Comptes Rendus
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

Naoufel Ben Abdallah1 ; Florian Méhats1 ; Nicolas Vauchelet1
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, Volume 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

Naoufel Ben Abdallah 1 ; Florian Méhats 1 ; Nicolas Vauchelet 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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Naoufel Ben Abdallah; Florian Méhats; Nicolas Vauchelet. A note on the long time behavior for the drift-diffusion-Poisson system. Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 683-688. doi : 10.1016/j.crma.2004.09.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.025/

[1] A. Arnold; P. Markowich; G. Toscani; A. Unterreiter On generalized Csiszár–Kullback inequalities, Monatsh. Math., Volume 131 (2000) no. 3, pp. 235-253

[2] A. Arnold; P.A. Markowich; G. Toscani; A. Unterreiter On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations, Volume 26 (2001) no. 1–2, pp. 43-100

[3] A. Arnold; P. Markowich; G. Toscani On large time asymptotics for drift-diffusion-Poisson systems, Transport Theory Statist. Phys., Volume 29 (2000) no. 3–5, pp. 571-581

[4] N. Ben Abdallah Weak solutions of the initial-boundary value problem for the Vlasov–Poisson system, Math. Methods Appl. Sci., Volume 17 (1994) no. 6, pp. 451-476

[5] N. Ben Abdallah, F. Méhats, N. Vauchelet, Diffusive transport of partially quantized particle: existence, uniqueness and long time behavior, submitted for publication

[6] P. Biler; J. Dolbeault Long time behavior of solutions of Nernst–Planck and Debye–Hückel drift-diffusion systems, Ann. Henri Poincaré, Volume 1 (2000) no. 3, pp. 461-472

[7] P. Biler; J. Dolbeault; P.A. Markowich Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling, Transport Theory Statist. Phys., Volume 30 (2001) no. 4–6, pp. 521-536

[8] J. Dolbeault Stationary states in plasma physics: Maxwellian solutions of the Vlasov–Poisson system, Math. Models Methods Appl. Sci., Volume 1 (1991) no. 2, pp. 183-208

[9] H. Gajewski On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech., Volume 65 (1985) no. 2, pp. 101-108

[10] H. Gajewski; K. Gröger Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi–Dirac statistics, Math. Nachr., Volume 140 (1989), pp. 7-36

[11] I. Gasser; C.D. Levermore; P.A. Markowich; C. Schmeiser The initial time layer problem and the quasineutral limit in the semiconductor drift-diffusion model, European J. Appl. Math., Volume 12 (2001) no. 4, pp. 497-512

[12] P.A. Markowich; C.A. Ringhofer; C. Schmeiser Semiconductor Equations, Springer-Verlag, Vienna, 1990

[13] F. Poupaud Boundary value problems for the stationary Vlasov–Maxwell systems, Forum Math., Volume 4 (1992), pp. 499-527

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

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