Abstract and Applied Analysis

On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System

Yujuan Wang, Yongduan Song, and Hamid Reza Karimi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm shallow water system after wave breaking was studied in this paper. The underlying approach is based on a skillfully defined characteristic and a set of newly introduced variables which transform the original system into a Lagrangian semilinear system. It is the transformation, together with the associated properties, that allows for the continuity of the solution beyond collision time to be established, leading to a uniquely global dissipative solution, which constructs a semigroup, and the multipeakon dissipative solution.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 348695, 16 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412278524

Digital Object Identifier
doi:10.1155/2014/348695

Mathematical Reviews number (MathSciNet)
MR3208530

Zentralblatt MATH identifier
07022199

Citation

Wang, Yujuan; Song, Yongduan; Karimi, Hamid Reza. On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 348695, 16 pages. doi:10.1155/2014/348695. https://projecteuclid.org/euclid.aaa/1412278524


Export citation

References

  • R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
  • A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.
  • A. Constantin, “The Hamiltonian structure of the Camassa-Holm equation,” Expositiones Mathematicae, vol. 15, no. 1, pp. 53–85, 1997.
  • A. Constantin, “On the scattering problem for the Camassa-Holm equation,” Proceedings of the the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, vol. 457, no. 2008, pp. 953–970, 2001.
  • A. Constantin, “Global existence of solutions and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.
  • A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009.
  • A. Bressan and A. Constantin, “Global conservative solutions of the Camassa–Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007.
  • H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation–-a lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10, pp. 1511–1549, 2007.
  • H. Holden and X. Raynaud, “Global conservative multipeakon solutions of the Camassa-Holm equation,” Journal of Hyperbolic Differential Equations, vol. 4, no. 1, pp. 39–64, 2007.
  • A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007.
  • H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 24, no. 4, pp. 1047–1112, 2009.
  • H. Holden and X. Raynaud, “Global dissipative multipeakon solutions of the Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 33, no. 11, pp. 2040–2063, 2008.
  • A. Constantin and R. Ivanov, “On an integrable two-component Camassa–Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.
  • M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006.
  • G. Falqui, “On a Camassa–Holm type equation with two dependent variables,” Journal of Physics A: Mathematical and General, vol. 39, no. 2, pp. 327–342, 2006.
  • P. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, Article ID 1900, 1996.
  • R. S. Johnson, “Camassa–Holm, Korteweg–de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.
  • G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa–Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011.
  • J. Escher, O. Lechtenfeld, and Z. Y. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 493–513, 2007.
  • Y. Wang and Y. Song, “Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system,” Boundary Value Problems, vol. 2013, article 165, 2013.
  • C. Guan and Z. Y. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010.
  • G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa–Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.
  • C. Guan and Z. Yin, “Global weak solutions for a two-component Camassa–Holm shallow water system,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1132–1154, 2011.
  • Y. Wang and Y. Song, “On the global existence of dissipative solutions for the modified coupled Camassa–Holm system,” Soft Computing, vol. 17, no. 1, pp. 2007–2019, 2013.
  • Z. Shen, Y. Wang, H. Karimi, and Y. Song, “On the multipeakon dissipative behavior of the modiied coupled Camassa-Holm model for shallow water system,” Mathematical Problems in Engineering, vol. 2013, Article ID 107450, 11 pages, 2013. \endinput