Wave-breaking phenomena for a weakly dissipative shallow water equation

Abstract

Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms \(u^{2n+1}\) and \(u^{2m}u_{xx}\), the equation loses the conservation law \(E=\int _{{\mathbb {S}}} (u^2+u^2_x)\mathrm{d}x.\) This difficulty has been dealt with by establishing the energy inequality.

This is a preview of subscription content, log in to check access.

References

  1. 1.

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

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

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

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. 266, 6954–6897 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Brandolese, L.: Local-in-space criteria for blowup in shallow water and dispersive rod equations. Commun. Math. Phys. 330, 401–414 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

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

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    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  Article  Google Scholar 

  7. 7.

    Chen, M., Guo, F., Liu, Y., Qu, C.: Analysis on the blow-up of solutions to a class of integrable peakon equations. J. Funct. Anal. 270, 2343–2374 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Google Scholar 

  11. 11.

    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  Article  Google Scholar 

  12. 12.

    Constantin, A., Ivanov, R.I., Lenells, J.: Inverse scattering trandform for the Gegasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Constantin, A., Johnson, R.: The dynamics of waves interacting with the equatorial undervurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Constantin, A., Kolev, H.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

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

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

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

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

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

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Escher, J., Liu, Y., Yin, Z.: Global weak solutions and blow-up structure for the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–486 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

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

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Gui, G., Liu, Y., Sun, J.: A nonlocal shallow-water model arising from the full water waves with the coriolis effect. J. Math. Fluid Mech. 21, 27 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

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

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Himonas, A.A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Hu, T., Liu, Y.: On the modeling of equational shallow-water waves with the Coriolis effect. Phys. D 391, 87–110 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mckean, H.P.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. LVII, 0416–0418 (2004)

    MathSciNet  Article  Google Scholar 

  26. 26.

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

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

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

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Misołek, G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ming, S., Lai, S., Su, Y.: The cauchy problem of a weakly dissipative shallow water equation. Appl. Anal. 98, 1387–1402 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Novruzova, E., Hagverdiyevb, A.: On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient. J. Differ. Equ. 257, 4525–4541 (2014)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ott, E., Sudan, R.: Damping of solitary waves. Phys. Fluids 13, 1432–1434 (1970)

    Article  Google Scholar 

  33. 33.

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

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Yin, Z.: On the Cauchy problem for an integrable equation with peakon solutions. Illinois J. Math. 47, 649–666 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Yin, Z.: On the blow-up of solutions of the periodic Camassa–Holm equation. Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal. 12, 375–381 (2005)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Wu, X., Yin, Z.: Well-posedness and global existence for the Novikov equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 707–727 (2012)

    MathSciNet  MATH  Google Scholar 

  37. 37.

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

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A Math. Theor. 44, 055202 (2011). 17pp

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Wu, S., Yin, Z.: Global existence and blow-up phenomena for the weakly dissipative Camassa–Holm equation. J. Differ. Equ. 246, 4309–4321 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Lai, S., Wu, Y.: Local well-posedness and weak solution of the dissipative Camassa–Holm equation. Sci. Sin. Math. 40, 901–920 (2010). Chiness

    Article  Google Scholar 

  41. 41.

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

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Xin, Z., Zhang, P.: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Commun. Partial Differ. Equ. 27, 1815–1844 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Zhu, M., Wang, Y.: Blow-up of solutions to the periodic generalized modified Camassa–Holm equation with varying linear dispersion. Discrete Cont. Dyn. S. 37, 645–661 (2017)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The work of Zhu is partially supported by the NSF of China under the grand 11401309. The work of Wang is partially supported by ZYGX2015J096 and NSF of China under the ground 11571063.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Min Zhu.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Zhu, M., Wang, Y. Wave-breaking phenomena for a weakly dissipative shallow water equation. Z. Angew. Math. Phys. 71, 96 (2020). https://doi.org/10.1007/s00033-020-01317-5

Download citation

Keywords

  • A weakly dissipative shallow water
  • Blow up
  • Wave breaking

Mathematics Subject Classification

  • 35B44
  • 35G25