Sharp well-posedness and ill-posedness of a higher-order modified Camassa-Holm equation.
type: Item
instance of: scholarly article
title: Sharp well-posedness and ill-posedness of a higher-order modified Camassa-Holm equation.
zbMATH Open document ID: 1274.35342
author: Wei Yan
author: Yongsheng Li
author: Shiming Li
published in: Differential and Integral Equations
publication date: 2013-03-21T00:00:00Z
review text: The well-posedness and ill-posedness of the following modified Camassa-Holm equation \[ \partial _t u + \partial _x^{2n+1}u + \frac {1}{2} \partial _x( u^2)+(\text{id} - \partial _x^2)^{-1} \partial _x \left [ u^2 + \frac {1}{2}( \partial _x u)^2 \right ] = 0, \quad (t,x)\in (0,\infty )\times \mathbb {R}, \] with initial condition \(u_0\) are studied for integers \(n\geq 2\). It is shown that, given \(u_0\in H^{-n+(5/4)}(\mathbb {R})\) small enough, the above Cauchy problem is well-posed, thereby extending the well-posedness in \(H^s(\mathbb {R})\) established in [\textit{Y. S. Li, W. Yan} and \textit{X. Y. Yang}, J. Evol. Equ. 10, No. 2, 465--486 (2010; Zbl 1239.35141)] for \(s>-n+(5/4)\). It is also proved that, if \(s<-n+(5/4)\), the map \(u_0\longmapsto [t\mapsto u(t)]\) associating the solution to the modified Camassa-Holm equation to its initial condition cannot be \(C^2\)-smooth at zero as a map from \(\dot {H}^s(\mathbb {R})\) to \(C([0,T];\dot {H}^s(\mathbb {R}))\).
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 37L50
zbMATH DE Number: 6148067
zbMATH Keywords: well-posedness
zbMATH Keywords: modified Camassa-Holm equation
zbMATH Keywords: ill-posedness
zbMATH Keywords: dyadic bilinear estimate
reviewed by: Philippe Laurençot
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6148067