Skip to main content
SpringerLink
Log in

Large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductor devices

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We are concerned with the global existence and large time behavior of entropy solutions to the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations in a bounded interval. In this paper, we first prove the global existence of entropy solution by vanishing viscosity and compensated compactness framework. In particular, the solutions are uniformly bounded with respect to space and time variables by introducing modified Riemann invariants and the theory of invariant region. Based on the uniform estimates of density, we further show that the entropy solution converges to the corresponding unique stationary solution exponentially in time. No any smallness condition is assumed on the initial data and doping profile. Moreover, the novelty in this paper is about the unform bound with respect to time for the weak solutions of the isentropic Euler–Poisson system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, G.Q.: Convergence of the Lax–Friedrichs scheme for ientropic gas dynamics (III). Acta Math. Sci. 6, 75–120 (1986)

    Article  Google Scholar 

  2. Chen, G.Q.: Remarks on Diperna’s paper convergence of the viscosity method for isentropic gas dynamics. Proc. Am. Math. Soc. 125, 2981–2986 (1997)

    Article  MATH  Google Scholar 

  3. Chen, G.Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, 89–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, G.Q., Glimm, J.: Global solutions to the compressible Euler equations with geometrical structure. Commin. Math. Phys. 180, 153–193 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Degond, P., Markowich, P.A.: On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl. Math. Lett. 3, 25–29 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding, X.X.: On a lemma of Diperna and Chen. Acta Math. Sci. 26, 188–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, X.X., Chen, G.Q., Luo, P.Z.: Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics (I)–(II). Acta Math. Sci. 5, 415-432–433-472 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Ding, X.X., Chen, G.Q., Luo, P.Z.: Convergence of the fractional step Lax–Friedrichs scheme and Godunov scheme for isentropic system of gas dynamics. Commun. Math. Phys. 121, 63–84 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. DiPerna, R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. DiPerna, R.: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82, 27–70 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gasser, I., Marcati, P.: A quasi-neutral limit in hydrodynamic model for charged fluids. Monatshefte fr Mathematik 138, 189–208 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gasser, I., Natalini, R.: The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors. Q. Appl. Math. 57, 269–282 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hsiao, L., Yang, T.: Asymptotics of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors. J. Differ. Equ. 170, 472–493 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, F.M., Li, T.H., Yu, H.M.: Weak solutions to isothermal hydrodynamic model for semi-conductors. J. Differ. Equ. 247, 3070–3099 (2009)

    Article  MATH  Google Scholar 

  15. Huang, F.M., Li, T.H., Yuan, D.F.: Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry. arXiv:1711.04430 (2017)

  16. Huang, F.M., Pan, R.H.: Convergence rate for compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 166, 359–376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, F.M., Pan, R.H., Yu, H.M.: Large time behavior of Euler–Poisson system for semiconductor. Sci. China Math. 51, 965–972 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, F.M., Wang, Z.: Convergence of viscosity solutions for isentropic gas dynamics. SIAM J. Math. Anal. 34, 595–610 (2003)

    Article  Google Scholar 

  19. Lions, P.L., Perthame, B., Souganidis, P.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49, 599–638 (1996). (Press, 2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lions, P.L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163, 415–431 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, T.H.: Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices. Z. Angew. Math. Phys. 57, 1–20 (2006)

    Article  MathSciNet  Google Scholar 

  22. Marcati, P., Natalini, R.: Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem. Proc. R. Soc. Edinb. 125A, 115–131 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Marcati, P., Natalini, R.: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion eqation. Arch. Ration. Mech. Anal. 129, 129–145 (1995)

    Article  MATH  Google Scholar 

  24. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductors Equations. Springer, New York (1990)

    Book  MATH  Google Scholar 

  25. Michele, F., Marcati, P., Rubino, B.: Steady states and interface transmission conditions for heterogeneous quantum-classical 1-D hydrodynamic model of semiconductor devices. Physica D 243, 1–13 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Michele, F., Marcati, P., Rubino, B.: Stationary solution for transient quantum hydrodynamics with bohmenian-type boundary conditions. Comput. Appl. Math. 36, 459–479 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nishida, T., Smoller, J.: Solutions in the large for some nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math. 26, 183–200 (1973); 44–65 (1998)

  28. Poupaud, F.: Derivation of a hydrodynamic systems hierarchy from the Boltzmann equation. Appl. Math. Lett. 4, 75–79 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1994)

    Book  MATH  Google Scholar 

  30. Tsuge, N.: Existence and stability of solutions to the compressible Euler equations with an outer force. Nonlinear Anal. Real World Appl. 27, 203–220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yu, H.M.: Large time behavior of entropy solutions to a unipolar hydrodynamic model of semiconductors. Commun. Math. Sci. 14, 69–82 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, B.: On a local existence theroem for a simplified one-dimensional hydrodynamic model for semiconductor devices. SIAM J. Math. Anal. 25, 941–947 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, B.: Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices. Commun. Math. Phys. 157, 1–22 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Feimin Huang & Difan Yuan

  2. Academy of Mathematics and System Sciences, CAS, Beijing, 100190, China

    Feimin Huang & Difan Yuan

  3. Hua Loo-Keng Key Laboratory of Mathematics, AMSS, CAS, Beijing, 100190, China

    Tianhong Li

  4. Department of Mathematics, Shandong Normal University, Jinan, 250014, China

    Huimin Yu

Authors
  1. Feimin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huimin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Difan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Difan Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, F., Li, T., Yu, H. et al. Large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductor devices. Z. Angew. Math. Phys. 69, 69 (2018). https://doi.org/10.1007/s00033-018-0968-z

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-018-0968-z

Mathematics Subject Classification

Keywords

Search

Navigation