Quasineutral limit of the pressureless Euler-Poisson equation for ions
Authors:
Xueke Pu and Boling Guo
Journal:
Quart. Appl. Math. 74 (2016), 245-273
MSC (2010):
Primary 35Q35, 35B25; Secondary 35C20
DOI:
https://doi.org/10.1090/qam/1424
Published electronically:
March 29, 2016
MathSciNet review:
3505603
Full-text PDF Free Access
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Abstract: In this paper, we consider the quasineutral limit of the Euler-Poisson equation for cold ions when the Debye length tends to zero. In the cold ion case the Euler-Poisson equation is pressureless and hence fails to be Friedrich symmetrizable, excluding the application of the PsDO energy estimates method of Grenier to obtain uniform estimates independent of $\varepsilon$. To overcome this difficulty, we use $\varepsilon$-weighted norms which combine energy estimates in different levels with weights depending on $\varepsilon$. Finally, that the quasineutral regimes are the compressible Euler equations is proven for well prepared initial data. As a natural extension, we also obtain the zero temperature limit of the Euler-Poisson equation.
References
- Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 737–754. MR 1748352, DOI 10.1080/03605300008821529
- Stéphane Cordier and Emmanuel Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations 25 (2000), no. 5-6, 1099–1113. MR 1759803, DOI 10.1080/03605300008821542
- P. Degond, H. Liu, D. Savelief, and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit, J. Sci. Comput. 51 (2012), no. 1, 59–86. MR 2891946, DOI 10.1007/s10915-011-9495-1
- David Gérard-Varet, Daniel Han-Kwan, and Frédéric Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J. 62 (2013), no. 2, 359–402. MR 3158514, DOI 10.1512/iumj.2013.62.4900
- E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9) 76 (1997), no. 6, 477–498. MR 1465607, DOI 10.1016/S0021-7824(97)89959-X
- E. Grenier, Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math. 50 (1997), no. 9, 821–865. MR 1459589, DOI 10.1002/(SICI)1097-0312(199709)50:9<821::AID-CPA2>3.3.CO;2-3
- Y. Guo, A.D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D. arXiv: 1303.1060v1.
- Yan Guo and Benoit Pausader, Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys. 303 (2011), no. 1, 89–125. MR 2775116, DOI 10.1007/s00220-011-1193-1
- Yan Guo and Xueke Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 673–710. MR 3149069, DOI 10.1007/s00205-013-0683-z
- Daniel Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations 36 (2011), no. 8, 1385–1425. MR 2825596, DOI 10.1080/03605302.2011.555804
- Daniel Han-Kwan, From Vlasov-Poisson to Korteweg–de Vries and Zakharov-Kuznetsov, Comm. Math. Phys. 324 (2013), no. 3, 961–993. MR 3123542, DOI 10.1007/s00220-013-1825-8
- Song Jiang, QiangChang Ju, HaiLiang Li, and Yong Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math. 53 (2010), no. 12, 3099–3114. MR 2746309, DOI 10.1007/s11425-010-4114-4
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
- N. Krall and A. Trivelpiece, Principles of plasma physics, San Francisco Press, 1986.
- David Lannes, Felipe Linares, and Jean-Claude 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, Progr. Nonlinear Differential Equations Appl., vol. 84, Birkhäuser/Springer, New York, 2013, pp. 181–213. MR 3185896, DOI 10.1007/978-1-4614-6348-1_{1}0
- Grégoire Loeper, Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1141–1167. MR 2180297, DOI 10.1080/03605300500257545
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
- Yue-Jun Peng and Shu Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations 33 (2008), no. 1-3, 349–376. MR 2398233, DOI 10.1080/03605300701318989
- Xueke Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal. 45 (2013), no. 2, 834–878. MR 3045650, DOI 10.1137/120875648
- Xueke Pu, Quasineutral limit of the pressureless Euler-Poisson equation, Appl. Math. Lett. 30 (2014), 33–37. MR 3162389, DOI 10.1016/j.aml.2013.12.008
- M. Slemrod and N. Sternberg, Quasi-neutral limit for Euler-Poisson system, J. Nonlinear Sci. 11 (2001), no. 3, 193–209. MR 1852940, DOI 10.1007/s00332-001-0004-9
- Shlomo Engelberg, Hailiang Liu, and Eitan Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 109–157. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855666, DOI 10.1512/iumj.2001.50.2177
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Shu Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations 29 (2004), no. 3-4, 419–456. MR 2041602, DOI 10.1081/PDE-120030403
- Shu Wang and Song Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 31 (2006), no. 4-6, 571–591. MR 2233033, DOI 10.1080/03605300500361487
References
- Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 737–754. MR 1748352 (2001c:76124), DOI 10.1080/03605300008821529
- Stéphane Cordier and Emmanuel Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations 25 (2000), no. 5-6, 1099–1113. MR 1759803 (2001c:82078), DOI 10.1080/03605300008821542
- P. Degond, H. Liu, D. Savelief, and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit, J. Sci. Comput. 51 (2012), no. 1, 59–86. MR 2891946, DOI 10.1007/s10915-011-9495-1
- David Gérard-Varet, Daniel Han-Kwan, and Frédéric Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J. 62 (2013), no. 2, 359–402. MR 3158514, DOI 10.1512/iumj.2013.62.4900
- E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9) 76 (1997), no. 6, 477–498. MR 1465607 (98h:35189), DOI 10.1016/S0021-7824(97)89959-X
- E. Grenier, Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math. 50 (1997), no. 9, 821–865. MR 1459589 (98g:35014), DOI 10.1002/(SICI)1097-0312(199709)50:9$\langle$821::AID-CPA2$\rangle$3.3.CO;2-3
- Y. Guo, A.D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D. arXiv: 1303.1060v1.
- Yan Guo and Benoit Pausader, Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys. 303 (2011), no. 1, 89–125. MR 2775116 (2012e:82075), DOI 10.1007/s00220-011-1193-1
- Yan Guo and Xueke Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 673–710. MR 3149069, DOI 10.1007/s00205-013-0683-z
- Daniel Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations 36 (2011), no. 8, 1385–1425. MR 2825596 (2012g:35349), DOI 10.1080/03605302.2011.555804
- Daniel Han-Kwan, From Vlasov-Poisson to Korteweg–de Vries and Zakharov-Kuznetsov, Comm. Math. Phys. 324 (2013), no. 3, 961–993. MR 3123542, DOI 10.1007/s00220-013-1825-8
- Song Jiang, QiangChang Ju, HaiLiang Li, and Yong Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math. 53 (2010), no. 12, 3099–3114. MR 2746309 (2011h:35223), DOI 10.1007/s11425-010-4114-4
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744 (90f:35162), DOI 10.1002/cpa.3160410704
- N. Krall and A. Trivelpiece, Principles of plasma physics, San Francisco Press, 1986.
- David Lannes, Felipe Linares, and Jean-Claude 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, Progr. Nonlinear Differential Equations Appl., vol. 84, Birkhäuser/Springer, New York, 2013, pp. 181–213. MR 3185896, DOI 10.1007/978-1-4614-6348-1_10
- Grégoire Loeper, Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1141–1167. MR 2180297 (2007b:35267), DOI 10.1080/03605300500257545
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308 (85e:35077), DOI 10.1007/978-1-4612-1116-7
- Yue-Jun Peng and Shu Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations 33 (2008), no. 1-3, 349–376. MR 2398233 (2009b:35339), DOI 10.1080/03605300701318989
- Xueke Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal. 45 (2013), no. 2, 834–878. MR 3045650, DOI 10.1137/120875648
- Xueke Pu, Quasineutral limit of the pressureless Euler-Poisson equation, Appl. Math. Lett. 30 (2014), 33–37. MR 3162389, DOI 10.1016/j.aml.2013.12.008
- M. Slemrod and N. Sternberg, Quasi-neutral limit for Euler-Poisson system, J. Nonlinear Sci. 11 (2001), no. 3, 193–209. MR 1852940 (2002f:35189), DOI 10.1007/s00332-001-0004-9
- Shlomo Engelberg, Hailiang Liu, and Eitan Tadmor, Critical thresholds in Euler-Poisson equations, Special Issue dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000), Indiana Univ. Math. J. 50 (2001), 109–157. MR 1855666 (2002i:35149), DOI 10.1512/iumj.2001.50.2177
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 \#7280)
- Shu Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations 29 (2004), no. 3-4, 419–456. MR 2041602 (2005i:35225), DOI 10.1081/PDE-120030403
- Shu Wang and Song Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 31 (2006), no. 4-6, 571–591. MR 2233033 (2008f:35319), DOI 10.1080/03605300500361487
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Additional Information
Xueke Pu
Affiliation:
Department of Mathematics, Chongqing University, Chongqing 401331, People’s Republic of China
Email:
xuekepu@cqu.edu.cn
Boling Guo
Affiliation:
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, People’s Republic of China, 100088
MR Author ID:
189874
Email:
gbl@iapcm.ac.cn
Keywords:
Euler-Poisson equation,
quasineutral limit,
compressible Euler equation
Received by editor(s):
August 14, 2014
Published electronically:
March 29, 2016
Additional Notes:
The first author was supported in part by NSFC (11471057) and Natural Science Foundation Project of CQ CSTC (cstc2014jcyjA50020).
Article copyright:
© Copyright 2016
Brown University