Skip to main content
SpringerLink
Log in

Energy conservation and well-posedness of the Camassa–Holm equation in Sobolev spaces

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we study the Cauchy problem for the Camassa–Holm equation in Sobolev spaces. Firstly, we establish the energy conservation for weak solutions of the Camassa–Holm equation in \(H^{1}(\mathbb {R})\cap B^{1}_{3,2}(\mathbb {R})\) and prove that every weak solution in \(H^{\frac{7}{6}}(\mathbb {R})\) is unique by the embedding \(H^{\frac{7}{6}}(\mathbb {R})\hookrightarrow B^{1}_{3,2}(\mathbb {R}).\) Then, we obtain the local well-posedness for the Camassa–Holm equation in \(W^{2,1}(\mathbb {R}).\) It is worth noting that \(B^{2}_{1,1}(\mathbb {R})\hookrightarrow W^{2,1}(\mathbb {R})\) and the Camassa–Holm equation is well-posed in \(B^{2}_{1,1}(\mathbb {R})\) and is ill-posed in \(W^{1+\frac{1}{p},p}(\mathbb {R})~(1<p\le \infty )\) by the pervious papers. Our result implies that \(W^{2,1}(\mathbb {R})\) is the critical Sobolev spaces for the well-posedness of the Camassa–Holm equation.

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

Data availability

Data sharing and materials are not applicable to this article as no new data and materials were created or analyzed in this study.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa-Holm equation. Inverse Problems 22, 2197–2207 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. El Dika, K., Molinet, L.: Stability of multipeakons. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1517–1532 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Guo, Y., Ye, W., Yin, Z.: Ill-posedness for the Cauchy problem of the Camassa-Holm equation in \(B_{\infty ,1} ^1(\mathbb{R} )\). J. Differ. Eq. 327, 127–144 (2022)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Ye, W., Yin, Z., Guo, Y.: The well-posedness for the Camassa-Holm type equations in critical Besov spaces \({B}^{1+1/p}_{p,1}\) with \(1\le p<+\infty \). J. Differ. Eq. 367, 729–748 (2023)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Li, J., Wu, X., Yu, Y., Zhu, W.: Non-uniform dependence on initial data for the Camassa-Holm equation in the critical besov space. J. Math. Fluid Mech. 23, 36 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, J., Yu, Y., Zhu, W.: Ill-posedness for the Camassa-Holm and related equations in Besov spaces. J. Differ. Eq. 306, 403–417 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, J., Yu, Y., Zhu, W.: Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces. J. Differ. Eq. 269, 8686–8700 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, W., Yin, Z.: Gevrey regularity and analyticity for Camassa-Holm type systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. 18, 1061–1079 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Coclite, G.M., di Ruvo, L.: Dispersive and diffusive limits for Ostrovsky-Hunter type equations. NoDEA Nonlinear Differ. Eq. Appl. 22, 1733–1763 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Coclite, G.M., di Ruvo, L.: A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law. Discret. Contin. Dyn. Syst. 36, 2981–2990 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Coclite, G.M., di Ruvo, L.: A note on the convergence of the solution of the Novikov equation. Discret. Contin. Dyn. Syst. Ser. B 24, 2865–2899 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Coclite, G.M., di Ruvo, L.: On \(H^2\)-solutions for a Camassa-Holm type equation. Open Math. 21, 20220577 (2023)

    Article  MATH  Google Scholar 

  21. Hu, Q., Yin, Z.: Well-posedness and blow-up phenomena for a periodic two-component Camassa-Holm equation. Proc. Roy. Soc. Edinb. Sect. A 141, 93–107 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, Q., Yin, Z.: Global existence and blow-up phenomena for a periodic 2-component Camassa-Holm equation. Monatsh. Math. 165, 217–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, X.: On the periodic Cauchy problem for a coupled Camassa-Holm system with peakons. Z. Angew. Math. Phys. 67, 14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guan, C., Yin, Z.: Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. J. Differ. Eq. 248, 2003–2014 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guan, C., Yin, Z.: Global weak solutions for a two-component Camassa-Holm shallow water system. J. Funct. Anal. 260, 1132–1154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tan, W., Yin, Z.: Global conservative solutions of a modified two-component Camassa-Holm water system. J. Differ. Eq. 251, 3558–3582 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tan, W., Yin, Z.: Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation. J. Funct. Anal. 261, 1204–1226 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cai, H., Chen, G., Shen, Y.: Lipschitz metric for conservative solutions of the two-component Camassa-Holm system. Z. Angew. Math. Phys. 68, 12 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Holden, H., Raynaud, X.: Global conservative multipeakon solutions of the Camassa-Holm equation. J. Hyperbolic Differ. Equ. 4, 39–64 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Bressan, A., Chen, G., Zhang, Q.: Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete Contin. Dyn. Syst. 35, 25–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Cheskidov, A., Constantin, P., Friedlander, S., Shvydkoy, R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, 1233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Constantin, P., Weinan, E., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. 165, 207–209 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Linares, F., Ponce, G., Sideris, T.: Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons. In Asymptotic analysis for nonlinear dispersive and wave equations, pp. 197–246. Mathematical Society of Japan (2019)

  39. Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011)

Download references

Acknowledgements

We would like to express our gratitude to the editorial board, reviewers and funding institutions mentioned above.

Funding

Guo was supported by the National Natural Science Foundation of China (No. 12161004), the Basic and Applied Basic Research Foundation of Guangdong Province (No. 2020A1515111092) and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano-Optoelectronic Technology (No. 2020B1212030010). Ye was supported by the general project of Natural Science Foundation of Guangdong Province (No. 2021A1515010296).

Author information

Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, China

    Yingying Guo

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Weikui Ye

Authors
  1. Yingying Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weikui Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both Guo and Ye designed research and performed research. Guo wrote the main manuscript text and Ye checked the manuscript. All authors reviewed the manuscript.

Corresponding author

Correspondence to Weikui Ye.

Ethics declarations

Consent for publication

All authors approved the final manuscript and the submission to this journal.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, Y., Ye, W. Energy conservation and well-posedness of the Camassa–Holm equation in Sobolev spaces. Z. Angew. Math. Phys. 74, 184 (2023). https://doi.org/10.1007/s00033-023-02081-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-02081-y

Keywords

Mathematics Subject Classification

Search

Navigation