Skip to main content
SpringerLink
Log in

On the Cauchy problem for the two-component Camassa–Holm system

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper we establish the local well-posedness for the two-component Camassa–Holm system in a range of the Besov spaces. We also derive a wave-breaking mechanism for strong solutions. In addition, we determine the exact blow-up rate of such solutions to the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aratyn, H., Gomes, J.F., Zimerman, A.H.: On a negative flow of the AKNS hierarchy and its relation to a two-Component Camassa–Holm equation. Symmetry Integrability Geom. Methods Appl. 2, (2006), paper 070, 12 pp.

  2. Bony J.M.: Calcul symbolique et propagation des singularités pour les q́uations aux drivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14(4), 209–246 (1981)

    MathSciNet  MATH  Google Scholar 

  3. Bressan A., Constantin A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bressan A., Constantin A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen M., Liu S., Zhang Y.: A 2-Component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75, 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chemin, J.-Y.: Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations. Proceedings, CRM series, Pisa, pp. 53–136 (2004)

  8. Chemin J.-Y.: Perfect Incompressible Fluids. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  9. Constantin A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin A., Escher J.: Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 44, 423–431 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Constantin A., Ivanov R.I.: On an integrable two-component Camassa Holm shallow water system. Phys. Lett. A 372, 7129–7132 (2008)

    Article  MathSciNet  Google Scholar 

  14. Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis-Procesi equations. Arch. Rantion. Mech. Anal. 192, 165–186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Constantin A., McKean H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)

    Article  MathSciNet  Google Scholar 

  17. Danchin R.: A few remarks on the Camassa–Holm equation. Differ. Integr. Equ. 14, 953–988 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Danchin, R.: Fourier Analysis Methods for PDEs. Lecture Notes, 14 November (2005)

  19. Holm D.D., Tronci C.: Geodesic ows on semidirect-product Lie groups: geometry of singular measure-valued solutions. Proc. R. Soc. London Ser. A 465, 457–476 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Escher J., Lechtenfeld O., Yin Z.: Well-posedness and blow-up phenomena for the 2-Component Camassa–Holm equation. Discret. Contin. Dyn. Syst. 19, 493–513 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Falqui G.: On a Camassa–Holm type equation with two dependent variables. J. Phys. A Math. Gen. 39, 327–342 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fokas A., Fuchssteiner B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47–66 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math., vol. 448. Springer, Berlin, pp. 25–70 (1975)

  24. Kolev B.: Poisson brackets in hydrodynamics. Discrete Contin. Dyn. Syst. 19, 555–574 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Popowwicz Z.: A 2-component or N = 2 supersymmetric Camassa–Holm equation. Phys. Lett. A 354, 110–114 (2006)

    Article  MathSciNet  Google Scholar 

  26. Shabat, A., Martínez Alonso, L.: On the prolongation of a hierarchy of hydrodynamic chains. In: Shabat, A.B. et al. (eds.) New trends in integrability and partial solvability. Proceedings of the NATO advanced research workshop, Cadiz, Spain 2002, NATO Science Series, Kluwer Academic Publishers, Dordrecht, pp. 263–280 (2004)

  27. Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Triebel H.: Theory of Function Spaces. Monograph in mathematics, vol. 78. Birkhauser Verlag, Basel (1983)

    Google Scholar 

  29. Whitham G.B.: Linear and nonlinear waves. Wiley, New York (1980)

    Google Scholar 

  30. Yin Z.: On the Cauchy problem for an integrable equation with peakonsolutions. Illinois J. Math. 47, 649–666 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, 212013, Zhenjiang, Jiangsu, China

    Guilong Gui

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    Guilong Gui

  3. Department of Mathematics, University of Texas, Arlington, TX, 76019, USA

    Yue Liu

Authors
  1. Guilong Gui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yue Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yue Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gui, G., Liu, Y. On the Cauchy problem for the two-component Camassa–Holm system. Math. Z. 268, 45–66 (2011). https://doi.org/10.1007/s00209-009-0660-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-009-0660-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation