Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces

Lorenzo Brandolese (1)

1 ICJ - Institut Camille Jordan

We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa--Holm equation: a wave breaking will occur as soon as the initial data decay faster at infinity than the solitons. In the case of data decaying slower than solitons we provide persistence results for the solution in weighted $L^p$-spaces, for a large class of moderate weights. Explicit asymptotic profiles illustrate the optimality of these results.

Mots clés: Moderate weight; Unique continuation; Blowup; Persistence; Water wave equation; Shallow water

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brandolese. Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces. International Mathematics Research Notices, 2012, 22, pp.5161-5181. ⟨10.1093/imrn/rnr218⟩. ⟨hal-00666127⟩