Abstract
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.
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References
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)
Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. (to appear). arXiv:1311.5170
Bressan A., Constantin A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)
Bressan A., Constantin A.: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 5, 1–27 (2007)
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)
Camassa R., Holm L.: An integrable shallow–water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Camassa R., Holm L., Hyman J.M.: A new integrable shallow–water equation. Adv. Appl. Mech. 31, 1–33 (1994)
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)
Constantin A.: Geometrical methods in hydrodynamics. J. Équations Dérivées Partielles 2, 14 (2001)
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)
Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Pisa. 26(2), 303–328 (1998)
Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)
Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009)
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)
Dai H.-H.: Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mech. 127(1-4), 193–207 (1998)
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)
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)
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)
Himonas A.A., Misiołek G.: The Cauchy problem for an integrable shallow-water equation. Diff. Integr. Eqn. 14(7), 821–831 (2001)
Holden H., Raynaud X.: Global conservative solutions of the generalized hyperelastic-rod wave equation. J. Diff. Eqn. 233(2), 448–484 (2007)
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)
Holden H., Raynaud X.: Dissipative solutions for the Camassa–Holm equation. Discr. Cont. Dyn. Syst. 24, 1047–1112 (2009)
Liu Y.: Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 335(3), 717–735 (2006)
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)
McKean H.P.: Breakdown of the Camassa-Holm equation. Comm. Pure Appl. Math. 57(3), 416–418 (2004)
Molinet L.: On well-posedness results for Camassa-Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11(4), 521–533 (2004)
Wahlén E.: On the blow-up of solutions to the periodic Camassa–Holm equation. NoDEA 13, 643–653 (2007)
Zhou Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57(1), 137–152 (2004)
Zhou Y.: Local well-posedness and blow-up criteria of solutions for a rod equation. Math. Nachr. 278(14), 1726–1739 (2005)
Zhou Y.: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Part. Diff. Eq. 25(1), 63–77 (2006)
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Brandolese, L. Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations. Commun. Math. Phys. 330, 401–414 (2014). https://doi.org/10.1007/s00220-014-1958-4
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DOI: https://doi.org/10.1007/s00220-014-1958-4