We shall discuss quantum hydrodynamic models (QHM)-Poisson
systems in bounded domains in the context of charged transport
to induced by an electric field for a rather general
termalization closure. We show non-existence of weak solutions
to stationary states for a large set of boundary conditions,
however the problem is solvable when a nonlinear friction term
is added.
We discuss comparisons corresponding Wigner-Poisson
systems, both in the case of collision of collisionless regimes.
In addition, we present numerical approximations to solutions
of the Wigner equation and discuss the relation to QHD models.
These problems are related to approximations to charged
non-linear Shroedinger transport.
The first part of this lecture is in cllaboration with
A. Junguel. The numerical simulations are part of an ongoing
project with Jing Shi.
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