Communications in Mathematical Physics

, Volume 330, Issue 1, pp 401–414 | Cite as

Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations

  • Lorenzo BrandoleseEmail author


We unify a few of the best known results on wave breaking for the Camassa–Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that \({u_0'+|u_0|}\) is strictly negative in at least one point \({x_0 \in \mathbb{R}}\). Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean’s necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.


Solitary Wave Wave Breaking Shallow Water Equation Differential Inequality Blowup Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Benjamin T.B., Bona J.L., Mahony J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272(1220), 47–78 (1972)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. (to appear). arXiv:1311.5170
  3. 3.
    Bressan A., Constantin A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bressan A., Constantin A.: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 5, 1–27 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brandolese L.: Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces. Int. Math. Res. Not. IMRN 22, 5161–5181 (2012)MathSciNetGoogle Scholar
  6. 6.
    Camassa R., Holm L.: An integrable shallow–water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Camassa R., Holm L., Hyman J.M.: A new integrable shallow–water equation. Adv. Appl. Mech. 31, 1–33 (1994)CrossRefGoogle Scholar
  8. 8.
    Coclite G.M., Holden H., Karlsen K.H.: Global weak solutions to a generalized hyperelastic-rod wave equation. SIAM J. Math. Anal. 37(4), 1044–1069 (2005) (electronic)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Constantin A.: Geometrical methods in hydrodynamics. J. Équations Dérivées Partielles 2, 14 (2001)MathSciNetGoogle Scholar
  10. 10.
    Constantin A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50(2), 321–362 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Pisa. 26(2), 303–328 (1998)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Constantin A., Strauss W.A.: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A. 270(3-4), 140–148 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Dai H.-H.: Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mech. 127(1-4), 193–207 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dai H.-H., Huo Y.: Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. R. Soc. Lond. Proc. Ser A. Math. Phys. Eng. Sci. 456, 331–363 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Glass O., Sueur F.: Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations. Discrete Continuous Dyn. Syst. A 33(7), 2791–2808 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Grunert K., Holden H., Raynaud X.: Global conservative solutions of the Camassa–Holm for initial data with nonvanishing asymptotics, Discr. Cont. Dyn. Syst. 32, 4209–4227 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Himonas A.A., Misiołek G.: The Cauchy problem for an integrable shallow-water equation. Diff. Integr. Eqn. 14(7), 821–831 (2001)zbMATHGoogle Scholar
  20. 20.
    Holden H., Raynaud X.: Global conservative solutions of the generalized hyperelastic-rod wave equation. J. Diff. Eqn. 233(2), 448–484 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Holden H., Raynaud X.: Global conservative solutions of the Camassa–Holm equation. A Lagrangian point of view. Comm. Part. Diff. Eq. 32, 1–27 (2007)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Holden H., Raynaud X.: Dissipative solutions for the Camassa–Holm equation. Discr. Cont. Dyn. Syst. 24, 1047–1112 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Liu Y.: Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 335(3), 717–735 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    McKean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2(4), 867–874 (1998); Correction to “Breakdown of a shallow water equation”, Asian J. Math. 3, (1999)Google Scholar
  25. 25.
    McKean H.P.: Breakdown of the Camassa-Holm equation. Comm. Pure Appl. Math. 57(3), 416–418 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Molinet L.: On well-posedness results for Camassa-Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11(4), 521–533 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Wahlén E.: On the blow-up of solutions to the periodic Camassa–Holm equation. NoDEA 13, 643–653 (2007)CrossRefzbMATHGoogle Scholar
  28. 28.
    Zhou Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57(1), 137–152 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Zhou Y.: Local well-posedness and blow-up criteria of solutions for a rod equation. Math. Nachr. 278(14), 1726–1739 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Zhou Y.: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Part. Diff. Eq. 25(1), 63–77 (2006)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNRS UMR 5208 Institut Camille JordanUniversité de Lyon, Université Lyon 1Villeurbanne CedexFrance

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