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

Lorenzo Brandolese (1)

1 ICJ - Institut Camille Jordan

We unify a few of the best know results on wave breaking for the Camassa--Holm equation (by A.~Constantin, J.~Escher, R. Camassa, L. Holm, J. Hyman and others) in a single theorem: a breakdown will occur as soon as $u_0'+|u_0|$ is strictly negative in at least one point $x_0\in\R$. 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. Contrarily to McKean's necessary and sufficient condition for blowup, our approach easily extends 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.

Mots clés: Camassa--Holm; rod equation; blowup; wave breaking; nonlinear waves; dispersive waves

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brandolese. Local-in-space criteria for blowup in shallow water and dispersive rod equations. 2012. ⟨hal-00746853v1⟩