Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations

Yue-Jun Peng

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 2, page 295-312
  • ISSN: 0764-583X

Abstract

top
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 L2.

How to cite

top

Peng, Yue-Jun. "Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis 35.2 (2010): 295-312. <http://eudml.org/doc/197394>.

@article{Peng2010,
abstract = { 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 $O(\varepsilon^\frac\{1\}\{2\})$ to the quasi-neutral limit in L2. },
author = {Peng, Yue-Jun},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Asymptotic analysis; boundary layers; optimal convergence rate; drift-diffusion equations.; asymptotic analysis; mixed boundary conditions},
language = {eng},
month = {3},
number = {2},
pages = {295-312},
publisher = {EDP Sciences},
title = {Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations},
url = {http://eudml.org/doc/197394},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Peng, Yue-Jun
TI - Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 2
SP - 295
EP - 312
AB - 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 $O(\varepsilon^\frac{1}{2})$ to the quasi-neutral limit in L2.
LA - eng
KW - Asymptotic analysis; boundary layers; optimal convergence rate; drift-diffusion equations.; asymptotic analysis; mixed boundary conditions
UR - http://eudml.org/doc/197394
ER -

References

top
  1. J.P. Aubin, Un théorème de compacité. C. R. Acad. Sci. Paris256 (1963) 5042-5044.  
  2. Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations25 (2000) 737-754.  Zbl0970.35110
  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. Paris321 (1995) 953-959.  Zbl0839.76096
  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.  Zbl0849.35105
  5. S. Cordier, E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics. Comm. Partial Differential Equations25 (2000) 1099-1113.  Zbl0978.82086
  6. P.C. Fife, Semilinear elliptic boundary value problems with small parameters. Arch. Rational Mech. Anal.52 (1973) 205-232.  Zbl0268.35007
  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.  Zbl0801.35133
  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).  Zbl0980.35158
  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).  Zbl1018.82024
  10. A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr.185 (1997) 85-110.  Zbl1157.35406
  11. A. Jüngel, Y.J. Peng, A hierarchy of hydrodynamic models for plasmas. Zero-relaxation-time limits. Comm. Partial Differential Equations24 (1999) 1007-1033.  Zbl0946.35074
  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 Linéaire17 (2000) 83-118.  Zbl0956.35010
  13. A. Jüngel, Y.J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisited. Z. Angew. Math. Phys.51 (2000) 385-396.  Zbl0963.35115
  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).  Zbl1045.76058
  15. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villard, Paris (1969).  
  16. P.A. Markowich, A singular perturbation analysis of the fundamental semiconductor device equations. SIAM J. Appl. Math.44 (1984) 896-928.  Zbl0568.35007
  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.  Zbl0639.34016
  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.  Zbl0965.65113
  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.  
  21. A. Visintin, Strong convergence results related to strict convexity. Comm. Partial Differential Equations9 (1984) 439-466.  Zbl0545.49019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.