Skip to main content
SpringerLink
Log in

The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, a one-dimensional bipolar Euler-Poisson system (a hydrodynamic model) from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L ∞ entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary effects and a vacuum.

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. and Frid, H., Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147, 1999, 294–307.

    Article  MathSciNet  Google Scholar 

  2. Cordier, S. and Grenier, E., Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25, 2000, 1099–1113.

    Article  MathSciNet  MATH  Google Scholar 

  3. Gasser, I., Hsiao, L. and Li, H. L., Large-time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192, 2003, 326–359.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gasser, I. and Marcati, P., The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Meth. Appl. Sci., 24, 2001, 81–92.

    Article  MathSciNet  MATH  Google Scholar 

  5. Hsiao, L. and Zhang, K. J., The global weak solution and relaxation limits of the initial-boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10, 2000, 1333–1361.

    Article  MathSciNet  Google Scholar 

  6. Huang, F. M. and Li, Y. P., Large-time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Dis. Cont. Dyn. Sys. Sec. A, 24, 2009, 455–470.

    Article  MATH  Google Scholar 

  7. Huang, F. M., Mei, M. and Wang, Y., Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43, 2011, 1595–1630.

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, F. M. and Pan, R. H., Convergence rate for compressible Euler equations with damping and vacuum, Arch. Rational Mech. Anal., 166, 2003, 359–376.

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, F. M. and Pan, R. H., Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations, 220, 2006, 207–233.

    Article  MathSciNet  MATH  Google Scholar 

  10. Jüngel, A., Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Differential Equations, Birkhäuser Verlag, Basel, 2001.

    Book  Google Scholar 

  11. Li, Y. P., Global existence and asymptotic behavior of smooth solutions for a bipolar Euler-Poisson system in the quarter plane, Boundary Value Problems, 21, 2012, 1–21.

    Google Scholar 

  12. Markowich, P. A., Ringhofev, C. A. and Schmeiser, C., Semiconductor Equations, Springer-Verlag, Wien, New York, 1990.

    Book  MATH  Google Scholar 

  13. Natalini, R., The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198, 1996, 262–281.

    Article  MathSciNet  MATH  Google Scholar 

  14. Pan, R. H. and Zhao, K., Initial boundary value problem for compressible Euler equations with damping, Indiana Univ. J. Math., 57, 2008, 2257–2282.

    Article  MathSciNet  MATH  Google Scholar 

  15. Peng, Y. J. and Wang, Y. G., Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41, 2005, 141–160.

    MathSciNet  MATH  Google Scholar 

  16. Sitnko, A. and Malnev, V., Plasma Physics Theory, Chapman & Hall, London, 1995.

    Google Scholar 

  17. Tsuge, N., Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. TMA, 73, 2010, 779–787.

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, S., Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29, 2004, 419–456.

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhou, F. and Li, Y. P., Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351, 2009, 480–490.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhu, C. and Hattori, H., Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166, 2000, 1–32.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yeping Li

Authors
  1. Yeping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yeping Li.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11171223) and the Innovation Program of Shanghai Municipal Education Commission (No. 13ZZ109).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Y. The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum. Chin. Ann. Math. Ser. B 34, 529–540 (2013). https://doi.org/10.1007/s11401-013-0782-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-013-0782-z

Keywords

2000 MR Subject Classification

Search

Navigation