# American Institute of Mathematical Sciences

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

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

 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.
Citation: Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & 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. 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. 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. 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. doi: 10.1142/S0218202503002647. [5] Y. Guo and X. Pu, KdV limit of the Euler-Poisson system,, Arch. Rational Mech. Anal., 211 (2014), 673. 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. 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. 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, 84 (2013), 181. 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, (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. [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. doi: 10.1137/100786927. [12] X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions,, SIAM J. Math. Anal., 45 (2013), 834. 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. 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. 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. doi: 10.1080/03605300500361487.

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##### References:
 [1] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737. 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. 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. 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. doi: 10.1142/S0218202503002647. [5] Y. Guo and X. Pu, KdV limit of the Euler-Poisson system,, Arch. Rational Mech. Anal., 211 (2014), 673. 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. 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. 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, 84 (2013), 181. 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, (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. [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. doi: 10.1137/100786927. [12] X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions,, SIAM J. Math. Anal., 45 (2013), 834. 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. 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. 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. doi: 10.1080/03605300500361487.
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