Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2017 (2017), Article ID 3276182, 12 pages
https://doi.org/10.1155/2017/3276182
Research Article

The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States

School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong Province 264025, China

Correspondence should be addressed to Chun Shen

Received 1 June 2017; Accepted 27 July 2017; Published 29 August 2017

Academic Editor: Pavel Kurasov

Copyright © 2017 Yuhao Jiang and Chun Shen. 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. 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
  2. H. Freistühler, “Rotational degeneracy of hyperbolic systems of conservation laws,” Archive for Rational Mechanics and Analysis, vol. 113, no. 1, pp. 39–64, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  3. G.-Q. Chen, “Hyperbolic Systems of Conservation Laws with a Symmetry,” Communications in Partial Differential Equations, vol. 16, no. 8-9, pp. 1461–1487, 1991. View at Publisher · View at Google Scholar · View at Scopus
  4. N. H. Risebro and F. Weber, “A note on front tracking for the Keyfitz-Kranzer system,” Journal of Mathematical Analysis and Applications, vol. 407, no. 2, pp. 190–199, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. J. Kearsley and A. M. Reiff, “Existence of weak solutions to a class of nonstrictly hyperbolic conservation laws with non-interacting waves,” Pacific Journal of Mathematics, vol. 205, no. 1, pp. 153–170, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. 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 MathSciNet
  7. Y.-g. Lu, “Existence of global entropy solutions to general system of Keyfitz-Kranzer type,” Journal of Functional Analysis, vol. 264, no. 10, pp. 2457–2468, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Y.-g. Lu, “Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 235–240, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. H. Cheng, “Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type,” Advances in Mathematical Physics, Article ID 958120, 2013. View at Publisher · View at Google Scholar · View at Scopus
  10. H. Cheng and H. Yang, “On a nonsymmetric keyfitz-kranzer system of conservation laws with generalized and modified chaplygin gas pressure law,” Advances in Mathematical Physics, Article ID 187217, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. 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 MathSciNet · View at Scopus
  12. H. Yang and Y. Zhang, “Delta shock waves with Dirac delta function in both components for systems of conservation laws,” Journal of Differential Equations, vol. 257, no. 12, pp. 4369–4402, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. B. Temple, “Systems of conservation laws with invariant submanifolds,” Transactions of the American Mathematical Society, vol. 280, no. 2, pp. 781–795, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. B. Temple, “Systems of conservation laws with coinciding shock and rarefaction curves,” in Nonlinear Partial Differential Equations, vol. 17 of Contemporary Mathematics, pp. 143–152, American Mathematical Society, Providence, RI, USA, 1983. View at Publisher · View at Google Scholar
  15. T. Chang and L. Hsiao, “The Riemann problem and interaction of waves in gas dynamics,” in Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41 of Longman Scientific and Technical, John Wiley & Sons, Inc., New York, NY USA, 1989. View at Google Scholar
  16. T. Raja Sekhar and V. D. Sharma, “Riemann problem and elementary wave interactions in isentropic magnetogasdynamics,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 619–636, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  17. T. R. Sekhar and V. D. Sharma, “Interaction of shallow water waves,” Studies in Applied Mathematics, vol. 121, no. 1, pp. 1–25, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. M. Sun, “Interactions of elementary waves for the Aw-Rascle model,” SIAM Journal on Applied Mathematics, vol. 69, no. 6, pp. 1542–1558, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. M. Sun, “Singular solutions to the Riemann problem for a macroscopic production model,” ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 97, no. 8, pp. 916–931, 2017. View at Publisher · View at Google Scholar
  20. L. Guo, Y. Zhang, and G. Yin, “Interactions of delta shock waves for the Chaplygin gas equations with split delta functions,” Journal of Mathematical Analysis and Applications, vol. 410, no. 1, pp. 190–201, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. G. Lai and W. Sheng, “Elementary wave interactions to the compressible Euler equations for Chaplygin gas in two dimensions,” SIAM Journal on Applied Mathematics, vol. 76, no. 6, pp. 2218–2242, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. A. Qu and Z. Wang, “Stability of the Riemann solutions for a Chaplygin gas,” Journal of Mathematical Analysis and Applications, vol. 409, no. 1, pp. 347–361, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. C. Shen, “The Riemann problem for the Chaplygin gas equations with a source term,” ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 96, no. 6, pp. 681–695, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. L. Guo, L. Pan, and G. Yin, “The perturbed Riemann problem and delta contact discontinuity in chromatography equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 106, pp. 110–123, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  25. C. Shen, “Wave interactions and stability of the Riemann solutions for the chromatography equations,” Journal of Mathematical Analysis and Applications, vol. 365, no. 2, pp. 609–618, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. C. Shen, “The asymptotic behaviors of solutions to the perturbed Riemann problem near the singular curve for the chromatography system,” Journal of Nonlinear Mathematical Physics, vol. 22, no. 1, pp. 76–101, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. M. Sun, “Interactions of delta shock waves for the chromatography equations,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 26, no. 6, pp. 631–637, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. M. Sun, “Delta shock waves for the chromatography equations as self-similar viscosity limits,” Quarterly of Applied Mathematics, vol. 69, no. 3, pp. 425–443, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, NY, USA, 1994. View at MathSciNet