I. Gamba, A. Jüngel, A. Vasseur:

"Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations";

in: "ASC Report 17/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

The existence of global-in-time weak solutions to the one-dimensional

viscous quantum hydrodynamic equations is proved. The model consists

of the conservation laws for the particle density and particle current density,

including quantum corrections from the Bohm potential and viscous stabilizations

arising from quantum Fokker-Planck interaction terms in the Wigner

equation. The model equations are coupled self-consistently to the Poisson

equation for the electric potential and are supplemented with periodic boundary

and initial conditions. When a diffusion term linearly proportional to the

velocity is introduced in the momentum equation, the positivity of the particle

density is proved. This term, which introduces a strong regularizing effect,

may be viewed as a classical conservative friction term due to particle interactions

with the background temperature. Without this regularizing viscous

term, only the nonnegativity of the density can be shown. The existence proof

relies on the Faedo-Galerkin method together with a priori estimates from the

energy functional.

http://www.asc.tuwien.ac.at/preprint/2009/asc17x2009.pdf

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