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

TL;DR

This paper presents a unified, local-in-space criterion for wave breaking in the Camassa--Holm equation and extends the approach to non-integrable dispersive wave equations, using Lyapunov functions to identify blowup conditions.

Contribution

It introduces a natural, purely local blowup criterion applicable to both integrable and non-integrable equations, unifying previous results and extending them to elastic rod models.

Findings

A sufficient local-in-space condition for wave breaking in the Camassa--Holm equation.

New blowup criteria for nonlinear dispersive waves in elastic rods.

Method based on Lyapunov functions applicable beyond integrable systems.

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 $x_0$ 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.