (2003), 297.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [12] M. A. Lieberman and A. J. Lichtenberg, "Principles of Plasma Discharges and Materials Processing,'', 2$^{nd}$ edition, (2005).  doi: 10.1002/0471724254.  Google Scholar [13] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differ. Equ., 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar [14] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639.   Google Scholar [15] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors,, Arch. Ration. Mech. Anal., 192 (2009), 187.  doi: 10.1007/s00205-008-0129-1.  Google Scholar [16] S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors,, J. Differ. Equ., 249 (2010), 1385.  doi: 10.1016/j.jde.2010.06.008.  Google Scholar [17] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar [18] M. Slemrod, The radio-frequency driven plasma sheath: asymptotics and analysis,, SIAM J. Appl. Math., 63 (2003), 1737.  doi: 10.1137/S0036139902411831.  Google Scholar [19] N. Sternberg and V. A. Godyak, Solving the mathematical model of the electrode sheath in symmetrically driven rf discharges,, J. Comput. Phys., 111 (1994), 347.  doi: 10.1006/jcph.1994.1068.  Google Scholar [20] Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows,, Nonlinearity, 17 (2004), 835.  doi: 10.1088/0951-7715/17/3/006.  Google Scholar [21] K.-U. Riemann, The Bohm criterion and sheath formation. Initial value problems,, J. Phys. D: Appl. Phys., 24 (1991), 493.  doi: 10.1088/0022-3727/24/4/001.  Google Scholar [22] M.-H. Vignal, A boundary layer problem for an asymptotic preserving scheme in the quasi-neutral limit for the Euler-Poisson system,, SIAM J. Appl. Math., 70 (2010), 1761.  doi: 10.1137/070703272.  Google Scholar
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