Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, p. 295-312

We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate O(ε 1 2 ) to the quasi-neutral limit in L 2 .

On étudie les couches limites et les limites de quasi-neutralité aux systèmes de dérivée-diffusion. On montre d’abord que cette limite est unique et déterminée par un système découplé avec données initiales et aux limites. On établit ensuite les équations des couches limites et montre l’existence et l’unicité de solutions avec l’atténuation exponentielle. Ceci implique un résultat de convergence globale (par rapport au domaine) de la suite de solutions et un taux de convergence optimale O(ε 1 2 ) dans la limite de quasi-neutralité dans L 2 .

Classification:  35B25,  35B40,  35K57
Keywords: asymptotic analysis, boundary layers, optimal convergence rate, drift-diffusion equations
@article{M2AN_2001__35_2_295_0,
     author = {Peng, Yue-Jun},
     title = {Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     pages = {295-312},
     zbl = {0994.35020},
     mrnumber = {1825700},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_2_295_0}
}
Peng, Yue-Jun. Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, pp. 295-312. http://www.numdam.org/item/M2AN_2001__35_2_295_0/

[1] J.P. Aubin, Un théorème de compacité. C. R. Acad. Sci. Paris 256 (1963) 5042-5044. | Zbl 0195.13002

[2] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000) 737-754. | Zbl 0970.35110

[3] H. Brézis, F. Golse, R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas. C. R. Acad. Sci. Paris 321 (1995) 953-959. | Zbl 0839.76096

[4] S. Cordier, P. Degond, P. Markowich, C. Schmeiser, Traveling wave analysis and jump relations for Euler-Poisson model in the quasineutral limit. Asymptot. Anal. 11 (1995) 209-224. | Zbl 0849.35105

[5] S. Cordier, E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics. Comm. Partial Differential Equations 25 (2000) 1099-1113. | Zbl 0978.82086

[6] P.C. Fife, Semilinear elliptic boundary value problems with small parameters. Arch. Rational Mech. Anal. 52 (1973) 205-232. | Zbl 0268.35007

[7] H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. Math. Models Methods Appl. Sci. 4 (1994) 121-133. | Zbl 0801.35133

[8] I. Gasser, The initial time layer problem and the quasi-neutral limit in a nonlinear drift diffusion model for semiconductors. Nonlinear Differential Equations Appl. (to appear). | MR 1841258 | Zbl 0980.35158

[9] I. Gasser, D. Levermore, P. Markowich, C. Schmeiser, The initial time layer problem and the quasi-neutral limit in the drift-diffusion model (submitted). | Zbl 1018.82024

[10] A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr. 185 (1997) 85-110. | Zbl pre01019611

[11] A. Jüngel, Y.J. Peng, A hierarchy of hydrodynamic models for plasmas. Zero-relaxation-time limits. Comm. Partial Differential Equations 24 (1999) 1007-1033. | Zbl 0946.35074

[12] A. Jüngel, Y.J. Peng, A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 17 (2000) 83-118. | Numdam | Zbl 0956.35010

[13] A. Jüngel, Y.J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisited. Z. Angew. Math. Phys. 51 (2000) 385-396. | Zbl 0963.35115

[14] A. Jüngel, Y.J. Peng, A hierarchy of hydrodynamic models for plasmas. Quasi-neutral limits in the drift-diffusion equations. Asymptot. Anal. (to appear). | MR 1865570 | Zbl 1045.76058

[15] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villard, Paris (1969). | MR 259693 | Zbl 0189.40603

[16] P.A. Markowich, A singular perturbation analysis of the fundamental semiconductor device equations. SIAM J. Appl. Math. 44 (1984) 896-928. | Zbl 0568.35007

[17] P.A. Markowich, C. Ringhofer, C. Schmeiser, An asymptotic analysis of one-dimensional models for semiconductor devices. IMA J. Appl. Math. 37 (1986) 1-24. | Zbl 0639.34016

[18] Y.J. Peng, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for a nonlinear Euler-Poisson system. Nonlinear Anal. TMA 42 (2000) 1033-1054. | Zbl 0965.65113

[19] P. Raviart, On singular perturbation problems for the nonlinear Poisson equation or: A mathematical approach to electrostatic sheaths and plasma erosion, Lect. Notes of the Summer school in Ile d'Oléron, France (1997) 452-539.

[20] L. Tartar, Compensated compactness and applications to partial differential equations. In: Nonlinear analysis and mechanics: Heriot-Watt Symp. Vol. 4 and Res. Notes Math. 3 (1979) 136-212. | Zbl 0437.35004

[21] A. Visintin, Strong convergence results related to strict convexity. Comm. Partial Differential Equations 9 (1984) 439-466. | Zbl 0545.49019