Local-in-space criteria for blowup in shallow water and dispersive rod equations

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 $u_0^{\prime }+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. 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.

To appear on Communications in Mathematical Physics. Final draft post-refereeing

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Cited by in corpus (7)

• A nonlocal shallow-water model arising from the full water waves with the Coriolis effect
• Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations
• Stability of the train of $N$ solitary waves for the two-component Camassa-Holm shallow water system
• The Cauchy problem for fractional Camassa-Holm equation in Besov space
• Stability of peakons of the Camassa-Holm equation beyond wave breaking
• The modified Camassa-Holm equation in Lagrangian coordinates
• Dynamical behavior near self-similar blowup waves for the generalized b-equation