On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation
type: Item
instance of: scholarly article
title: On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation
zbMATH Open document ID: 1178.35081
author: Ognyan Christov
author: Sevdzhan Hakkaev
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-09-10T00:00:00Z
review text: The authors make an interesting study of the periodic \(b\)-family of (1 + 1)-dimensional nonlinear partial differential equation for the fluid velocity \(u (x, t)\) in the form \[ m_t + u m_x + b u_x m = 0, \] where \(m = u - u_{xx}\) and \(b\) is a balance or bifurcation parameter. It is well known that the above equation is integrable for \(b = 2\) (the Camassa-Holm (CH) equation) and for \(b = 3\) (the Degasperis-Procesi (DP) equation). The Cauchy problem for both CH and DP equations in both periodic and non-periodic case is investigated in some detail. Both CH and DP equations are mathematical models of wave propagation on the free surface of water. In both equations, the blow-up occurs as wave breaking in nite time. In other words, the solution remains bounded but its slope becomes infinite in finite time. The major objective of this paper is to study the Cauchy problem for the \(b\)-family in the periodic case. They investigate the global existence of solution and the blow-up criteria for the periodic case. They also prove the well-posedness of the initial value problem of the \(b\)-family of equations. These nonlinear equations have solutions which exist globally in time provided the initial conditions satisfy certain conditions. Finally, they obtain a family of smooth travelling wave solutions for the DP equation and prove the non-uniform continuity of the equation. This paper contains several nice theorems and lemmas in order to describe the local well-posedness and the blow-up of the solutions in the Sobolev space \(H^s (\mathbb R)\) for \(s > \frac32\). They also prove a theorem concerning non-uniform well-posedness for the periodic Cauchy problem for the DP equation.
reviewed by: Lokenath Debnath
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35C07
zbMATH DE Number: 5602696
zbMATH Keywords: Camassa-Holm equation' wave propagation
zbMATH Keywords: free surface of water
zbMATH Keywords: non-uniform well-posedness
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.06.035
OpenAlex ID: W2017853637
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DOI: 10.1016/J.JMAA.2009.06.035
description: scientific article; zbMATH DE number 5602696