Abstract and Applied Analysis

On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System

Xiuming Li

Full-text: Open access

Abstract

We establish the local well-posedness for the viscous two-component Camassa-Holm system. Moreover, applying the energy identity, we obtain a globalexistence result for the system with ( u 0 , η 0 ) H 1 ( ) × L 2 ( ) .

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 327572, 15 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174040

Digital Object Identifier
doi:10.1155/2012/327572

Mathematical Reviews number (MathSciNet)
MR2922905

Zentralblatt MATH identifier
1237.35008

Citation

Li, Xiuming. On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 327572, 15 pages. doi:10.1155/2012/327572. https://projecteuclid.org/euclid.aaa/1365174040


Export citation

References

  • A. Constantin and R. I. Ivanov, “On an integrable two-component Camassa-Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.
  • P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900–1906, 1996.
  • R. Ivanov, “Two-component integrable systems modelling shallow water waves: the constant vorticity case,” Wave Motion, vol. 46, no. 6, pp. 389–396, 2009.
  • A. Constantin and W. Strauss, “Exact steady periodic water waves with vorticity,” Communications on Pure and Applied Mathematics, vol. 57, no. 4, pp. 481–527, 2004.
  • R. S. Johnson, “Nonlinear gravity waves on the surface of an arbitrary shear flow with variable depth,” in Nonlinear Instability Analysis, vol. 12 of Adv. Fluid Mech., pp. 221–243, Comput. Mech., Southampton, UK, 1997.
  • R. S. Johnson, “On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer),” Geophysical and Astrophysical Fluid Dynamics, vol. 57, no. 1–4, pp. 115–133, 1991.
  • P. Popivanov and A. Slavova, Nonlinear Waves, vol. 4 of Series on Analysis, Applications and Computation, World Scientific, Hackensack, NJ, USA, 2011.
  • 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.
  • D. D. Holm, L. Nraigh, and C. Tronci, “Singular solutions of a modified two-component Camassa-Holm equation,” Physical Review E, vol. 85, no. 1, Article ID 016601, 5 pages, 2012.
  • J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems. Series A, vol. 19, no. 3, pp. 493–513, 2007.
  • 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.
  • 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.
  • W. K. Lim, “Global well-posedness for the viscous Camassa-Holm equation,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 432–442, 2007.