Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations
By Lorenzo Brandolese. Communications in Mathematical Physics, 2012
144 citations recorded by the source corpus.
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 u0′+|u0|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_0'+|u_0|}$$\end{document} is strictly negative in at least one point x0∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x_0 \in \mathbb{R}}$$\end{document}. 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.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/defc0c449108a6a72584b3de28233ff87e6d4740
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In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Analysis on the blow-up of solutions to a class of integrable peakon equations (2016, background): “[4] L. Brandolese and M. F. Cortez, On permanent and breaking waves in hyperelastic rods and rings, J. Funct.”
- Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion (2015, background): “It is noticed that a new type of blow-up criteria in the sense of the so called “local-in-space” is recently established in [2, 3, 4], which illustrates that the blowup condition is merely imposed on a small neighborhood of a single point in space variable, and hence local perturbation of data aroun”