On the Cauchy problem for the periodic Camassa-Holm equation
type: Item
instance of: scholarly article
title: On the Cauchy problem for the periodic Camassa-Holm equation
zbMATH Open document ID: 0889.35022
author: A. Constantin
published in: Journal of Differential Equations
publication date: 1998-06-18T00:00:00Z
review text: We consider the Cauchy problem for the recently derived shallow water equation \[ u_t- u_{txx} +3uu_x =2u_xu_{xx} +uu_{xxx},\;t>0,\;x\in\mathbb{R}, \quad u(0,x) =u_0(x), \] where \(u\) describes the free surface of the water above a flat bottom. With \(m=u- u_{xx}\), the equation can be written as \[ m_t= -2mu_x- m_xu,\;t>0,\;x\in\mathbb{R}, \quad m (0)= \varphi. \tag{1} \] We look for solutions of (1) which are spatially of period 1. We will prove a local existence theorem for (1) in the Sobolev space \(H^2\) of functions with period 1, using Kato's method for abstract quasilinear equations. If the initial data \(\varphi\in H^2_p\) do not change sign \((\varphi\geq 0\) or \(\varphi\leq 0)\), we prove that (1) is globally well-posed in \(H^2_p\). We also show that for a large class of initial data \(\varphi\in H^2_p\) taking both strictly positive and strictly negative values, the solution of (1) blows up in finite time.
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35F25
zbMATH DE Number: 1109510
zbMATH Keywords: blow up
zbMATH Keywords: shallow water equation
Wikidata QID: Q58870102
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3333
OpenAlex ID: W2001206262
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DOI: 10.1006/JDEQ.1997.3333
description: scientific article; zbMATH DE number 1109510