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Curvature Blow-up for the Higher-Order Camassa–Holm Equations

  • Changzheng Qu
  • Ying FuEmail author
Article
  • 25 Downloads

Abstract

This paper is devoted to understanding how higher-order nonlinearities affect the dispersive dynamics. As a prototype, a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation is studied. The local well-posedness of the Cauchy problem in Besov spaces and Sobolev spaces is established. Furthermore, a delicate analysis is employed to investigate the formation of singularities, and some sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solutions are provided.

Keywords

Higher-order Camassa–Holm equation Peaked solitary wave Curvature blow-up Local well-posedness Wave-breaking 

Mathematics Subject Classification.

Primary: 35B30 35G25 70H07 

Notes

Acknowledgements

The work of Qu is supported by the National Science Foundation of China grant-11631007 and grant-11971251. The work of Fu is supported by the National Science Foundation of China grant-11471259 and grant-11631007, and the National Science Basic Research Program of Shaanxi (Program No. 2019JM-007).

References

  1. 1.
    Brandolese, L.: Local-in-space criteria for blowup in shallow water and dispersive rod equations. Commun. Math. Phys. 330, 401–414 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. 266, 6954–6987 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chemin, J.Y.: Localization in Fourier space and Navier–Stokes system. In: Proceedings of Phase Space Analysis of Partial Differential Equations 2004, CRM series, Pisa, pp. 53–136 (2004)Google Scholar
  8. 8.
    Chen, R.M., Guo, F., Liu, Y., Qu, C.Z.: Analysis on the blow-up of solutions to a class of integrable peakon equations. J. Funct. Anal. 270, 2343–2374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, R.M., Liu, Y., Qu, C.Z., Zhang, S.H.: Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion. Adv. Math. 272, 225–251 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa 26, 303–328 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integr. Equ. 14, 953–988 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Danchin, R.: Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2005)Google Scholar
  17. 17.
    Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192, 429–444 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Degasperis, A., Procesi, M.: Asymptotic integrability, Symmetry and Perturbation Theory (Rome, 1998), 23. World Scientific Publishing, River Edge, NJ (1999)zbMATHGoogle Scholar
  19. 19.
    Fokas, A.S.: On a class of physically important integrable equations. Phys. D 87, 145–150 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fu, Y., Gui, G.L., Liu, Y., Qu, C.Z.: On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity. J. Differ. Equ. 255, 1905–1938 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Phys. D 95, 229–243 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/1982)Google Scholar
  23. 23.
    Gui, G.L., Liu, Y., Olver, P.J., Qu, C.Z.: Wave-breaking and peakons for a modified Camassa–Holm equation. Commun. Math. Phys. 319, 731–759 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Himonas, A., Misiolek, G.: The Cauchy problem for an integrable shallow water equation. Differ. Integr. Equ. 14, 821–831 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Himonas, A., Mantzavinos, D.: The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation. Nonlinear Anal. 95, 499–529 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Himonas, A., Mantzavinos, D.: An \(ab\)-family of equations with peakon traveling waves. Proc. Am. Math. Soc. 144, 3797–3811 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Himonas, A., Mantzavinos, D.: The Cauchy problem for a 4-parameter family of equations with peakon traveling waves. Nonlinear Anal. 133, 161–199 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Holden, H., Raynaud, X.: Periodic conservative solutions of the Camassa–Holm equation. Annales de l’institut Fourier 58, 945–988 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jiang, Z.H., Ni, L.D.: Blow-up phenomenon for the integrable Novikov equation. J. Math. Anal. Appl. 385, 551–558 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math. 27, 337–405 (1974)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kang, J., Liu, X.C., Olver, P.J., Qu, C.Z.: Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy. J. Nonlinear Sci. 26, 141–170 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. 39, 422–442 (1895)CrossRefzbMATHGoogle Scholar
  33. 33.
    Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, T.P.: Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations. J. Differ. Equ. 33, 92–111 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Liu, Y., Olver, P.J., Qu, C.Z., Zhang, S.H.: On the blow-up of solutions to the integrable modified Camassa–Holm equation. Anal. Appl. 12, 355–368 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liu, Y., Yin, Z.Y.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lundmark, H.: Formation and dynamics of shock waves in the Degasperis–Procesi equation. J. Nonlinear Sci. 17, 169–198 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mckean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 867–874 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Novikov, V.: Generalizations of the Camassa–Holm equation. J. Phys. A: Math. Theor. 42, 342002 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Qu, C.Z., Fu, Y., Liu, Y.: Well-posedness, wave breaking and peakons for a modified \(\mu \)-Camassa–Holm equation. J. Funct. Anal. 266, 433–477 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Qu, C.Z., Fu, Y., Liu, Y.: Blow-up solutions and peakons to a generalized \(\mu \)-Camassa–Holm integrable equation. Commun. Math. Phys. 331, 375–416 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Recio, E., Anco, S.C.: A general family of multi-peakon equations and their properties. J. Phys. A: Math. Theor. 52, 125203 (2019)CrossRefGoogle Scholar
  44. 44.
    Tan, W.K., Yin, Z.Y.: Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation. J. Funct. Anal. 261, 1204–1226 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Tiğlay, F.: The periodic Cauchy problem for Novikov’s equation. Int. Math. Res. Not. 2011, 4633–4648 (2011)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1980)zbMATHGoogle Scholar
  47. 47.
    Zhou, Y.: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Partial Differ. Equ. 25, 63–77 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Nonlinear StudiesNingbo UniversityNingboPeople’s Republic of China
  2. 2.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

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