Skip to main content
SpringerLink
Log in

Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Considered herein is a modified two-component periodic Camassa–Holm system with peakons. The local well-posedness and low regularity result of solutions are established. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail and the exact blow-up rate is also obtained.

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. Alber M.S., Camassa R., Holm D.D., Marsden J.E.: On the link between umbilic geodesics and soliton solutions of nonlinear PDEs. Proc. R. Soc. Lond. Ser. A 450, 677–692 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beals R., Sattinger D.H., Szmigielski J.: Multi-peakons and a theorem of Stieltjes. Inv. Probl. 15, 1–4 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Bona J.L., Smith R.: The initial value problem for the Korteweg-de Vries equation. Phil. Trans. R. Soc. Lond. A 278, 555–601 (1975)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Coclite G.M., Karlsen K.H.: On the well-posdeness of the Degasperis–Procesi equation. J. Funct. Anal. 233, 60–91 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Constantin A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ. Equ. 141, 218–235 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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)

    MATH  MathSciNet  Google Scholar 

  11. Constantin A.: On the blow-up of solutions of a periodic shallow water equation. J. Nonlinear Sci. 10, 391–399 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Constantin A., Escher J.: On the structure of a family of quasilinear equations arising in shallow water theory. Math. Ann. 312, 403–416 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Constantin A., Escher J.: Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Commun Pure Appl. Math. 51, 475–504 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Constantin A., Escher J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Constantin A., Johnson R.S.: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 40, 175–211 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Constantin A., Strauss W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Degasperis, A., Procesi, M.: Asymptotic integrability. Symmetry and Perturbation Theory (Rome, 1998), vol. 23. World Scientific Publishing, River Edge (1999)

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

    MATH  MathSciNet  Google Scholar 

  26. Escher J., Liu Y., Yin Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fu Y., Qu C.Z.: Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons. J. Math. Phys. 50, 012906 (2009)

    Article  MathSciNet  Google Scholar 

  28. Fuchssteiner B., Fokas A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/1982)

    Article  MathSciNet  Google Scholar 

  29. Holden H., Raynaud X.: Periodic conservative solutions of the Camassa–Holm equation. Ann. Inst. Fourier 58, 945–988 (2008)

    MATH  MathSciNet  Google Scholar 

  30. Holm D.D., Náraigh L.Ó., Tronci C.: Singular solutions of a modified two-component Camassa–Holm equation. Phys. Rev. E 79, 016601 (2009)

    Article  MathSciNet  Google Scholar 

  31. Holm D.D., Marsden J.E., Ratiu T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Iorio R.J., de Magãlhaes Iorio V.: Fourier Analysis and Partial Differential Equations. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  33. Johnson R.S.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Johnson R.S.: The Camassa–Holm equation for water waves moving over a shear flow. Fluid Dyn. Res. 33, 97–111 (2003)

    Article  MATH  Google Scholar 

  35. 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, pp. 25–70. Springer, Berlin (1975)

  36. Kenig C., Ponce G., Vega L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  38. Kouranbaeva S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Lakshmanan, M.: Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Kundu, A. (ed.) Tsunami and Nonliner Waves, pp. 31–49. Springer, Berlin (2007)

  40. Lenells J.: Stability of periodic peakons. Int. Math. Res. Not. 10, 485–499 (2004)

    Article  MathSciNet  Google Scholar 

  41. Li Y., Olver P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier-Villars, Paris (1969)

  43. Liu Y., Yin Z.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  44. Mckean H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 867–874 (1998)

    MATH  MathSciNet  Google Scholar 

  45. Misiolek G.: Classical solutions of the periodic Camassa–Holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  46. Shabat, A., Martínez Alonso, L.: On the prolongation of a hierarchy of hydrodynamic chains. In: Proceedings of the NATO advanced research workshop, Cadiz, Spain 2002, NATO Science Series, pp. 263–280. Kluwer Academic Publishers, Dordrecht (2004)

  47. Segur H.: Waves in shallow water, with emphasis on the tsunami of 2004. In: Kundu, A. (eds) Tsunami and Nonlinear Waves, pp. 3–29. Springer, Berlin (2007)

    Chapter  Google Scholar 

  48. Tao, T.: Low-regularity global solutions to nonlinear dispersive equations. Surveys in Analysis and Operator Theory, Canberra, 2001. In: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, pp. 19–48. Austral. Nat. Univ., Canberra (2002)

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

    MATH  MathSciNet  Google Scholar 

  50. Wahlén E.: On the blow-up of solutions to the periodic Camassa–Holm equation. NoDEA Nonlinear Differ. Equ. Appl. 13, 643–653 (2007)

    Article  MATH  Google Scholar 

  51. Walter W.: Differential and Integral Inequalities. Springer, New York (1970)

    MATH  Google Scholar 

  52. Whitham G.B.: Linear and Nolinear Waves. John Wiley & Sons, New York (1974)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwest University, 710069, Shanxi, People’s Republic of China

    Ying Fu & Changzheng Qu

  2. Center for Nonlinear Studies, Northwest University, 710069, Shanxi, People’s Republic of China

    Ying Fu & Changzheng Qu

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

    Yue Liu

Authors
  1. Ying Fu
    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

  3. Changzheng Qu
    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

Fu, Y., Liu, Y. & Qu, C. Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons. Math. Ann. 348, 415–448 (2010). https://doi.org/10.1007/s00208-010-0483-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0483-9

Mathematics Subject Classification (2000)

Search

Navigation