Skip to main content
SpringerLink
Log in

On the Cauchy problem for a modified Camassa–Holm equation

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

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. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  2. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5(01), 1–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantin, A.: The Hamiltonian structure of the Camassa–Holm equation. Expo. Math. 15(1), 53–85 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond Ser A: Math. Phys. Eng. Sci. 457, 953–970 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 26, 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chern, S.S., Tenenblat, K.: Pseudo-spherical surfaces and evolution equations. Stud. Appl. Math. 74, 55–83 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192, 429–444 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D. 4(1), 47–66 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, Z., Liu, X., Molinet, L., et al.: Ill-posedness of the Camassa-Holm and related equations in the critical space. J. Differ. Equ. 266, 1698–1707 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gorka, P., Reyes, E.G.: The modified Camassa–Holm equation. Int. Math. Res. Not. 2011, 2617–2649 (2010)

    MATH  Google Scholar 

  20. He, H., Yin, Z.: On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient. Appl. Anal. 96(4), 679–701 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luo, W., Yin, Z.: Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space. Nonlinear Anal. Theory Methods Appl. 122, 1–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, J., Yin, Z.: Well-posedness and global existence for a generalized Degasperis-Procesi equation. Nonlinear Anal. Real World Appl. 28, 72–90 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J., Yin, Z.: Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces. J. Differ. Equ. 261(11), 6125–6143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, J., Yin, Z.: Well-posedness and analytic solutions of the two-component Euler–Poincare system. Monatsh. Math. 183, 509–537 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qiao, Z.J.: The Camassa–Holm hierarchy, related \(N\)-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys. 239, 309–341 (2003)

    Article  MATH  Google Scholar 

  26. Rodríguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. Theory Methods Appl. 46, 309–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Reyes, E.G.: Geometric integrability of the Camassa–Holm equation. Lett. Math. Phys 59, 117–131 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zheng, R., Yin, Z.: The Cauchy problem for a generalized Novikov equation. Discrete Contin. Dyn. Syst. 37(6), 3503–3519 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53, 1411–1433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by NNSFC (No. 11671407), FDCT (No. 0091/2018/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A03031104). The author Qiao thanks the UT President Endowed Professorship (Project # 450000123) for its partial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Zhaonan Luo & Zhaoyang Yin

  2. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, 78539, USA

    Zhijun Qiao

  3. Faculty of Information Technology, Macau University of Science and Technology, Macau, China

    Zhaoyang Yin

Authors
  1. Zhaonan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhijun Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhaoyang Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaoyang Yin.

Additional information

Communicated by Adrian Constantin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, Z., Qiao, Z. & Yin, Z. On the Cauchy problem for a modified Camassa–Holm equation. Monatsh Math 193, 857–877 (2020). https://doi.org/10.1007/s00605-020-01426-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-020-01426-3

Keywords

Mathematics Subject Classification

Search

Navigation