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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 187217, 14 pages
http://dx.doi.org/10.1155/2013/187217
Research Article

On a Nonsymmetric Keyfitz-Kranzer System of Conservation Laws with Generalized and Modified Chaplygin Gas Pressure Law

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 2 June 2013; Revised 20 October 2013; Accepted 25 October 2013

Academic Editor: Stephen C. Anco

Copyright © 2013 Hongjun Cheng and Hanchun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. C. Kranzer and B. L. Keyfitz, “A strictly hyperbolic system of conservation laws admitting singular shocks,” in Nonlinear Evolution Equations that Change Type, vol. 27, pp. 107–125, Springer, New York, NY, USA, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. LeFloch, “An existence and uniqueness result for two nonstrictly hyperbolic systems,” in Nonlinear Evolution Equations that Change Type, vol. 27, pp. 126–138, Springer, New York, NY, USA, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. C. Tan and T. Zhang, “Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I. Four-J cases, II,” Journal of Differential Equations, vol. 111, no. 2, pp. 203–254, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  4. D. C. Tan, T. Zhang, and Y. X. Zheng, “Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws,” Journal of Differential Equations, vol. 112, no. 1, pp. 1–32, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Li, T. Zhang, and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, vol. 98 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, 1998. View at MathSciNet
  6. H. Yang, “Riemann problems for a class of coupled hyperbolic systems of conservation laws,” Journal of Differential Equations, vol. 159, no. 2, pp. 447–484, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G.-Q. Chen and H. Liu, “Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,” SIAM Journal on Mathematical Analysis, vol. 34, no. 4, pp. 925–938, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  8. G.-Q. Chen and H. Liu, “Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,” Physica D, vol. 189, no. 1-2, pp. 141–165, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. V. M. Shelkovich, “The Riemann problem admitting δ-, δ'-shocks, and vacuum states (the vanishing viscosity approach),” Journal of Differential Equations, vol. 231, no. 2, pp. 459–500, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. L. Guo, W. Sheng, and T. Zhang, “The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,” Communications on Pure and Applied Analysis, vol. 9, no. 2, pp. 431–458, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Cheng and H. Yang, “Delta shock waves in chromatography equations,” Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 475–485, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. Cheng and H. Yang, “Riemann problem for the isentropic relativistic Chaplygin Euler equations,” Zeitschrift für Angewandte Mathematik und Physik, vol. 63, no. 3, pp. 429–440, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Yang and Y. Zhang, “New developments of delta shock waves and its applications in systems of conservation laws,” Journal of Differential Equations, vol. 252, no. 11, pp. 5951–5993, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. L. Keyfitz and H. C. Kranzer, “A system of nonstrictly hyperbolic conservation laws arising in elasticity theory,” Archive for Rational Mechanics and Analysis, vol. 72, no. 3, pp. 219–241, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  15. Y.-g. Lu, “Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type,” Journal of Functional Analysis, vol. 261, no. 10, pp. 2797–2815, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Aw and M. Rascle, “Resurrection of “second order” models of traffic flow,” SIAM Journal on Applied Mathematics, vol. 60, no. 3, pp. 916–938, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. D. Lax, “Hyperbolic systems of conservation laws. II,” Communications on Pure and Applied Mathematics, vol. 10, pp. 537–566, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, vol. 11 of CBMS Regional Conference Series in Applied Mathematics, Society for industrial and Applied Mathematics, Philadelphia, 1973. View at Publisher · View at Google Scholar
  19. H. Cheng, “Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type,” Advances in Mathematical Physics, vol. 2013, Article ID 958120, 10 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H. Nessyahu and E. Tadmor, “Nonoscillatory central differencing for hyperbolic conservation laws,” Journal of Computational Physics, vol. 87, no. 2, pp. 408–463, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  21. S. A. Chaplygin, “On gas jets,” Scientific Memoirs, Moscow University Mathematic Physics, vol. 21, no. 1063, pp. 1–121, 1904.
  22. H.-S. Tsien, “Two-dimensional subsonic flow of compressible fluids,” Journal of the Aeronautical Sciences, vol. 6, pp. 399–407, 1939. View at Zentralblatt MATH · View at MathSciNet
  23. M. Bento, O. Bertolami, and A. Sen, “Generalized Chaplygin gas, accelerated expansion and dark energy-matter unification,” Physical Review D, vol. 66, Article ID 043507, 5 pages, 2002. View at Publisher · View at Google Scholar
  24. H. Benaoum, “Accelerated universe from modified Chaplygin gas and tachyonic fluid,” High Energy Physics, 11 pages, 2002, http://arxiv.org/abs/hep-th/0205140.
  25. H. B. Benaoum, “Modified Chaplygin gas cosmology,” Advances in High Energy Physics, Article ID 357802, 12 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  26. Y. Brenier, “Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations,” Journal of Mathematical Fluid Mechanics, vol. 7, no. suppl. 3, pp. S326–S331, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet