The Cauchy problem for the Novikov equation

  • Wei Yan
  • Yongsheng Li
  • Yimin ZhangEmail author


In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces \({H^s(\mathfrak{R})}\) with \({s < \frac{3}{2}}\) in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in \({H^{s}(\mathfrak{R})}\) with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in \({H^{s}(\mathfrak{R})}\) with s > 3/2.

Mathematics Subject Classification (2010)

Primary 35G25 35L05 Secondary 35R25 


Cauchy problem Novikov equation Blow-up 


  1. 1.
    Chemin, J.Y.: Localization in Fourier space and Navier–Stokes. In: Phase Space Analysis of Partial Differential Equations, CRM Series, pp. 53–136. Scuola Norm. Sup. Pisa, Pisa (2004)Google Scholar
  2. 2.
    Constantin A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Constantin A., Escher J.: Analyticity of periodic travelling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Constantin A., Escher J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Danchin R.: A note on well-posedness for Camassa–Holm equation. J. Diff. Eqns. 192, 429–444 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Himonas A.A., Misiolek G.: The Cauchy problem for an integrable shallow water equation. Diff. Int. Eqns. 14, 821–831 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hone A.N.W., Lundmark H., Szmigielski J.: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa–Holm equation. Dyn. Partial Differ. Equ. 6, 253–289 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hone, A.N.W., Wang, J.P.: Integrable peakon equations with cubic nonlinearity. J. Phys. Appl. Math. Theor. 41 (2008), 372002Google Scholar
  11. 11.
    Kato, T.: Quasi-linear equations with applications to partial differential equations. Lect. Notes Math., vol. 448. Springer, Berlin (1975)Google Scholar
  12. 12.
    Kato T., Ponce G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ni L., Zhou Y.: Well-posedness and persistence properties for the Novikov equation. J. Diff. Eqns. 250, 3002–3021 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Novikov, V.S.: Generalizations of the Camassa–Holm equation. J. Phys. A 42 (2009), 342002Google Scholar
  15. 15.
    Jiang Z.H., Ni L.D.: Blow-up phenomenon for the integrable Novikov equation. J. Math. Appl. Anal. 385, 551–558 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.Department of MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China
  3. 3.Wuhan Institute of Physics and MathematicsChinese Academy of SciencesWuhanPeople’s Republic of China

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