logo École polytechnique
logo Journal de l'École Polytechnique
Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II
David Gérard-Varet1; Daniel Han-Kwan2; Frédéric Rousset3
1 Institut de Mathématiques de Jussieu (UMR 7586), Université Paris-Diderot Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
2 CNRS & Centre de Mathématiques Laurent Schwartz (UMR 7640), École polytechnique 91128 Palaiseau Cedex, France
3 Laboratoire de Mathématiques d’Orsay (UMR 8628), Université Paris-Sud et Institut Universitaire de France Bâtiment 425, 91405 Orsay Cedex, France
Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 343-386.

In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work [5], devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different.

Dans cet article, nous étudions la limite quasineutre du système d’Euler-Poisson pour les ions dans un domaine à bord. Il s’agit de la suite de notre travail précédent [5], qui était consacré aux cas de conditions limites de type non-pénétration ou sortantes subsoniques. Nous nous focalisons ici sur le cas des vitesses sortantes supersoniques. La structure des couches limites ainsi que le mécanisme de stabilisation sont différents.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.13
Classification: 76N20, 76X05
Keywords: Isothermal Euler-Poisson equations, quasineutral limit, boundary layers, supersonic boundary conditions
Mot clés : Équations d’Euler-Poisson isothermes, limite quasineutre, couches limites, conditions aux limites supersoniques

David Gérard-Varet 1; Daniel Han-Kwan 2; Frédéric Rousset 3

1 Institut de Mathématiques de Jussieu (UMR 7586), Université Paris-Diderot Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
2 CNRS & Centre de Mathématiques Laurent Schwartz (UMR 7640), École polytechnique 91128 Palaiseau Cedex, France
3 Laboratoire de Mathématiques d’Orsay (UMR 8628), Université Paris-Sud et Institut Universitaire de France Bâtiment 425, 91405 Orsay Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JEP_2014__1__343_0,
     author = {David G\'erard-Varet and Daniel Han-Kwan and Fr\'ed\'eric Rousset},
     title = {Quasineutral limit of the {Euler-Poisson} system for ions in a domain with boundaries {II}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {343--386},
     publisher = {\'Ecole polytechnique},
     volume = {1},
     year = {2014},
     doi = {10.5802/jep.13},
     mrnumber = {3322792},
     zbl = {06389894},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.13/}
}
TY  - JOUR
AU  - David Gérard-Varet
AU  - Daniel Han-Kwan
AU  - Frédéric Rousset
TI  - Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2014
SP  - 343
EP  - 386
VL  - 1
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.13/
DO  - 10.5802/jep.13
LA  - en
ID  - JEP_2014__1__343_0
ER  - 
%0 Journal Article
%A David Gérard-Varet
%A Daniel Han-Kwan
%A Frédéric Rousset
%T Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II
%J Journal de l’École polytechnique — Mathématiques
%D 2014
%P 343-386
%V 1
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.13/
%R 10.5802/jep.13
%G en
%F JEP_2014__1__343_0
David Gérard-Varet; Daniel Han-Kwan; Frédéric Rousset. Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 343-386. doi : 10.5802/jep.13. https://jep.centre-mersenne.org/articles/10.5802/jep.13/

[1] A. Ambroso - “Stability for solutions of a stationary Euler-Poisson problem”, Math. Models Methods Appl. Sci. 16 (2006) no. 11, p. 1817-1837 | DOI | MR | Zbl

[2] A. Ambroso, F. Méhats & P. A. Raviart - “On singular perturbation problems for the nonlinear Poisson equation”, Asymptot. Anal. 25 (2001) no. 1, p. 39-91 | MR | Zbl

[3] S. Benzoni-Gavage & D. Serre - Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007, First-order systems and applications | Zbl

[4] S. Cordier & E. Grenier - “Quasineutral limit of an Euler-Poisson system arising from plasma physics”, Comm. Partial Differential Equations 25 (2000) no. 5-6, p. 1099-1113 | DOI | MR | Zbl

[5] D. Gérard-Varet, D. Han-Kwan & F. Rousset - “Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries”, Indiana Univ. Math. J. 62 (2013), p. 359-402 | DOI | MR | Zbl

[6] J. Goodman - “Nonlinear asymptotic stability of viscous shock profiles for conservation laws”, Arch. Rational Mech. Anal. 95 (1986) no. 4, p. 325-344 | DOI | MR | Zbl

[7] M. Lieberman & A. Lichtenberg - Principles of plasma discharges and materials processing, Cambridge University Press, 1994

[8] S. Nishibata, M. Ohnawa & M. Suzuki - “Asymptotic stability of boundary layers to the Euler-Poisson equations arising in plasma physics”, SIAM J. Math. Anal. 44 (2012) no. 2, p. 761-790 | DOI | MR | Zbl

[9] K.-U. Riemann - “The Bohm criterion and sheath formation”, J. Phys. D: Applied Physics 24 (1991) no. 4, 493 pages

[10] M. Slemrod & N. Sternberg - “Quasi-neutral limit for Euler-Poisson system”, J. Nonlinear Sci. 11 (2001) no. 3, p. 193-209 | DOI | MR | Zbl

[11] M. Suzuki - “Asymptotic stability of stationnary solutions to the Euler-Poisson equations arising in plasma physics”, Kinet. and Relat. Mod. 4 (2011) no. 2, p. 569-588 | DOI

Cited by Sources: