no. 10
Partial Differential Equations
A note on the long time behavior for the drift-diffusion-Poisson system
[Une note sur le comportement en temps long du système dérive-diffusion-Poisson.]

Présenté par : Pierre-Louis Lions

Ben Abdallah, Naoufel1 ; Méhats, Florian1 ; Vauchelet, Nicolas1
1 Mathématiques pour l'industrie et la physique, UMR 5640, université Paul Sabatier, UFR MIG, 118, route de Narbonne, 31032 Toulouse cedex 4, France
Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 683-688.

Dans cette note, nous analysons le comportement en temps long des solutions du système couplé dérive-diffusion-Poisson avec une matrice de diffusion définie positive et soumis à des conditions aux limites de Dirichlet. Ce système modélise le transport de charges dans des dispositifs à semiconducteurs ou à plasmas. En utilisant l'entropie relative développée à l'ordre 2, nous prouvons la convergence exponentielle des solutions vers l'équilibre.

In this note we analyze the long time behavior of a drift-diffusion-Poisson system with a symmetric definite positive diffusion matrix, subject to Dirichlet boundary conditions. This system models the transport of electrons in semiconductor or plasma devices. By using a quadratic relative entropy obtained by keeping the lowest order term of the logarithmic relative entropy, we prove the exponential convergence to the equilibrium.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.09.025
Ben Abdallah, Naoufel 1 ; Méhats, Florian 1 ; Vauchelet, Nicolas 1

1 Mathématiques pour l'industrie et la physique, UMR 5640, université Paul Sabatier, UFR MIG, 118, route de Narbonne, 31032 Toulouse cedex 4, France 
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Ben Abdallah, Naoufel; Méhats, Florian; Vauchelet, Nicolas. A note on the long time behavior for the drift-diffusion-Poisson system. Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 683-688. doi : 10.1016/j.crma.2004.09.025. http://www.numdam.org/articles/10.1016/j.crma.2004.09.025/

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* Support by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282, is acknowledged.