Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 787–827 | Cite as

Stability of Transonic Shock Solutions for One-Dimensional Euler–Poisson Equations

  • Tao Luo
  • Jeffrey Rauch
  • Chunjing Xie
  • Zhouping Xin


In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler–Poisson systems are investigated. First, a steady transonic shock solution with a supersonic background charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with a supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and on a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler–Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role.


Poisson Equation Dynamical Stability Shock Location Shock Position Background Charge 


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  1. 1.
    Ascher U.M., Markowich P.A., Pietra P., Schmeiser C.: A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 1(3), 347–376 (1991)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chen D.P., Eisenberg R.S., Jerome J.W., Shu C.W.: A hydrodynamic model of temperature change in open ionic channels. Biophys. J. 69, 2304–2322 (1995)ADSCrossRefGoogle Scholar
  3. 3.
    Chen G.-Q., Feldman M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Am. Math. Soc. 16(3), 461–494 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chen G.Q., Wang D.: Convergence of shock capturing schemes for the compressible Euler-Poisson equations. Commun. Math. Phys. 179(2), 333–364 (1996)ADSMATHCrossRefGoogle Scholar
  5. 5.
    Degond P., Markowich P.A.: On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl. Math. Lett. 3(3), 25–29 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Degond P., Markowich P.A.: A steady state potential flow model for semiconductors. Ann. Mat. Pura Appl. (4) 165, 87–98 (1993)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Embid P., Goodman J., Majda A.: Multiple steady states for 1-D transonic flow. SIAM J. Sci. Stat. Comput. 5(1), 21–41 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, 1998Google Scholar
  9. 9.
    Gamba I.M.: Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors. Commun. Partial Differ. Equ. 17(3–4), 553–577 (1992)MathSciNetMATHGoogle Scholar
  10. 10.
    Gamba I.M., Morawetz C.S.: A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: existence theorem for potential flow. Commun. Pure Appl. Math. 49(10), 999–1049 (1996)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ha S.-Y.: L 1 stability for systems of conservation laws with a nonresonant moving source. SIAM J. Math. Anal. 33(2), 411–439 (2001)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ha S.-Y., Yang T.: L 1 stability for systems of hyperbolic conservation laws with a resonant moving source. SIAM J. Math. Anal. 34(5), 1226–1251 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Huang F., Pan R., Yu H.: Large time behavior of Euler-Poisson system for semiconductor. Sci. China Ser. A 51(5), 965–972 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Lax, P.D.: Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2002Google Scholar
  15. 15.
    Li H., Markowich P., Mei M.: Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations. Q. Appl. Math. 60(4), 773–796 (2002)MathSciNetMATHGoogle Scholar
  16. 16.
    Li J., Xin Z., Yin H.: On transonic shocks in a nozzle with variable end pressures. Commun. Math. Phys. 291(1), 111–150 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Li J., Xin Z., Yin H.: A free boundary value problem for the Euler system and 2-D transonic shock in a large variable nozzle. Math. Res. Lett. 16(5), 777–796 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    Li, T.T., Yu, W.C.: Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985Google Scholar
  19. 19.
    Lien W.-C.: Hyperbolic conservation laws with a moving source. Commun. Pure Appl. Math. 52(9), 1075–1098 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Liu T.P.: Transonic gas flow in a duct of varying area. Arch. Rational Mech. Anal. 80(1), 1–18 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Liu T.P.: Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83(2), 243–260 (1982)ADSMATHCrossRefGoogle Scholar
  22. 22.
    Liu T.P.: Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28(11), 2593–2602 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Luo T., Natalini R., Xin Z.: Large time behavior of the solutions to a hydrodynamic model for semiconductors. SIAM J. Appl. Math. 59(3), 810–830 (1999)MathSciNetMATHGoogle Scholar
  24. 24.
    Luo, T., Xin, Z.: Transonic shock solutions for a system of Euler-Poisson equations. IMS Research Report, Vol. 167, 2009.
  25. 25.
    Majda, A.: The existence of multidimensional shock fronts. Mem. Am. Math. Soc. 43(281) (1983)Google Scholar
  26. 26.
    Markowich P.A.: On steady state Euler-Poisson models for semiconductors. Z. Angew. Math. Phys. 42(3), 389–407 (1991)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Markowich P.A., Ringhofer C.A., Schmeiser C.: Semiconductor Equations. Springer, Vienna (1990)MATHCrossRefGoogle Scholar
  28. 28.
    Métivier, G.: Stability of multidimensional shocks. Advances in the Theory of Shock Waves. Progr. Nonlinear Differential Equations Appl., Vol. 47. Birkhäuser Boston, 25–103, 2001Google Scholar
  29. 29.
    Pao C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)MATHGoogle Scholar
  30. 30.
    Peng Y.-J., Violet I.: Example of supersonic solutions to a steady state Euler-Poisson system. Appl. Math. Lett. 19(12), 1335–1340 (2006)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rauch J.: Qualitative behavior of dissipative wave equations on bounded domains. Arch. Rational Mech. Anal. 62(1), 77–85 (1976)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Rauch J., Massey F.: Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Am. Math. Soc. 189, 303–318 (1974)MathSciNetMATHGoogle Scholar
  33. 33.
    Rauch J., Taylor M.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Rauch, J., Xie, C., Xin, Z.: Global stability of transonic shock solutions in quasi-one-dimensional nozzles. Preprint (2010)Google Scholar
  35. 35.
    Rosini M.D.: Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors. J. Differ. Equ. 199(2), 326–351 (2004)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Rosini M.D.: A phase analysis of transonic solutions for the hydrodynamic semiconductor model. Q. Appl. Math. 63(2), 251–268 (2005)MathSciNetMATHGoogle Scholar
  37. 37.
    Wang D., Chen G.-Q.: Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation. J. Differ. Equ. 144(1), 44–65 (1998)MATHCrossRefGoogle Scholar
  38. 38.
    Xin Z., Yin H.: Transonic shock in a nozzle. I. Two-dimensional case. Commun. Pure Appl. Math. 58(8), 999–1050 (2005)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Xin Z., Yin H.: The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. J. Differ. Equ. 245(4), 1014–1085 (2008)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Zhang B.: Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices. Commun. Math. Phys. 157(1), 1–22 (1993)ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Tao Luo
    • 1
  • Jeffrey Rauch
    • 2
  • Chunjing Xie
    • 2
  • Zhouping Xin
    • 3
  1. 1.Department of Mathematics and StatisticsGeorgetown UniversityWashingtonUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.The Institute of Mathematical Sciences and Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

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