Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 6, p. 1071-1090
In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.
DOI : https://doi.org/10.1051/m2an:2003006
Classification:  76X05,  76N99,  81Q99,  82D10,  35Q40
Keywords: quasi-neutral plasmas, semi-classical limit, modulated energy
@article{M2AN_2002__36_6_1071_0,
     author = {Puel, Marjolaine},
     title = {Convergence of the Schr\"odinger-Poisson system to the Euler equations under the influence of a large magnetic field},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {6},
     year = {2002},
     pages = {1071-1090},
     doi = {10.1051/m2an:2003006},
     zbl = {1137.76836},
     mrnumber = {1958659},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_6_1071_0}
}
Puel, Marjolaine. Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 6, pp. 1071-1090. doi : 10.1051/m2an:2003006. http://www.numdam.org/item/M2AN_2002__36_6_1071_0/

[1] A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case. Math. Methods Appl. Sci. 14 (1991) 595-613. | Zbl 0742.35078

[2] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000) 737-754. | Zbl 0970.35110

[3] T. Cazenave, An introduction to nonlinear Schrödinger equations, in: Textos de méthodos Mathemàticas 26. Universidad Federal do Rio de Janeiro (1993).

[4] C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique quantique. Hermann (1973).

[5] P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-379. | Zbl 0881.35099

[6] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthiers-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[7] P.-L. Lions, Mathematical topics in fluid mechanics, Vol. 1. Incompressible models. Oxford Lecture in Mathematics and its Applications. Oxford University Press, New York (1996). | MR 1422251 | Zbl 0866.76002

[8] P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618. | Zbl 0801.35117

[9] P.A. Markowich and N.J. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3D. Math. Models Methods Appl. Sci. 3 (1993) 109-124. | Zbl 0772.35061

[10] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations. Preprint LAN, Université Paris VI (2001). | MR 1944031 | Zbl 1040.35076

[11] M. Puel, Études variationnelle et asymptotique de problèmes en mécanique des fluides et des plasmas. Ph.D. thesis, Université Paris VI (2001).