Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



From a multidimensional quantum hydrodynamic model to the classical drift-diffusion equation

Author: Yeping Li
Journal: Quart. Appl. Math. 67 (2009), 489-502
MSC (2000): Primary 35B25, 35M20, 35L45
DOI: https://doi.org/10.1090/S0033-569X-09-01156-7
Published electronically: May 6, 2009
MathSciNet review: 2547643
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Abstract: In the paper, we discuss the combined semiclassical and relaxation-time limits of a multidimensional isentropic quantum hydrodynamical model for semiconductors with small momentum relaxation time and Planck constant. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohn potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the Planck constant and the relaxation time tend to zero, periodic initial-value problems of a scaled isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion models have smooth solutions.

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Additional Information

Yeping Li
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Email: ypleemei@yahoo.com.cn

DOI: https://doi.org/10.1090/S0033-569X-09-01156-7
Keywords: Relaxation limit, isentropic quantum models semiconductors, energy estimates
Received by editor(s): February 17, 2008
Published electronically: May 6, 2009
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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