I. Gamba, A. Jüngel, A. Vasseur:

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

Journal of Differential Equations,247(2009), 3117 - 3135.

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.

Siehe englisches Abstract.

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