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Convergence of the Schrödinger-Poisson system to the incompressible Euler equations. (English) Zbl 1040.35076
The asymptotic behavior of the solutions to the Schrödinger-Poisson system is studied. One proves that if the quasi-neutral assumption is satisfied and if the initial current defined by the superposition of different states converges (when the Planck constant and the permittivity converge independently to 0) then the current converges to a dissipative solution of the Euler equations. Examples are given and an estimate of the error is obtained in the case that smooth solutions to the Euler equations exist. The main result of the paper and the method of proof are similar to those in [Y. Brenier, Commun. Partial Differ. Equ. 25, 737–754 (2000; Zbl 0970.35110)].

MSC:
35Q35 PDEs in connection with fluid mechanics
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