Skip to main content
Log in

Generalised Riemann problem for Euler system

  • Reviews
  • Progress of Projects Supported by NSFC
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves (shock, rarefaction wave and contact discontinuity) is also introduced.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Alinhac S. Existence d’ondes de rarefaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm Partial Differential Equations, 1989, 14: 173–230

    Article  MathSciNet  MATH  Google Scholar 

  2. Artola M, Majda A. Nonlinear development of instabilities in supersonic vortex sheets I: The basic kink modes. Phys D, 1987, 28: 253–281

    Article  MathSciNet  MATH  Google Scholar 

  3. Artola M, Majda A. Nonlinear development of instabilities in supersonic vortex sheets II: Resonant interaction among kink modes. SIAM J Appl Math, 1989, 49: 1310–1349

    Article  MathSciNet  MATH  Google Scholar 

  4. Bui A T, Li D. The double shock front solutions for hyperbolic conservation laws in multi-dimensional space. Trans Amer Math Soc, 1989, 316: 233–250

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang T, Chen G, Yang S. On the 2-D Riemann problem for the compressible Euler equations, I: Interaction of shocks and rarefaction waves. Discrete Contin Dyn Syst, 1995, 1: 555–584

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang T, Chen G, Yang S. On the 2-D Riemann problem for the compressible Euler equations, II: Interaction of contact discontinuities. Discrete Contin Dyn Syst, 2000, 6: 419–430

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen S X. Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary. Front Math China, 2007, 2: 87–102

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen S X. Study of multidimensional systems of conservation laws: Problems, difficulties and progress. In: Proceedings of the International Congress of Mathematicians, vol. 3. New Delhi: Hindustan Book Agency, 2010, 1884–1900

    Google Scholar 

  9. Chen S X. Mult-dimensional nonlinear systems of conservation laws (in Chinese). Sci Sin Math, 2013, 43: 317–332

    Article  Google Scholar 

  10. Chen S X, Li D. Cauchy problem with general discontinuous initial data along a smooth curve for 2-D Euler system. J Differential Equations, 2015, 257: 1939–1988

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen S X, Qu A F. Two-dimensional Riemann problems for Chaplygin gas. SIAM J Math Anal, 2012, 44: 2146–2178

    Article  MathSciNet  MATH  Google Scholar 

  12. Coulombel J-F, Secchi P. The stability of compressible vortex sheets in two-space dimension. Indiana Univ Math J, 2004, 53: 941–1012

    Article  MathSciNet  MATH  Google Scholar 

  13. Coulombel J-F, Secchi P. Nonlinear compressible vortex sheets in two space dimensions. Ann Sciéc Norm Supér, 2008, 41: 85–139

    Article  MathSciNet  MATH  Google Scholar 

  14. Coulombel J-F, Secchi P. Uniqueness of 2-D compressible vortex sheets. Comm Pure Appl Anal, 2009, 8: 1439–1450

    Article  MathSciNet  MATH  Google Scholar 

  15. Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Springer-Verlag, 1949

    MATH  Google Scholar 

  16. Dafermos C. Hyperbolic Conservation Laws in Coninuum Physics. New York: Springer, 2010

    Book  MATH  Google Scholar 

  17. Douglis A. Some existence theorems for hyperbolic systems of partial differential equations intwo independent variables. Comm Pure Appl Math, 1952, 5: 119–154

    Article  MathSciNet  MATH  Google Scholar 

  18. Francheteau J, Metivier G. Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels. C R Math Acad Sci Ser I, October 1998, 327: 725–728

    MATH  Google Scholar 

  19. Friedrichs K O. Nonlinear hyperbolic differential equations for functions of two independent variables. Amer J Math, 1948, 70: 555–589

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedrichs K O. Symmetric positive linear differential equations. Comm Pure Appl Math, 1958, 11: 333–418

    Article  MathSciNet  MATH  Google Scholar 

  21. Gu C H. Differentiable solutions of symmetric positive partial differential equations. Chin Math Acta, 1964, 5: 541–555

    MathSciNet  MATH  Google Scholar 

  22. Gu C H, Lee D, Hou Z. The Cauchy problem of quasi-linear hyperbolic system with discontinuous initial values I. Acta Math Sinica, 1961, 11: 314–323

    Google Scholar 

  23. Gu C H, Lee D, Hou Z. The Cauchy problem of quasi-linear hyperbolic system with discontinuous initial values II. Acta Math Sinica, 1961, 11: 324–327

    Google Scholar 

  24. Gu C H, Lee D, Hou Z. The Cauchy problem of quasi-linear hyperbolic system with discontinuous initial values III. Acta Math Sinica, 1962, 12: 132–143

    Google Scholar 

  25. Hamilton R S. The inverse function theorem of Nash and Moser. Bull Amer Math Soc, 1982, 7: 65–222

    Article  MathSciNet  MATH  Google Scholar 

  26. Hormander L. On the Nash-Moser implicit function theorem. Ann Acad Sci Fenn Math, 1985, 10: 255–259

    Article  MathSciNet  MATH  Google Scholar 

  27. Klainerman S. Long-time behavior of solutions to nonlinear evolution equations. Arch Ration Mech Anal, 1982, 78: 73–98

    Article  MathSciNet  MATH  Google Scholar 

  28. Kreiss H-O. Initial boundary value problems for hyperbolic systems. Comm Pure Appl Math, 1970, 23: 277–298

    Article  MathSciNet  MATH  Google Scholar 

  29. Lax P D. Hyperbolic systems of conservation laws. Comm Pure Appl Math, 1957, 10: 537–566

    Article  MathSciNet  MATH  Google Scholar 

  30. Lax P D, Phillips R S. Local boundary conditions for dissipative symmetric linear differential operators. Comm Pure Appl Math, 1960, 13: 427–455

    Article  MathSciNet  MATH  Google Scholar 

  31. Lee D T, Yu W T. Some existence theorems for quasi-linear hyperbolic systems of partial differential equations in two independent variables, I: Typical boundary value problems. Sci Sinica, 1964, 13: 529–549

    MathSciNet  MATH  Google Scholar 

  32. Lee D T, Yu W T. Some existence theorems for quasi-linear hyperbolic systems of partial differential equations in two independent variables, II: Typical boundary value problems of functional formal and typical free boundary problems. Sci Sinica, 1964, 13: 551–562

    MathSciNet  MATH  Google Scholar 

  33. Lee D T, Yu W T. Some existence theorems for quasi-linear hyperbolic systems of partial differential equations in two independent variables, III: General boundary vaue problems and general free boundary problems. Sci Sinica, 1965, 14: 1065–1067

    MathSciNet  Google Scholar 

  34. Lee D T, Yu W T. Boundary value problem for the first-order quasi-linear hyperbolic systems and their applications. J Differential Equations, 1981, 41: 1–26

    Article  MathSciNet  MATH  Google Scholar 

  35. Li D. Rarefaction and shock waves for multi-dimensional hyperbolic conservation laws. Comm Partial Differential Equations, 1991, 16: 425–450

    Article  MathSciNet  MATH  Google Scholar 

  36. Li D. Compatibility of jump Cauchy data for non-isentropic Euler equations. J Math Anal Appl, 2015, 425: 565–587

    Article  MathSciNet  MATH  Google Scholar 

  37. Li D. Initial-boundary value problem for non-isentropic Euler equations with incompatible data. Http://www.math. wvu.edu/~li/research/pub/33 ibvp-shock.pdf

  38. Li J. On the two-dimensional gas expansion for compressible Euler equations. SIAM J Appl Math, 2001, 62: 831–852

    Article  MathSciNet  MATH  Google Scholar 

  39. Li J, Zheng Y. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch Rat Mech Anal, 2009, 193: 523–657

    Article  MathSciNet  Google Scholar 

  40. Li T T, Yu W C. Boundary Value Problem for Quasi-Linear Hyperbolic Systems. Durham: Duke University Press, 1985

    Google Scholar 

  41. Majda A. The existence and stability of multi-dimensional shock front. Bull Amer Math Soc (NS), 1981, 4: 342–344

    Article  MATH  Google Scholar 

  42. Majda A. The stability of multi-dimensional shock front. Mem Amer Math Soc, 1983, 275: 1–96

    Google Scholar 

  43. Majda A, Osher S. Initial-Boundary value problems for hyperbolic equations with uniformly characteristic boundary. Comm Pure Appl Math, 1975, 28: 607–675

    Article  MathSciNet  MATH  Google Scholar 

  44. Majda A, Thomann E. Multidimensional shock fronts for second order wave equations. Comm Partial Differential Equations, 1990, 15: 983–1028

    Article  MathSciNet  Google Scholar 

  45. Metivier G. Interaction de deux chocs pour un systeme de deux lois de conservation en dimension deux d’espace. Trans Amer Math Soc, 1986, 296: 431–479

    Article  MathSciNet  MATH  Google Scholar 

  46. Metivier G. Ondes soniques. J Math Pure Appl, 1991, 70: 197–268

    MathSciNet  MATH  Google Scholar 

  47. Nirenberg L. Variational and topological methods in nonlinear problems. Bull Amer Math Soc (NS), 1981, 4: 267–302

    Article  MathSciNet  MATH  Google Scholar 

  48. Rauch J, Massey F. Differentiability of solutions to hyperbolic initial-boundary value problems. Trans Amer Math Soc, 1974, 189: 303–318

    MathSciNet  MATH  Google Scholar 

  49. Smoller J. Shock Waves and Reaction-Diffusion Equations. NewYork: Springer-Verlag, 1983

    Book  MATH  Google Scholar 

  50. Zhang T, Zheng Y. Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems. SIAM J Math Anal, 1990, 21: 593–630

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11031001, 11101101 and 11421061).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to DeNing Li.

Additional information

Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, S., Li, D. Generalised Riemann problem for Euler system. Sci. China Math. 60, 581–592 (2017). https://doi.org/10.1007/s11425-016-0437-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-0437-x

Keywords

MSC(2010)

Navigation