Simple waves for two-dimensional compressible pseudo-steady Euler system

Abstract

A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that “the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow pattern” see Courant and Friedrichs’s chassical book “Supersonic Flow and Shock Waves”. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems.

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References

  1. [1]

    Li, J., Zhang, T., and Yang, S. The Two-Dimensional Riemann Problem in Gas Dynamics, Addison Wesley Longman limited, London (1998)

    Google Scholar 

  2. [2]

    Zheng, T. and Zheng, Y. Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics system. SIAM J. Math. Anal. 21(3), 593–630 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. [3]

    Zheng, Y. Systems of Conservation Laws: Two-Dimensional Riemann Problems, Bikhäuser Boston, Boston (2001)

    Google Scholar 

  4. [4]

    John, F. Partial Differential Equations, Springer-Verlag, New York (1982)

    Google Scholar 

  5. [5]

    Courant, R. and Friedrichs, K. O. Supersonic Flow and Shock Waves, Interscience, New York (1948)

    Google Scholar 

  6. [6]

    Li, J., Zhang, T., and Zheng, Y. Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commu. Math. Phys. 267(1), 1–12 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    Bang, S. Rarefaction Wave Interaction of Pressure Gradient System, Ph. D. dissertation, Pennsylvania State University (2007)

  8. [8]

    Dai, Z. and Zhang, T. Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch. Ration. Mech. Anal. 155(4), 277–298 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. [9]

    Lei, Z. and Zheng, Y. A complete global solution to the pressure gradient equation. Journal of Differential Equations 236(1), 280–292 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. [10]

    Song, K. and Zheng, Y. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete and Continuous Dynamic System 24(4), 1365–1380 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. [11]

    Li, J. On the two-dimensional gas expansion for the compressible Euler equations. SIAM J. Appl. Math. 62(3), 831–852 (2002)

    Article  Google Scholar 

  12. [12]

    Li, J. Global solutions of an initial value problem for two-dimensional compressible Euler equations. Journal of Differential Equations 179(1), 178–194 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. [13]

    Li, J. and Zheng, Y. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Rat. Mech. Anal. 193(3), 623–657 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. [14]

    Chang, T., Chen, G., and Yang, S. On the 2-D Riemann problem for the compressible Euler equation, I. interaction of contact discontinuities. Discrete and Continuous Dynamical Systems 6(2), 419–430 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. [15]

    Glimm, J., Ji, X., Li, J., Li, X., Zhang, P., Zhang, T., and Zheng, Y. Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations. SIAM J. Appl. Math. 69(3), 720–742 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Wan-cheng Sheng 盛万成.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10971130) and the Shanghai Leading Academic Discipline Project (No. J50101)

Communicated by Xing-ming GUO

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Lai, G., Sheng, Wc. Simple waves for two-dimensional compressible pseudo-steady Euler system. Appl. Math. Mech.-Engl. Ed. 31, 827–838 (2010). https://doi.org/10.1007/s10483-010-1317-7

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Key words

  • self-similar Euler system
  • simple wave
  • generalized characteristic analysis
  • pseudo-stream line
  • sonic circle
  • sonic edge

Chinese Library Classification

  • O175.27
  • O175.29
  • O351.3

2000 Mathematics Subject Classification

  • 35L65
  • 35J70
  • 35R35