Derivation of the ion equation
Authors:
E. Grenier, Y. Guo, B. Pausader and M. Suzuki
Journal:
Quart. Appl. Math. 78 (2020), 305-332
MSC (2010):
Primary 35Q31; Secondary 35Q32
DOI:
https://doi.org/10.1090/qam/1558
Published electronically:
September 23, 2019
MathSciNet review:
4077465
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Additional Information
Abstract: We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion $m_e/M_i\ll 1$ is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on $\mathbb {R}^3$, the validity of such an expansion is established even with “ill-prepared” Cauchy data, by including an additional initial layer correction.
References
- Thomas Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 1–73. MR 2211706, DOI https://doi.org/10.1007/s00205-005-0393-2
- Jean-François Bony and Dietrich Häfner, Local energy decay for several evolution equations on asymptotically Euclidean manifolds, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 2, 311–335 (English, with English and French summaries). MR 2977621, DOI https://doi.org/10.24033/asens.2166
- Bin Cheng, Qiangchang Ju, and Steve Schochet, Three-scale singular limits of evolutionary PDEs, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 601–625. MR 3803773, DOI https://doi.org/10.1007/s00205-018-1233-5
- E. Grenier, Methodology à la Métivier-Schochet, Expository notes.
- Yan Guo, Alexandru D. Ionescu, and Benoit Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D, Ann. of Math. (2) 183 (2016), no. 2, 377–498. MR 3450481, DOI https://doi.org/10.4007/annals.2016.183.2.1
- Y. Guo, A. D. Ionescu, and B. Pausader, Global solutions of certain plasma fluid models in $3D$, J. Math. Phys. 55, 123102 (2014).
- 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 https://doi.org/10.1007/s00220-011-1193-1
- Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481–524. MR 615627, DOI https://doi.org/10.1002/cpa.3160340405
- Sergiu Klainerman and Andrew Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982), no. 5, 629–651. MR 668409, DOI https://doi.org/10.1002/cpa.3160350503
- Shinya Nishibata, Masashi Ohnawa, and Masahiro Suzuki, Asymptotic stability of boundary layers to the Euler-Poisson equations arising in plasma physics, SIAM J. Math. Anal. 44 (2012), no. 2, 761–790. MR 2914249, DOI https://doi.org/10.1137/110835657
- G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), no. 1, 61–90. MR 1834114, DOI https://doi.org/10.1007/PL00004241
- J. Metcalf, J. Sterbenz, and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., to appear.
- Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. London Ser. A 306 (1968), 291–296. MR 234136, DOI https://doi.org/10.1098/rspa.1968.0151
- Steve Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR 834481
- Masahiro Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinet. Relat. Models 4 (2011), no. 2, 569–588. MR 2786399, DOI https://doi.org/10.3934/krm.2011.4.569
- Daniel Tataru, Decay of linear waves on black hole space-times, Surveys in differential geometry 2015. One hundred years of general relativity, Surv. Differ. Geom., vol. 20, Int. Press, Boston, MA, 2015, pp. 157–182. MR 3467367, DOI https://doi.org/10.4310/SDG.2015.v20.n1.a7
- Seiji Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26 (1986), no. 2, 323–331. MR 849223, DOI https://doi.org/10.1215/kjm/1250520925
References
- Thomas Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 1–73. MR 2211706, DOI https://doi.org/10.1007/s00205-005-0393-2
- Jean-François Bony and Dietrich Häfner, Local energy decay for several evolution equations on asymptotically Euclidean manifolds, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 2, 311–335 (English, with English and French summaries). MR 2977621, DOI https://doi.org/10.24033/asens.2166
- Bin Cheng, Qiangchang Ju, and Steve Schochet, Three-scale singular limits of evolutionary PDEs, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 601–625. MR 3803773, DOI https://doi.org/10.1007/s00205-018-1233-5
- E. Grenier, Methodology à la Métivier-Schochet, Expository notes.
- Yan Guo, Alexandru D. Ionescu, and Benoit Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D, Ann. of Math. (2) 183 (2016), no. 2, 377–498. MR 3450481, DOI https://doi.org/10.4007/annals.2016.183.2.1
- Y. Guo, A. D. Ionescu, and B. Pausader, Global solutions of certain plasma fluid models in $3D$, J. Math. Phys. 55, 123102 (2014).
- 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 https://doi.org/10.1007/s00220-011-1193-1
- Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481–524. MR 615627, DOI https://doi.org/10.1002/cpa.3160340405
- Sergiu Klainerman and Andrew Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982), no. 5, 629–651. MR 668409, DOI https://doi.org/10.1002/cpa.3160350503
- Shinya Nishibata, Masashi Ohnawa, and Masahiro Suzuki, Asymptotic stability of boundary layers to the Euler-Poisson equations arising in plasma physics, SIAM J. Math. Anal. 44 (2012), no. 2, 761–790. MR 2914249, DOI https://doi.org/10.1137/110835657
- G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), no. 1, 61–90. MR 1834114, DOI https://doi.org/10.1007/PL00004241
- J. Metcalf, J. Sterbenz, and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., to appear.
- Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291–296. MR 0234136, DOI https://doi.org/10.1098/rspa.1968.0151
- Steve Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR 834481
- Masahiro Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinet. Relat. Models 4 (2011), no. 2, 569–588. MR 2786399, DOI https://doi.org/10.3934/krm.2011.4.569
- Daniel Tataru, Decay of linear waves on black hole space-times, Surveys in differential geometry 2015. One hundred years of general relativity, Surv. Differ. Geom., vol. 20, Int. Press, Boston, MA, 2015, pp. 157–182. MR 3467367, DOI https://doi.org/10.4310/SDG.2015.v20.n1.a7
- Seiji Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26 (1986), no. 2, 323–331. MR 849223, DOI https://doi.org/10.1215/kjm/1250520925
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Additional Information
E. Grenier
Affiliation:
École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France
Email:
egrenier@umpa.ens-lyon.fr
Y. Guo
Affiliation:
Brown University, Division of Applied Mathematics, 182 George Street, Providence, Rhode Island 02912
Email:
yan_guo@brown.edu
B. Pausader
Affiliation:
Brown University, Department of Mathematics, 151 Thayer Street, Providence Rhode Island, 02912
MR Author ID:
822827
Email:
benoit.pausader@math.brown.edu
M. Suzuki
Affiliation:
Nagoya Institute of Technology, Department of Computer Science, Gokiso-cho Showa-ku, Nagoya, 466-8555 Japan
MR Author ID:
844148
Email:
masahiro@nitech.ac.jp
Received by editor(s):
May 17, 2019
Received by editor(s) in revised form:
July 23, 2019
Published electronically:
September 23, 2019
Additional Notes:
The second author’s research was supported in part by NSF grant DMS1810868, BICMR, and a Chinese NSF grant through Beijing Capital Normal University.
The third author was supported by NSF grant DMS-1700282.
The fourth author was supported by JSPS KAKENHI Numbers 26800067 and 18K03364.
Dedicated:
Dedicated to Walter Strauss on the occasion of his 80th birthday
Article copyright:
© Copyright 2019
Brown University
