Local existence result in time for a drift-diffusion system with Robin boundary conditions

  • Arnaud HeibigEmail author
  • Adrien Petrov


This paper deals with a drift-diffusion system being subjected to Robin boundary conditions. Under appropriate hypotheses on the data, a local existence result in time is obtained by using a fixed-point argument combined with some a priori estimates.


Drift-diffusion system Robin boundary conditions Fixed-point theorem Existence result 

Mathematics Subject Classification

35B45 35A65 35D30 76R50 



  1. 1.
    Blanchet, A., Carrillo, J.A., Masmoudi, N.: Infinite time aggregation for the critical Patlak-Keller-Segel model in \({\mathbb{R}}^2\). Commun. Pure Appl. Math. 61(19), 1449–1481 (2008)CrossRefGoogle Scholar
  2. 2.
    Brezis, H.: Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris. Théorie et applications (1983)Google Scholar
  3. 3.
    Brezzi, F., Marini, L.D., Pietra, P.: Numerical simulation of semiconductor devices. Comput. Methods Appl. Mech. Eng. 75, 493–514 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ciuperca, I.S., Heibig, A., Palade, L.I.: Existence and uniqueness results for the Doi-Edwards polymer melt model: the case of the (full) nonlinear configurational probability density equation. Nonlinearity 25(4), 991–1009 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, Z., Cockburn, B.: Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case. Numer. Math. 71, 1–28 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Constantin, P., Masmoudi, N.: Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D. Commun. Math. Phys. 278(1), 179–191 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Constantin, P., Seregin, G.: Global regularity of solutions of coupled Navier–Stokes equations and nonlinear Fokker Planck equations. Discrete Contin. Dyn. Syst. 26(4), 1185–1196 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Da Veiga, H.B.: On semiconductor drift-diffusion equations. Differ. Int. Eqs. 9, 729–744 (1996)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fang, W., Ito, K.: Global solutions of the time-dependent drift-diffusion semiconductor equations. J. Differ. Equ. 123(2), 523–566 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fang, W., Ito, K.: On the time-dependent drift-diffusion model for semiconductors. J. Differ. Equ. 117(2), 245–280 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gajewski, H.: On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors. Z. Angew. Math. Mech. 65(2), 101–108 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gajewski, H.: On the uniqueness of solutions to the drift-diffusion model of semi-conductors devices. Math. Models Methods Appl. Sci. 4, 121–133 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heibig, A.: Well-posedness of a Debye type system endowed with a full wave equation. Appl. Math. Lett. 81, 27–34 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Marcati, P., Natalini, R.: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Ration. Mech. Anal. 129(2), 129–145 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Markowich, P.A.: The stationary semiconductor device equations, p. ix+193. Springer, Vienna (1990)CrossRefGoogle Scholar
  17. 17.
    Mock, M.S.: Analysis of mathematical models of semiconductor devices. In: Advances in Numerical Computation Series, vol 3, pp. viii+200. Boole Press (1983)Google Scholar
  18. 18.
    Mizoguchi, N.: Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane. Calc. Var. Partial Differ. Equ. 48, 491–505 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Scharfetter, D.L., Gummel, H.K.: Large signal analysis of a silicon read diode oscillator. IEEE Trans. Electron. Dev. 16, 64–77 (1969)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Lyon, CNRS, INSA de Lyon & Institut Camille Jordan UMR 5208VilleurbanneFrance

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