# American Institute of Mathematical Sciences

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:
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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. Google Scholar [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. Google Scholar [23] Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible Euler equations, Asymptot. Analysis, 41 (2005), 141-160. Google Scholar [24] Y. J. Peng, Y. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] G. Alì, D. 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] D. Gérard-Varet, D. 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. Google Scholar [5] D. Gérard-Varet, D. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] L. Hsiao, P. 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. Google Scholar [10] S. Jiang, Q. C. Ju, H. 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. Google Scholar [11] Q. C. Ju, H. L. Li, Y. 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. Google Scholar [12] Q. C. Ju and Y. Li, Quasineutral limit of the two-fluid Euler-Poisson system in a boundary domain of $R^3$, submitted.Google Scholar [13] T. Kato, Nonstationary flow of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305. Google Scholar [14] Y. C. Li, Y. J. Peng and Y. G. Wang, From two-fluid Euler-Poisson equations to one-fluid Euler equations, Asymptot. Anal., 85 (2013), 125-148. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] Y. J. Peng, Some asymptotic analysis in steady-state Euler-Poisson equations for potential flow, Asymptot. Anal., 36 (2003), 75-92. Google Scholar [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. Google Scholar [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. Google Scholar [23] Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible Euler equations, Asymptot. Analysis, 41 (2005), 141-160. Google Scholar [24] Y. J. Peng, Y. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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