Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2014, Article ID 354349, 11 pages
http://dx.doi.org/10.1155/2014/354349
Research Article

Delta Shock Wave for the Suliciu Relaxation System

1School of Mathematics and Statistics, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia
2Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia

Received 13 March 2014; Accepted 27 May 2014; Published 18 June 2014

Academic Editor: Manuel De León

Copyright © 2014 Richard De la cruz et al. 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. F. Bouchut and S. Boyaval, “A new model for shallow viscoelastic fluids,” Mathematical Models and Methods in Applied Sciences, vol. 23, no. 8, pp. 1479–1526, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Y.-G. Lu, C. Klingenberg, L. Rendon, and D.-Y. Zheng, “Global solutions for a simplified shallow elastic fluids model,” Abstract and Applied Analysis, vol. 2014, Article ID 920248, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  3. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  4. G. Carbou, B. Hanouzet, and R. Natalini, “Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation,” Journal of Differential Equations, vol. 246, no. 1, pp. 291–319, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. I. Suliciu, “On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure,” International Journal of Engineering Science, vol. 28, no. 8, pp. 829–841, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. Baiti and A. Bressan, “The semigroup generated by a Temple class system with large data,” Differential and Integral Equations, vol. 10, no. 3, pp. 401–418, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Bianchini, “The semigroup generated by a Temple class system with non-convex flux function,” Differential and Integral Equations, vol. 13, no. 10-12, pp. 1529–1550, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Bianchini, “Stability of L solutions for hyperbolic systems with coinciding shocks and rarefactions,” SIAM Journal on Mathematical Analysis, vol. 33, no. 4, pp. 959–981, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, vol. 20 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, UK, 2000. View at MathSciNet
  10. A. Bressan and P. Goatin, “Stability of L solutions of Temple class systems,” Differential and Integral Equations, vol. 13, no. 10–12, pp. 1503–1528, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Heibig, “Existence and uniqueness of solutions for some hyperbolic systems of conservation laws,” Archive for Rational Mechanics and Analysis, vol. 126, no. 1, pp. 79–101, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y.-J. Peng, “Euler-Lagrange change of variables in conservation laws,” Nonlinearity, vol. 20, no. 8, pp. 1927–1953, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Serre, “Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation,” Journal of Differential Equations, vol. 68, no. 2, pp. 137–168, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. Chalons and F. Coquel, “Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes,” Numerische Mathematik, vol. 101, no. 3, pp. 451–478, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. G. Danilov and V. M. Shelkovich, “Delta-shock wave type solution of hyperbolic systems of conservation laws,” Quarterly of Applied Mathematics, vol. 63, no. 3, pp. 401–427, 2005. View at Google Scholar · View at MathSciNet
  16. J. Q. Li and T. Zhang, “Generalized Rankine-Hugoniot relations of deltashocks in solutions of transportation equations,” in Advance in Nonlinear PDE and Related Areas, pp. 219–232, World Scientific, Singapore, 1998. View at Google Scholar
  17. D. Serre, “Richness and the classification of quasilinear hyperbolic systems,” IMA Preprint Series 597, 1989. View at Google Scholar
  18. D. H. Wagner, “Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions,” Journal of Differential Equations, vol. 68, no. 1, pp. 118–136, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. Weinan and R. V. Kohn, “The initial value problem for measure-valued solutions of a canonical 2×2 system with linearly degenerate fields,” Communications on Pure and Applied Mathematics, vol. 44, no. 8-9, pp. 981–1000, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. T.-T. Li, Y.-J. Peng, and J. Ruiz, “Entropy solutions for linearly degenerate hyperbolic systems of rich type,” Journal de Mathématiques Pures et Appliquées, vol. 91, no. 6, pp. 553–568, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. C. Chalons and J.-F. Coulombel, “Relaxation approximation of the Euler equations,” Journal of Mathematical Analysis and Applications, vol. 348, no. 2, pp. 872–893, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. F. Coquel, E. Godlewski, and N. Seguin, “Relaxation of fluid systems,” Mathematical Models and Methods in Applied Sciences, vol. 22, no. 8, 52 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet