Asymptotic limit of nonlinear Schrödinger-Poisson system  withgeneral initial data

Qiangchang Ju; Fucai Li; Hailiang Li

doi:10.3934/krm.2011.4.767

Article Contents

2011, Volume 4, Issue 3: 767-783. Doi: 10.3934/krm.2011.4.767

This issue Previous Article Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime Next Article Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution

Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data

1.
Institute of Applied Physics and Computational Mathematics, Box 8009-28, Beijing 100088
2.
Department of Mathematics, Nanjing University, Nanjing 210093
3.
Department of Mathematics and Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing 100037

Received: December 31, 2010

Revised: April 30, 2011

Published: August 2011

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Abstract

The asymptotic limit of the nonlinear Schrödinger-Poisson system with general WKB initial data is studied in this paper. It is proved that the current, defined by the smooth solution of the nonlinear Schrödinger-Poisson system, converges to the strong solution of the incompressible Euler equations plus a term of fast singular oscillating gradient vector fields when both the Planck constant $\hbar$ and the Debye length $\lambda$ tend to zero. The proof involves homogenization techniques, theories of symmetric quasilinear hyperbolic system and elliptic estimates, and the key point is to establish the uniformly bounded estimates with respect to both the Planck constant and the Debye length.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.

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