G. Ali and A. , Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, Journal of Differential Equations, vol.190, issue.2, pp.663-685, 2003.
DOI : 10.1016/S0022-0396(02)00157-2

P. Amster, M. P. Varela, A. Jüngel, and M. C. Mariani, Subsonic Solutions to a One-Dimensional Non-isentropic Hydrodynamic Model for Semiconductors, Journal of Mathematical Analysis and Applications, vol.258, issue.1, pp.258-52, 2001.
DOI : 10.1006/jmaa.2000.7359

M. Ancona, DIFFUSION???DRIFT MODELING OF STRONG INVERSION LAYERS, COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol.6, issue.1, pp.11-18, 1987.
DOI : 10.1108/eb010295

F. Arimburgo, C. Baiocchi, and L. D. Marini, Numerical approximation of the 1-D nonlinear drift-diffusion model in semiconductors, Nonlinear Kinetic Theory and Mathematical Aspects of Hyperbolic Systems, pp.1-10, 1992.

H. Beirão and . Veiga, On the semiconductor drift-diffusion equations, Differential Integral Equations, vol.9, pp.729-744, 1996.

B. Abdallah and A. Unterreiter, On the stationary quantum drift-diffusion model, Zeitschrift f??r angewandte Mathematik und Physik, vol.49, issue.2, pp.251-275, 1998.
DOI : 10.1007/s000330050218

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, vol.83, issue.1, pp.83-179, 1990.
DOI : 10.1016/0022-0396(90)90074-Y
URL : http://doi.org/10.1016/0022-0396(90)90074-y

A. Bertozzi, The mathematics of moving contact lines in the thin liquid films, Notices Amer, Math. Soc, vol.45, pp.689-697, 1998.

M. Bertsch, R. Dal-passo, G. Garcke, and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Diff. Eqs, vol.3, pp.417-440, 1998.

P. Bleher, J. Lebowitz, and E. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Communications on Pure and Applied Mathematics, vol.44, issue.7, pp.47-923, 1994.
DOI : 10.1002/cpa.3160470702

L. Bonilla, V. Kochelap, and C. Velasco, Pattern formation and stability under bistable electro-optical absorption in quantum wells, Microelectronic Engineering, vol.43, issue.44, pp.11-6395, 1999.
DOI : 10.1016/S0167-9317(98)00158-0

Y. Brenier and E. Grenier, Limitesingulì ere du système de Vlasov-Poisson dans le régime de quasi neutralité: Le cas indépendant du temps, C. R. Acad. Sci, pp.318-121, 1994.

H. Brézis, F. Golse, and R. Sentis, Analyse asymptotique de l'´ equation de Poisson coupléè a la relation de Boltzmann. Quasi-neutralité des plasmas, C. R. Acad. Sci. Paris, pp.321-953, 1995.

F. Brezzi, I. Gasser, P. A. Markowich, and C. Schmeiser, Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension, Applied Mathematics Letters, vol.8, issue.1, pp.47-52, 1995.
DOI : 10.1016/0893-9659(94)00109-P

F. Brezzi, L. D. Marini, and P. Pietra, Numerical simulation of semiconductor devices, Computer Methods in Applied Mechanics and Engineering, vol.75, issue.1-3, pp.493-514, 1989.
DOI : 10.1016/0045-7825(89)90044-3

M. Cáceres, J. Carillo, and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Transactions of the American Mathematical Society, vol.357, issue.03, pp.1161-1175, 2004.
DOI : 10.1090/S0002-9947-04-03528-7

J. A. Carillo, A. Jüngel, and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation, Discrete Contin, Dynam Syst, vol.3, pp.1-20, 2003.

C. Chainais-hillairet, J. G. Liu, and Y. J. Peng, Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis, ESAIM: Mathematical Modelling and Numerical Analysis, vol.37, issue.2, pp.319-338, 2003.
DOI : 10.1051/m2an:2003028

C. Chainais-hillairet and Y. J. Peng, Convergence of a finite-volume scheme for the drift-diffusion equations in 1D, IMA Journal of Numerical Analysis, vol.23, issue.1, pp.81-108, 2003.
DOI : 10.1093/imanum/23.1.81

C. Chainais-hillairet and Y. J. Peng, FINITE VOLUME APPROXIMATION FOR DEGENERATE DRIFT-DIFFUSION SYSTEM IN SEVERAL SPACE DIMENSIONS, Mathematical Models and Methods in Applied Sciences, vol.14, issue.03, pp.461-481, 2004.
DOI : 10.1142/S0218202504003313

C. Chainais-hillairet and Y. J. , Peng and I.Violet, Numerical solutions of Euler-Poisson systems for the potential flows, 2006.

F. Chen, Introduction to plasma physics and controlled fusion, Vol1, 1984.

Z. X. Chen and B. Cockburn, Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case, Numerische Mathematik, vol.71, issue.1, pp.1-28, 1995.
DOI : 10.1007/s002110050134

L. Chen and Q. Ju, Existence of weak solution and semiclassical limit for quantum drift-diffusion model, Zeitschrift f??r angewandte Mathematik und Physik, vol.58, issue.1, 2005.
DOI : 10.1007/s00033-005-0051-4

B. Cockburn and I. Triandaf, Convergence of a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case, Mathematics of Computation, vol.59, issue.200, pp.383-401, 1992.
DOI : 10.1090/S0025-5718-1992-1145661-0

B. Cockburn and I. Triandaf, Error estimates for a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case, Mathematics of Computation, vol.63, issue.207, pp.51-76, 1994.
DOI : 10.1090/S0025-5718-1994-1226812-8

S. Cordier and E. Grenier, Quasineutral limit of an euler-poisson system arising from plasma physics, Communications in Partial Differential Equations, vol.100, issue.5-6, pp.25-1099, 2000.
DOI : 10.1002/(SICI)1097-0312(199709)50:9<821::AID-CPA2>3.0.CO;2-7

R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience , J. Willey and Sons, vol.21, 1967.
DOI : 10.1007/978-1-4684-9364-1

P. Crispel, P. Degond, and M. H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, Journal of Computational Physics, vol.205, issue.2, pp.408-438, 2005.
DOI : 10.1016/j.jcp.2004.11.011

P. Crispel, P. Degond, C. Parzani, and M. H. Vignal, Trois formulations d'un mod??le de plasma quasi-neutre avec courant non nul, Comptes Rendus Mathematique, vol.338, issue.4, pp.408-438, 2005.
DOI : 10.1016/j.crma.2003.11.029

P. Crispel, P. Degond, and M. H. Vignal, An asymptotically stable discretization for the Euler???Poisson system in the quasi-neutral limit, Comptes Rendus Mathematique, vol.341, issue.5, pp.323-328, 2005.
DOI : 10.1016/j.crma.2005.07.008

P. Crispel, P. Degond, and M. H. Vignal, An asymptotic preserving scheme for the two-fluid Euler???Poisson model in the quasineutral limit, Journal of Computational Physics, vol.223, issue.1, 2006.
DOI : 10.1016/j.jcp.2006.09.004
URL : https://hal.archives-ouvertes.fr/hal-00635602

P. Degond and P. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Applied Mathematics Letters, vol.3, issue.3, pp.3-25, 1990.
DOI : 10.1016/0893-9659(90)90130-4

P. Degond and P. Markowich, A steady state potential flow model for semiconductors, IV), pp.87-98, 1993.
DOI : 10.1007/BF01765842

P. Degond, F. Méhats, and C. Ringhofer, Quantum Energy-Transport and Drift-Diffusion Models, Journal of Statistical Physics, vol.126, issue.3-4, pp.625-665, 2005.
DOI : 10.1007/s10955-004-8823-3
URL : https://hal.archives-ouvertes.fr/hal-00378549

P. Degond, F. Méhats, and C. Ringhofer, Quantum hydrodynamic models derived from the entropy principle, 2005.
DOI : 10.1090/conm/371/06850
URL : https://hal.archives-ouvertes.fr/hal-00366178

P. Degond and P. A. Raviart, An analysis of the Darwin model aproximation to Maxwell's equations, Rapport interne, Centre de mathématiques appliqués, 1990.

B. Derrida, J. Lebowitz, E. Speer, and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Physical Review Letters, vol.67, issue.2, pp.165-168, 1991.
DOI : 10.1103/PhysRevLett.67.165

J. Dolbeault, I. Gentil, and A. , A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities, 2004.

J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, submitted for publication, 2005.

M. Emery and J. E. Yukich, A simple proof of the logarithmic Sobolev inequality on the circle, Séminaire de probabilités, XXI, Lecture Notes in Math, pp.1247-173, 1987.

R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, in Handbook of numerical analysis, pp.713-1020, 2000.

R. Eymard, T. Gallouët, and R. Herbin, A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA Journal of Numerical Analysis, vol.26, issue.2, pp.326-353, 2006.
DOI : 10.1093/imanum/dri036

P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Archive for Rational Mechanics and Analysis, vol.52, issue.3, pp.52-205, 1973.
DOI : 10.1007/BF00247733

I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Part. Diff. Eqs, vol.17, pp.553-577, 1992.

I. M. Gamba and C. S. Morawetz, A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow : existence theorem for potential flow, Comm. Pure Appl. Math. XLIX, pp.999-1049, 1996.

C. Gardner, The Quantum Hydrodynamic Model for Semiconductor Devices, SIAM Journal on Applied Mathematics, vol.54, issue.2, pp.409-427, 1994.
DOI : 10.1137/S0036139992240425

C. Gardner, J. Jerome, and D. Rose, Numerical methods for the hydrodynamic device model: subsonic flow, nonlinear Differential Equations and Applications, pp.501-507, 1989.
DOI : 10.1109/43.24878

I. Gasser, D. Levermore, P. A. Markowich, and C. Schmeiser, The initial time layer problem and the quasineutral limit in the semiconductor driftdiffusion model, European J. Appl. Math, vol.12, pp.497-512, 2001.

I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Mathematical Methods in the Applied Sciences, vol.146, issue.2, pp.81-92, 2001.
DOI : 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.0.CO;2-X

I. Gasser and P. Marcati, A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids, Monatshefte f???r Mathematik, vol.138, issue.3, pp.189-208, 2003.
DOI : 10.1007/s00605-002-0482-3

U. Gianazza, G. Savaré, and G. Toscani, A fourth-order nonlinear PDE as gradient flow of the Fisher information in Wasserstein spaces, Manuscript, 2005.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 1984.

R. Glowinski, Numerical methods for nonlinear variational problems, 1984.

E. Grenier, Oscillations in quasineutral plasmas, Communications in Partial Differential Equations, vol.26, issue.2, pp.363-394, 1996.
DOI : 10.1080/03605309608821189

M. Gualdani, A. Jüngel, and G. Toscani, A nonlinear fourth-order parabolic equation with non-homogeneous boundary conditions, 2004.

M. T. Gyi and A. , A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Diff. Eqs, vol.5, pp.773-800, 2000.

A. Jüngel, H. L. Li, and A. Matsumura, The relaxation-time limit in the quantum hydrodynamic equations for semiconductors, Preprint, 2004.

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, vol.19, issue.3, pp.633-659, 2006.
DOI : 10.1088/0951-7715/19/3/006

A. Jüngel and J. P. Mili?i´mili?i´c, Macroscopic quantum models with and without collisions, Proceedings of the Sixth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, 2005.

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas zero-relaxation-time limits, Communications in Partial Differential Equations, vol.5, issue.5-6, pp.24-1007, 1999.
DOI : 10.1006/jdeq.1995.1158

A. Jüngel and Y. J. Peng, Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited, Zeitschrift f??r angewandte Mathematik und Physik, vol.51, issue.3, pp.385-396, 2000.
DOI : 10.1007/s000330050004

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas Quasineutral limits in the drift-diffusion equations, Asymptotic Analysis, pp.49-73, 2001.

A. Jüngel and P. Pietra, A Discretization Scheme for a Quasi-Hydrodynamic Semiconductor Model, Mathematical Models and Methods in Applied Sciences, vol.07, issue.07, pp.935-955, 1997.
DOI : 10.1142/S0218202597000475

A. Jüngel and R. Pinnau, Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems, SIAM Journal on Mathematical Analysis, vol.32, issue.4, pp.760-777, 2000.
DOI : 10.1137/S0036141099360269

A. Jüngel and R. Pinnau, A Positivity-Preserving Numerical Scheme for a Nonlinear Fourth Order Parabolic System, SIAM Journal on Numerical Analysis, vol.39, issue.2, pp.385-406, 2001.
DOI : 10.1137/S0036142900369362

A. Jüngel and G. Toscani, Exponential time decay of solutions to a nonlinear fourth-order parabolic equation, Zeitschrift f???r Angewandte Mathematik und Physik (ZAMP), vol.54, issue.3, pp.377-386, 2003.
DOI : 10.1007/s00033-003-1026-y

A. Jüngel and I. Violet, The quasineutral limit in the quantum driftdiffusion equations, Preprint, 2005.

A. Jüngel and I. Violet, Regularity of solutions for a logarithmic fourth order parabolic equation, 2006.

C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discr. contin. dyn. syst, vol.5, pp.449-455, 1999.

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Archive for Rational Mechanics and Analysis, vol.157, issue.2, pp.129-145, 1995.
DOI : 10.1007/BF00379918

P. A. Markowich, The stationnary semiconductor device equations, 1986.

Y. J. Peng, Asymptotic limits of one-dimensional hydrodynamic model for plasmas and semiconductors, Chinese Ann, Math, pp.23-25, 2002.

Y. J. Peng, Some asymptotic analysis in steady state Euler-Poisson equations for potential flow, Asymptotic analysis, pp.75-92, 2003.

Y. J. Peng and I. Violet, ASYMPTOTIC EXPANSIONS IN A STEADY STATE EULER???POISSON SYSTEM AND CONVERGENCE TO INCOMPRESSIBLE EULER EQUATIONS, Mathematical Models and Methods in Applied Sciences, vol.15, issue.05, pp.15-717, 2005.
DOI : 10.1142/S0218202505000546

Y. J. Peng and I. Violet, Example of supersonic solutions to a steady state Euler???Poisson system, Applied Mathematics Letters, vol.19, issue.12, pp.1335-1340, 2006.
DOI : 10.1016/j.aml.2006.01.015
URL : https://hal.archives-ouvertes.fr/hal-00489009

Y. J. Peng and Y. G. Wang, Boundary layers and quasi-neutral limit in steady state Euler???Poisson equations for potential flows, Nonlinearity, vol.17, issue.3, pp.17-835, 2004.
DOI : 10.1088/0951-7715/17/3/006

Y. J. Peng and J. G. Wang, Quasineutral limit and boundary layers in Euler- Poisson system, Proceedings of the international conference on nonlinear partial differential equations and their applications, pp.239-252, 2003.

Y. J. Peng and Y. G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Analysis, pp.41-141, 2005.

R. Pinnau and . Scharfetter, A Scharfetter-Gummel Type Discretization of the Quantum Drift Diffusion Model, PAMM, vol.26, issue.1, pp.37-40, 2003.
DOI : 10.1002/pamm.200310010

P. Raviart, On singular perturbation problems for the nonlinear Poisson equation : A mathematical approach to electrostatic sheaths and plasma erosion, Lecture Notes of the Summer school in Ile d, pp.452-539, 1997.

O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Sturm-Liouville operators, Journal of Functional Analysis, vol.39, issue.1, pp.42-56, 1980.
DOI : 10.1016/0022-1236(80)90018-X

R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation, Numerical Methods for Partial Differential Equations, vol.13, issue.3, pp.215-236, 1997.
DOI : 10.1002/(SICI)1098-2426(199705)13:3<215::AID-NUM1>3.0.CO;2-Q

C. Schmeiser and S. Wang, QUASINEUTRAL LIMIT OF THE DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS WITH GENERAL INITIAL DATA, Mathematical Models and Methods in Applied Sciences, vol.13, issue.04, pp.463-470, 2003.
DOI : 10.1142/S0218202503002593

S. Selberherr, Analysis and simulation of semiconductor devices, 1984.
DOI : 10.1007/978-3-7091-8752-4

W. Shockley, The theory of p-n junctions in semiconductors and p-n junction transistors, Bell. Syst. Tech. J, vol.27, pp.435-489, 1949.

J. Simon, Compact sets in the spaceL p (O,T; B), Annali di Matematica Pura ed Applicata, vol.287, issue.1, pp.65-96, 1987.
DOI : 10.1007/BF01762360

A. Sitenko and V. Malnev, Plasma Physics Theory, 1995.

M. Slemrod and N. Sternberg, Quasi-Neutral Limit for Euler-Poisson System, Journal of Nonlinear Science, vol.11, issue.3, pp.193-209, 2001.
DOI : 10.1007/s00332-001-0004-9

I. Violet, High order expansions in quasineutral limit of the Euler-Poisson system for a potential flow, accepted for publication in the, Royal Society of Edinburgh Proceedings A (Mathematics), 2006.

S. Wang, Quasineutral Limit of Euler???Poisson System with and without Viscosity, Communications in Partial Differential Equations, vol.157, issue.3-4, pp.419-456, 2004.
DOI : 10.1081/PDE-120030403

S. Wang, Z. Xin, and P. Markowich, Quasineutral limit of the drift diffusion models for semiconductors: The case of general sign-changing doping profile, 2004.

F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, Journal of Functional Analysis, vol.37, issue.2, pp.218-234, 1980.
DOI : 10.1016/0022-1236(80)90042-7