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Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field. (English) Zbl 1137.76836
Summary: 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.

MSC:
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76N99 Compressible fluids and gas dynamics, general
81Q99 General mathematical topics and methods in quantum theory
82D10 Statistical mechanical studies of plasmas
35Q40 PDEs in connection with quantum mechanics
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References:
[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 · Zbl 0742.35078 · doi:10.1002/mma.1670140902
[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 · Zbl 0970.35110 · doi:10.1080/03605300008821529
[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 · Zbl 0881.35099 · doi:10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C
[6] J.-L. Lions , Quelques méthodes de résolution des problèmes aux limites non linéaires . Dunod, Gauthiers-Villars, Paris ( 1969 ). MR 259693 | Zbl 0189.40603 · 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 · 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 · Zbl 0801.35117 · doi:10.4171/RMI/143 · eudml:39445
[9] P.A. Markowich and N.J. Mauser , The classical limit of a self-consistent quantum-Vlasov equation in \(3\)D . Math. Models Methods Appl. Sci. 3 ( 1993 ) 109 - 124 . Zbl 0772.35061 · Zbl 0772.35061 · doi:10.1142/S0218202593000072
[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 · Zbl 1040.35076 · doi:10.1081/PDE-120016159
[11] M. Puel , Études variationnelle et asymptotique de problèmes en mécanique des fluides et des plasmas . Ph.D. thesis, Université Paris VI ( 2001 ).
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