Electron. J. Diff. Eqns., Vol. 2003(2003), No. 93, pp. 1-17.

Semiclassical limit and well-posedness of nonlinear Schrodinger-Poisson systems

Hailiang Li & Chi-Kun Lin

Abstract:
This paper concerns the well-posedness and semiclassical limit of nonlinear Schrodinger-Poisson systems. We show the local well-posedness and the existence of semiclassical limit of the two models for initial data with Sobolev regularity, before shocks appear in the limit system. We establish the existence of a global solution and show the time-asymptotic behavior of a classical solutions of Schrodinger-Poisson system for a fixed re-scaled Planck constant.

Submitted April 1, 2002. Published September 8, 2003.
Math Subject Classifications: 35A05, 35Q55.
Key Words: Schrodinger-Poisson system, quantum hydrodynamics, Euler-Poisson system, semiclassical limit, WKB expansion, quasilinear symmetric hyperbolic system.

An addendum was attached on August 17, 2006. The authors made the following two corrections:
On the sixth line of Theorem 2.1, the expression $A^\epsilon\in L^\infty([0,T];H^s(\mathbb{R}^N))$ should be replaced by $|A^\epsilon|-\sqrt{\mathcal{C}}\in L^\infty([0,T];H^s(\mathbb{R}^N))$.
On the fourteenth line of Theorem 2.2, the expression $A^\epsilon$ should be replaced by $|A^\epsilon|-\sqrt{\mathcal{C}}\in L^\infty([0,T];H^s(\mathbb{R}^N))$.
See last page of this manuscript for details.

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Hailiang Li
Institute of Mathematics, University of Vienna
A-1090 Vienna, Austria
and Institute of Mathematics, Academia Sinica
Beijing 100080, China
email: hailiang.li@univie.ac.at
Chi-Kun Lin
Department of Mathematics
National Cheng Kung University
Tainan 701, Taiwan
email: cklin@math.cku.edu.tw

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