Advertisement

Archive for Rational Mechanics and Analysis

, Volume 193, Issue 3, pp 623–657 | Cite as

Interaction of Rarefaction Waves of the Two-Dimensional Self-Similar Euler Equations

  • Jiequan LiEmail author
  • Yuxi Zheng
Article

Abstract

We construct classical self-similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic Euler equations in two space dimensions. The binary interaction represents a major type of interaction in the two-dimensional Riemann problems, and includes in particular the classical problem of the expansion of a wedge of gas into vacuum. Based on the hodograph transformation, the method employed here involves the phase space analysis of a second-order equation and the inversion back to (or development onto) the physical space.

Keywords

Rarefaction Wave Riemann Problem Simple Wave Euler System Riemann Invariant 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-dor G., Glass I.I.: Domains and boundaries of non-stationary oblique shock wave reflection, 1. J. Fluid Mech. 92, 459–496 (1979)ADSCrossRefGoogle Scholar
  2. 2.
    Ben-dor G., Glass I.I.: Domains and boundaries of non-stationary oblique shock wave reflection, 2. J. Fluid Mech. 96, 735–756 (1980)ADSCrossRefGoogle Scholar
  3. 3.
    Chang T., Chen G.Q., Yang S.L.: On the 2-D Riemann problem for the compressible Euler equations. I. Interaction of shock waves and rarefaction waves. Disc. Cont. Dyn. Syst. 1(4), 555–584 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen S.X., Xin Z.P., Yin H.C.: Global shock waves for the supersonic flow past a perturbed cone. Commun. Math. Phys. 228((1), 47–84 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Courant R., Friedrichs K.O.: Supersonic Flow and Shock Waves. Interscience Publishers, Inc., New York (1948)zbMATHGoogle Scholar
  6. 6.
    Dafermos C.: Hyperbolic Conservation Laws in Continuum Physics (Grundlehren der mathematischen Wissenschaften). Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Dai Z., Zhang T.: Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch. Ration. Mech. Anal. 155, 277–298 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Glaz H.M., Colella P., Glass I.I., Deschambault R.L.: A numerical study of oblique shock-wave reflections with experimental comparisons. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 398, 117–140 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    Glimm, G., Ji, X., Li, J., Li, X., Zhang, P., Zhang, T., Zheng, Y.: Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations. Preprint, submitted (2007)Google Scholar
  10. 10.
    Lax P.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. X, 537–566 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lax P., Liu X.: Solutions of two-dimensional Riemann problem of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319–340 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Levine L.E.: The expansion of a wedge of gas into a vacuum. Proc. Camb. Philol. Soc. 64, 1151–1163 (1968)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Li J.Q.: Global solution of an initial-value problem for two-dimensional compressible Euler equations. J. Differ. Equ. 179(1), 178–194 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li J.Q.: On the two-dimensional gas expansion for compressible Euler equations. SIAM J. Appl. Math. 62, 831–852 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, J.Q., Zhang, T., Yang, S.L.: The two-dimensional Riemann problem in gas dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 98. Addison Wesley Longman limited, Reading, 1998Google Scholar
  16. 16.
    Li J.Q., Zhang T., Zheng Y.X.: Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commun. Math. Phys. 267, 1–12 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, J.Q., Zheng, Y.X.: Interaction of bi-symmetric rarefaction waves of the two-dimensional Euler equations. (submitted) (2008)Google Scholar
  18. 18.
    Li T.T.: Global Classical Solutions for Quasilinear Hyperbolic Systems. Wiley, New York (1994)zbMATHGoogle Scholar
  19. 19.
    Li T.T., Yu W.C.: Boundary Value Problem for Quasilinear Hyperbolic Systems. Duke University, USA (1985)Google Scholar
  20. 20.
    Mackie A.G.: Two-dimensional quasi-stationary flows in gas dynamics. Proc. Camb. Philol. Soc. 64, 1099–1108 (1968)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Majda A., Thomann E.: Multi-dimensional shock fronts for second order wave equations. Comm. PDE. 12(7), 777–828 (1987)CrossRefzbMATHGoogle Scholar
  22. 22.
    Pogodin I.A., Suchkov V.A., Ianenko N.N.: On the traveling waves of gas dynamic equations. J. Appl. Math. Mech. 22, 256–267 (1958)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schulz-Rinne C.W., Collins J.P., Glaz H.M.: Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 4(6), 1394–1414 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Smoller J.: Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, Heidelberg (1994)CrossRefzbMATHGoogle Scholar
  25. 25.
    Suchkov V.A.: Flow into a vacuum along an oblique wall. J. Appl. Math. Mech. 27, 1132–1134 (1963)CrossRefGoogle Scholar
  26. 26.
    Wang, R., Wu, Z.: On mixed initial boundary value problem for quasilinear hyperbolic system of partial differential equations in two independent variables (in Chinese). Acta Sci. Nat. Jinlin Univ. 459–502 (1963)Google Scholar
  27. 27.
    Zhang T., Zheng Y.X.: Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21, 593–630 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zheng, Y.X.: Systems of Conservation Laws: Two-Dimensional Riemann Problems, vol. 38. PNLDE, Birkhäuser, Boston, 2001Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Mathematical ScienceCapital Normal UniversityBeijingChina
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations