Skip to main content
SpringerLink
Log in

Non-uniform Dependence for the Novikov Equation in Besov Spaces

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the dependence on initial data of solutions to the Novikov equation. We show that the solution map is not uniformly continuous dependence on the initial data in Besov spaces \(B^s_{p,r}({\mathbb {R}}),\ s>\max \{1+\frac{1}{p},\frac{3}{2}\}\).

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, Grundlehren der Mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011)

    MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166(3), 523–535 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26, 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. (2) 173(1), 559–568 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Ivanov, R., Lenells, J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23(10), 2559–2575 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  13. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52(8), 949–982 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integral equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)

    Google Scholar 

  18. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Gaeta, G. (ed.) Symmetry and Perturbation Theory (Rome 1998), pp. 23–37. World Science Publications, River Edge, NJ (1999)

    Google Scholar 

  19. Dullin, H.R., Gottwald, G.A., Holm, D.D.: On asymptotically equivalent shallow water wave equations. Phys. D 190, 1–14 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Escher, J., Liu, Y., Yin, Z.: Global weak solutions and blow-up structure foe the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–485 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–177 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Fokas, A., Fuchssteiner, B.: Symplectic structures, their Backlund transformation and hereditary symmetries. Physica D 4(1), 47–66 (1981/82)

  23. Gui, G., Liu, Y.: On the Cauchy problem for the Degasperis–Procesi equation. Q. Appl. Math. 69, 445–464 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Himonas, A.A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  25. Himonas, A., Holliman, C.: On well-posedness of the Degasperis–Procesi equation. Discrete Contin. Dyn. Syst. A 31, 469–484 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Himonas, A., Kenig, C.: Non-uniform dependence on initial data for the CH equation on the line. Differ. Integral Equ. 22, 201–224 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Himonas, A., Misiolek, G.: Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics. Commun. Math. Phys. 296, 285–301 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Hone, A.N.W., Wang, J.: Integrable peakon equations with cubic nonlinearity. J. Phys. A Math. Theor. 41, 372002 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Kenig, C., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. 106, 617–633 (2001)

    MathSciNet  MATH  Google Scholar 

  30. Lai, S.: Global weak solutions to the Novikov equation. J. Funct. Anal. 265, 520–544 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Lenells, J.: Traveling wave solutions of the Degasperis–Procesi equation. J. Math. Anal. Appl. 306, 72–82 (2005)

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  35. Li, J., Yu, Y., Zhu, W.: Non-uniform dependence on initial data for the Euler equations in Besov spaces. arXiv:2001.03301

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

    ADS  MathSciNet  MATH  Google Scholar 

  37. Lundmark, H.: Formation and dynamics of shock waves in the Degasperis–Procesi equation. J. Nonlinear. Sci. 17, 169–198 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  38. Novikov, V.: Generalization of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46, 309–327 (2001)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Vakhnenko, V.O., Parkes, E.J.: Periodic and solitary-wave solutions of the Degasperis–Procesi equation. Chaos Solitons Fractals 20, 1059–1073 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A Math. Theor. 44, 055202 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  43. Wu, X., Yin, Z.: Well-posedness and global existence for the Novikov equation. Ann. Scuola Norm. Super. Pisa Classe di Sci. Ser. V 11, 707–727 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Wu, X., Yin, Z.: A note on the Cauchy problem of the Novikov equation. Appl. Anal. 92, 1116–1137 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  46. Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 253, 298–318 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  47. Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. 20, 1157–1169 (2013)

    MathSciNet  MATH  Google Scholar 

  48. Yin, Z.: Global existence for a new periodic integrable equation. J. Math. Anal. Appl. 49, 129–139 (2003)

    MathSciNet  MATH  Google Scholar 

  49. Yin, Z.: Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 212, 182–194 (2004)

    MathSciNet  MATH  Google Scholar 

  50. Yin, Z.: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 53, 1189–1210 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

J. Li is supported by the National Natural Science Foundation of China (Grant No. 11801090). M. Li is supported by Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ190284). W. Zhu is supported by the National Natural Science Foundation of China (Grant No. 11901092) and Natural Science Foundation of Guangdong Province (No. 2017A030310634).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Jinlu Li & Weipeng Zhu

  2. School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000, China

    Jinlu Li

  3. School of information Technology, Jiangxi University of Finance and Economics, Nanchang, 330032, China

    Min Li

Authors
  1. Jinlu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weipeng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weipeng Zhu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by A. 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

Li, J., Li, M. & Zhu, W. Non-uniform Dependence for the Novikov Equation in Besov Spaces. J. Math. Fluid Mech. 22, 50 (2020). https://doi.org/10.1007/s00021-020-00511-9

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-020-00511-9

Keywords

Mathematics Subject Classification

Search

Navigation