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I. Gamba, A. Jüngel, A. Vasseur:
"Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations";
Journal of Differential Equations, 247 (2009), S. 3117 - 3135.

Siehe englisches Abstract.

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.