November 2017, 16(6): 2177-2199. doi: 10.3934/cpaa.2017108

Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System

College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

* Corresponding author

Received  January 2017 Revised  March 2017 Published  July 2017

Fund Project: Shu Wang is supported by NSF grant 11371042, Chundi Liu is supported by NSF grant 11471028,11601021

We study the boundary layer problem and the quasineutral limit of the compressible Euler-Poisson system arising from plasma physics in a domain with boundary. The quasineutral regime is the incompressible Euler equations. Compared to the quasineutral limit of compressible Euler-Poisson equations in whole space or periodic domain, the key difficulty here is to deal with the singularity caused by the boundary layer. The proof of the result is based on a λ-weighted energy method and the matched asymptotic expansion method.

Citation: Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108
References:
[1]

G. AlìD. Bini and S. Rionero, Global existence and relaxation limit for smooth solution to the Euler-Poisson model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[2]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Diff. Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[3]

S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113. doi: 10.1080/03605300008821542.

[4]

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

[5]

D. Gérard-VaretD. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries Ⅱ., J. Éc. polytech. Math., 1 (2014), 343-386.

[6]

Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in $R^{3+1}$, Comm. Math. Phys., 195 (1998), 249-265. doi: 10.1007/s002200050388.

[7]

Y. Guo and B. Pausader, Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys., 303 (2011), 89-125. doi: 10.1007/s00220-011-1193-1.

[8]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rat. Mech. and Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.

[9]

L. HsiaoP. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Diff. Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[10]

S. JiangQ. C. JuH. L. Li and Y. Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53 (2010), 3099-3114. doi: 10.1007/s11425-010-4114-4.

[11]

Q. C. JuH. L. LiY. Li and S. Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9 (2010), 1577-1590. doi: 10.3934/cpaa.2010.9.1577.

[12]

Q. C. Ju and Y. Li, Quasineutral limit of the two-fluid Euler-Poisson system in a boundary domain of $R^3$, submitted.

[13]

T. Kato, Nonstationary flow of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305.

[14]

Y. C. LiY. J. Peng and Y. G. Wang, From two-fluid Euler-Poisson equations to one-fluid Euler equations, Asymptot. Anal., 85 (2013), 125-148.

[15]

C. D. Liu and B. Y. Wang, Quasineutral limit for a model of three dimensional Euler-Poisson system with boundary Anal. Appl. accepted. doi: 10.1007/s11401-013-0782-z.

[16]

G. Loeper, Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampére systems, Comm. Partial Differential Equations, 30 (2005), 1141-1167. doi: 10.1080/03605300500257545.

[17]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables Springer-Verlag Wien New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[18]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations Springer-Verlag Wien New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[19]

Y. J. Peng, Asymptotic limits of one-dimensional hydrodynamic models for plasmas and semiconductors, Chinese Ann. Math. Ser. B, 23 (2002), 25-36. doi: 10.1142/S0252959902000043.

[20]

Y. J. Peng, Some asymptotic analysis in steady-state Euler-Poisson equations for potential flow, Asymptot. Anal., 36 (2003), 75-92.

[21]

Y. J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433. doi: 10.3934/dcds.2009.23.415.

[22]

Y. J. Peng and Y. G. Wang, Boundary layers and quasi-neutral limit in seady state Euler-Poisson equations for potential folws, Nonlinearity, 17 (2004), 835-849. doi: 10.1088/0951-7715/17/3/006.

[23]

Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible Euler equations, Asymptot. Analysis, 41 (2005), 141-160.

[24]

Y. J. PengY. G. Wang and W. A. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proc. Roy. Soc. Edinburgh Sect., A 136 (2006), 1013-1026. doi: 10.1017/S0308210500004856.

[25]

X. K. Pu, Quasineutral limit of the Euler-Poisson system under strong magnetic fields, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 2095-2111. doi: 10.3934/dcdss.2016086.

[26]

M. Slemrod and N. Sternberg, Quasi-neutral limit for Euler-Poisson system, J. Nonlinear Sci., 11 (2001), 193-209. doi: 10.1007/s00332-001-0004-9.

[27]

M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinet. Relat. Models, 4 (2011), 569-588. doi: 10.3934/krm.2011.4.569.

[28]

I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1101-1118. doi: 10.1017/S0308210505001216.

[29]

S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.

show all references

References:
[1]

G. AlìD. Bini and S. Rionero, Global existence and relaxation limit for smooth solution to the Euler-Poisson model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[2]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Diff. Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[3]

S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113. doi: 10.1080/03605300008821542.

[4]

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

[5]

D. Gérard-VaretD. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries Ⅱ., J. Éc. polytech. Math., 1 (2014), 343-386.

[6]

Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in $R^{3+1}$, Comm. Math. Phys., 195 (1998), 249-265. doi: 10.1007/s002200050388.

[7]

Y. Guo and B. Pausader, Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys., 303 (2011), 89-125. doi: 10.1007/s00220-011-1193-1.

[8]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rat. Mech. and Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.

[9]

L. HsiaoP. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Diff. Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[10]

S. JiangQ. C. JuH. L. Li and Y. Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53 (2010), 3099-3114. doi: 10.1007/s11425-010-4114-4.

[11]

Q. C. JuH. L. LiY. Li and S. Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9 (2010), 1577-1590. doi: 10.3934/cpaa.2010.9.1577.

[12]

Q. C. Ju and Y. Li, Quasineutral limit of the two-fluid Euler-Poisson system in a boundary domain of $R^3$, submitted.

[13]

T. Kato, Nonstationary flow of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305.

[14]

Y. C. LiY. J. Peng and Y. G. Wang, From two-fluid Euler-Poisson equations to one-fluid Euler equations, Asymptot. Anal., 85 (2013), 125-148.

[15]

C. D. Liu and B. Y. Wang, Quasineutral limit for a model of three dimensional Euler-Poisson system with boundary Anal. Appl. accepted. doi: 10.1007/s11401-013-0782-z.

[16]

G. Loeper, Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampére systems, Comm. Partial Differential Equations, 30 (2005), 1141-1167. doi: 10.1080/03605300500257545.

[17]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables Springer-Verlag Wien New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[18]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations Springer-Verlag Wien New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[19]

Y. J. Peng, Asymptotic limits of one-dimensional hydrodynamic models for plasmas and semiconductors, Chinese Ann. Math. Ser. B, 23 (2002), 25-36. doi: 10.1142/S0252959902000043.

[20]

Y. J. Peng, Some asymptotic analysis in steady-state Euler-Poisson equations for potential flow, Asymptot. Anal., 36 (2003), 75-92.

[21]

Y. J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433. doi: 10.3934/dcds.2009.23.415.

[22]

Y. J. Peng and Y. G. Wang, Boundary layers and quasi-neutral limit in seady state Euler-Poisson equations for potential folws, Nonlinearity, 17 (2004), 835-849. doi: 10.1088/0951-7715/17/3/006.

[23]

Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible Euler equations, Asymptot. Analysis, 41 (2005), 141-160.

[24]

Y. J. PengY. G. Wang and W. A. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proc. Roy. Soc. Edinburgh Sect., A 136 (2006), 1013-1026. doi: 10.1017/S0308210500004856.

[25]

X. K. Pu, Quasineutral limit of the Euler-Poisson system under strong magnetic fields, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 2095-2111. doi: 10.3934/dcdss.2016086.

[26]

M. Slemrod and N. Sternberg, Quasi-neutral limit for Euler-Poisson system, J. Nonlinear Sci., 11 (2001), 193-209. doi: 10.1007/s00332-001-0004-9.

[27]

M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinet. Relat. Models, 4 (2011), 569-588. doi: 10.3934/krm.2011.4.569.

[28]

I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1101-1118. doi: 10.1017/S0308210505001216.

[29]

S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.

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