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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