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 and 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.

[1]

Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022020

[2]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086

[3]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[4]

Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic and Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008

[5]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[6]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[7]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345

[8]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[9]

Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049

[10]

Léo Bois, Emmanuel Franck, Laurent Navoret, Vincent Vigon. A neural network closure for the Euler-Poisson system based on kinetic simulations. Kinetic and Related Models, 2022, 15 (1) : 49-89. doi: 10.3934/krm.2021044

[11]

Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201

[12]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[13]

Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448

[14]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085

[15]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011

[16]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[17]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292

[18]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046

[19]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775

[20]

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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (171)
  • HTML views (47)
  • Cited by (1)

Other articles
by authors

[Back to Top]