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This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. in [7]. The model is obtained as the diffusive limit of the quantum Liouville–BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.
@article{AIHPC_2019__36_3_811_0,
author = {Pinaud, Olivier},
title = {The quantum drift-diffusion model: {Existence} and exponential convergence to the equilibrium},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {811--836},
publisher = {Elsevier},
volume = {36},
number = {3},
year = {2019},
doi = {10.1016/j.anihpc.2018.10.002},
zbl = {1412.82022},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.002/}
}
TY - JOUR AU - Pinaud, Olivier TI - The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium JO - Annales de l'I.H.P. Analyse non linéaire PY - 2019 SP - 811 EP - 836 VL - 36 IS - 3 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.002/ DO - 10.1016/j.anihpc.2018.10.002 LA - en ID - AIHPC_2019__36_3_811_0 ER -
%0 Journal Article %A Pinaud, Olivier %T The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium %J Annales de l'I.H.P. Analyse non linéaire %D 2019 %P 811-836 %V 36 %N 3 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.002/ %R 10.1016/j.anihpc.2018.10.002 %G en %F AIHPC_2019__36_3_811_0
Pinaud, Olivier. The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 811-836. doi: 10.1016/j.anihpc.2018.10.002
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