Well-posedness and blow up phenomena for the periodic generalized Camassa-Holm equation
type: Item
instance of: scholarly article
title: Well-posedness and blow up phenomena for the periodic generalized Camassa-Holm equation
zbMATH Open document ID: 1072.35057
author: Zhaoyang Yin
published in: Communications on Pure and Applied Analysis
publication date: 2005-02-28T00:00:00Z
review text: The Cauchy problem for the periodic generalized Camassa-Holm equation \[ \begin{cases} u_t - u_{xx} + ku_x + [g(u)/2]_x = 2u_x u_{xx} + uu_{xxx} &t > 0, \quad x \in \mathbb{R}, \\ u({0,x}) = u_0(x), & x \in \mathbb{R}, \\ u({t,x + 1}) = u({t,x}), & t \geq 0, \quad x \in \mathbb{R}\end{cases} \] is considered in the paper. Here \(g:\mathbb{R} \to \mathbb{R}\) is a given \(C^m\)-function, \(m \geq 2\) , and \(k \in \mathbb{R}\) is a fixed constant. For \(g(u) = 3u^2\) and \(k \geq 0\) the periodic generalized equation becomes the periodic Camassa-Holm equation, modelling the unidirectional propagation of shallow water waves over a flat bottom, \(u\left( {t,x} \right)\) standing for the fluid velocity at time \(t > 0\) in the spatial \(x\) direction and \(k\) being a nonnegative parameter related to the critical shallow water speed. The Camassa-Holm equation has a bi-Hamiltonian structure and is completely integrable. For \(k = 0,\) the periodic Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group of the circle, fact that can be used to show that it satisfies the least action principle. That equation has global solutions and also solutions which blow up in finite time. The aim of the paper is to establish the local well-posedness of strong solutions to generalized Camassa-Holm equation for a large class of initial data, to give the precise blow up scenario, and to show that the equation has solutions which blow up, provided their initial data satisfy certain conditions. The paper is organized as follows. In Section 2, the author proves the local well-posedness of the initial value problem associated with the periodic generalized Camassa-Holm equation. In Section 3, the author gives an exact blow up scenario and gets an explosion result of strong solution to the equation.
reviewed by: Leonid B. Chubarov
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 2139415
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: strong solutions
zbMATH Keywords: local well-posedness
zbMATH Keywords: blow up scenario
zbMATH Keywords: periodic boundary condition
zbMATH Keywords: shallow water waves
zbMATH Keywords: bi-Hamiltonian structure
zbMATH Keywords: completely integrable
MaRDI profile type: Publication
full work available at URL: cpaa.2004.3.501
OpenAlex ID: W2048588535
DOI: 10.3934/CPAA.2004.3.501
description: scientific article; zbMATH DE number 2139415