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Stability of transonic shocks to the Euler-Poisson system with varying background charges

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Abstract

This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length. The background charge in the Poisson equation is a piecewise constant function. The structural stability of the steady transonic shock solution is obtained by the monotonicity argument. Furthermore, this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data. One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions.

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References

  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, 1991, 1: 347–376

    Article  MathSciNet  Google Scholar 

  2. Bae M, Chen G Q, Feldman M. Regularity of solutions to regular shock reflection for potential flow. Invent Math, 2009, 175: 505–543

    Article  MathSciNet  Google Scholar 

  3. Bae M, Duan B, Xiao J J, Xie C J. Structural stability of supersonic solutions to the Euler-Poisson system. Arch Ration Mech Anal, 2021, 239: 679–731

    Article  MathSciNet  Google Scholar 

  4. Bae M, Duan B, Xie C J. Subsonic solutions for steady Euler-Poisson system in two-dimensional nozzles. SIAM J Math Anal, 2014, 46: 3455–3480

    Article  MathSciNet  Google Scholar 

  5. Bae M, Duan B, Xie C J. Subsonic flow for the multidimensional Euler-Poisson system. Arch Ration Mech Anal, 2016, 220: 155–191

    Article  MathSciNet  Google Scholar 

  6. Bae M, Duan B, Xie C J. Two-dimensional subsonic flows with self-gravitation in bounded domain. Math Models Methods Appl Sci, 2015, 25: 2721–2747

    Article  MathSciNet  Google Scholar 

  7. Bae M, Feldman M. Transonic shocks in multidimensional divergent nozzles. Arch Ration Mech Anal, 2011, 201: 777–840

    Article  MathSciNet  Google Scholar 

  8. Chen G Q, Fang B X. Stability of transonic shocks in steady supersonic flow past multidimensional wedges. Adv Math, 2017, 314: 493–539

    Article  MathSciNet  Google Scholar 

  9. Courant R, Friendrichs K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers Inc, 1948

    Google Scholar 

  10. Degond P, Markowich P A. A steady state potential flow model for semiconductors. Ann Mat Pura Appl, 1993, 165: 87–98

    Article  MathSciNet  Google Scholar 

  11. Guo Y, Strauss W. Stability of semiconductor states with insulating and contact boundary conditions. Arch Ration Mech Anal, 2006, 179: 1–30

    Article  MathSciNet  Google Scholar 

  12. Li J, Xin Z P, Yin H C. On transonic shocks in a nozzle with variable end pressures. Comm Math Phys, 2009, 291: 111–150

    Article  MathSciNet  Google Scholar 

  13. Li T T, Yu W C. Boundary value problems for quasilinear hyperbolic systems. Duke Univeisity Mathematics Series V, Duke University, Mathematics Department, Durham, NC, 1985

    Google Scholar 

  14. Liu T P. Nonlinear stability and instability of transonic flows through a nozzle. Comm Math Phys, 1982, 83: 243–260

    Article  MathSciNet  Google Scholar 

  15. Liu T P. Transonic gas flow in a duct of varying area. Arch Rational Mech Anal, 1982, 80: 1–18

    Article  MathSciNet  Google Scholar 

  16. Luo T, Ranch J, Xie C J, Xin Z P. Stability of transonic shock solutions for one-dimensional Euler-Poisson equations. Arch Ration Mech Anal, 2011, 202: 787–827

    Article  MathSciNet  Google Scholar 

  17. Luo T, Xin Z P. Transonic shock solutions for a system of Euler-Poisson equations. Commun Math Sci, 2012, 10: 419–462

    Article  MathSciNet  Google Scholar 

  18. Majda A. The existence of multidimensional shock fronts. Mem Amer Math Soc, 1983, 41: 1–95

    MathSciNet  Google Scholar 

  19. Métivier G. Stability of multidimensional shocks//Freistuhler H, Szepessy A. Advances in the Theory of Shock Waves. Boston: Birkhäuser, 2001: 25–103

    Chapter  Google Scholar 

  20. Rosini M D. A phase analysis of transonic solutions for the hydrodynamic semiconductor model. Quart Appl Math, 2005, 63: 251–268

    Article  MathSciNet  Google Scholar 

  21. Rauch J. Qualitative behavior of dissipative wave equations on bounded domains. Arch Rational Mech Anal, 1976, 62: 77–85

    Article  MathSciNet  Google Scholar 

  22. Rauch J, Taylor M. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ Math J, 1974, 24: 79–86

    Article  MathSciNet  Google Scholar 

  23. Weng S K. On steady subsonic flows for Euler-Poisson models. SIAM J Math Anal, 2014, 46: 757–779

    Article  MathSciNet  Google Scholar 

  24. Xin Z P, Yan W, Yin H C. Transonic shock problem for the Euler system in a nozzle. Arch Ration Mech Anal, 2009, 194: 1–47

    Article  MathSciNet  Google Scholar 

  25. Xin Z P, Yin H C. Transonic shock in a nozzle I: Two-dimensional case. Comm Pure Appl Math, 2005, 58: 999–1050

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Yang Cao & Yuanyuan Xing

  2. School of mathematics and statistics, Wuhan University, Wuhan, 430070, China

    Na Zhang

Authors
  1. Yang Cao
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  2. Yuanyuan Xing
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  3. Na Zhang
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Corresponding author

Correspondence to Yuanyuan Xing.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

This work was supported by the National Natural Science Foundation of China (11871134,12171166) and the Fundamental Research Funds for the Central Universities (DUT23LAB303).

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Cao, Y., Xing, Y. & Zhang, N. Stability of transonic shocks to the Euler-Poisson system with varying background charges. Acta Math Sci 44, 1487–1506 (2024). https://doi.org/10.1007/s10473-024-0416-4

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  • DOI: https://doi.org/10.1007/s10473-024-0416-4

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