Quantum Semiconductor Models
L. Chen and M. Dreher
Abstract.
We give an overview of analytic investigations of quantum semiconductor
models, where we focus our attention on two classes of models: quantum
drift diffusion models, and quantum hydrodynamic models. The key feature of
those models is a quantum interaction term which introduces a perturbation
term with higher order derivatives into a system which otherwise might be
seen as a fluid dynamic system. After a discussion of the modeling, we
present the quantum drift diffusion model in detail, discuss various
versions of this model, list typical questions
and the tools how to answer them, and we give an account of the
state-of-the-art of concerning this model. Then we discuss the
quantum hydrodynamic model,
which figures as an application of the theory of mixed order
parameter-elliptic systems in the sense of Douglis, Nirenberg, and
Volevich. For various versions of this model, we give a unified proof of
the local existence of classical solutions. Furthermore, we present new
results on the existence as well as the exponential stability of steady
states, with explicit description of the decay rate.
Operator Theory: Advances and Applications, Vol. 211
Subseries: Advances in Partial Differential Equations.
(2011), 72 pp.
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