American Institute of Mathematical Sciences

Advanced
December  2016, 9(6): 2095-2111. doi: 10.3934/dcdss.2016086

Quasineutral limit of the Euler-Poisson system under strong magnetic fields

Xueke Pu 1,

1. 

Department of Mathematics, Chongqing University, Chongqing 401331

Received  July 2015 Revised  September 2016 Published  November 2016

The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.
Keywords: strong magnetic field., quasineutral limit, Euler-Poisson equation.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76X0.
Citation: 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
References:
[1]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529.

[2]

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.

[3]

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

[4]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.

[5]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Rational Mech. Anal., 211 (2014), 673-710. doi: 10.1007/s00205-013-0683-z.

[6]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.

[7]

Q. Ju, F. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224. doi: 10.1016/j.jde.2009.02.019.

[8]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in Phase Space Analysis with Applications to PDEs, in Progress in Nonlinear Differential Equations and Their Applications, 84 (2013), 181-213. doi: 10.1007/978-1-4614-6348-1_10.

[9]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York-Berlin, 1984. doi: 10.1007/978-1-4612-1116-7.

[10]

Y. Peng and S. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asympt. Anal., 41 (2005), 141-160.

[11]

Y. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970. doi: 10.1137/100786927.

[12]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878. doi: 10.1137/120875648.

[13]

X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273. doi: 10.1090/qam/1424.

[14]

S. Wang, Quasineutral limit of Euler-Poisson system with and withour viscosity, Commun. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.

[15]

S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591. doi: 10.1080/03605300500361487.

show all references

References:
[1]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529.

[2]

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.

[3]

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

[4]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.

[5]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Rational Mech. Anal., 211 (2014), 673-710. doi: 10.1007/s00205-013-0683-z.

[6]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.

[7]

Q. Ju, F. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224. doi: 10.1016/j.jde.2009.02.019.

[8]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in Phase Space Analysis with Applications to PDEs, in Progress in Nonlinear Differential Equations and Their Applications, 84 (2013), 181-213. doi: 10.1007/978-1-4614-6348-1_10.

[9]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York-Berlin, 1984. doi: 10.1007/978-1-4612-1116-7.

[10]

Y. Peng and S. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asympt. Anal., 41 (2005), 141-160.

[11]

Y. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970. doi: 10.1137/100786927.

[12]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878. doi: 10.1137/120875648.

[13]

X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273. doi: 10.1090/qam/1424.

[14]

S. Wang, Quasineutral limit of Euler-Poisson system with and withour viscosity, Commun. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403.

[15]

S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591. doi: 10.1080/03605300500361487.

[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]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic and Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[9]

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
[10]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure and Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579
[11]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

Lixi Wen, Wen Zhang. Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022109
[20]

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

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (170)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy