Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation
type: Item
instance of: scholarly article
title: Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation
zbMATH Open document ID: 1180.35005
author: Xiaolian Ai
author: Guilong Gui
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-10T00:00:00Z
review text: This paper deals with the Cauchy problem of the viscous Degasperis-Procesi (VDP) equation \[ m_{t} + u\, m_{x} + 3\, u_{x}m = m_{xx}, \quad t>0, \quad x \in \mathbb{R}, \] \[ \qquad \qquad \quad \qquad m (0,x) = u_{0} (x) - u_{0,xx} (x), \quad x \in \mathbb{R}, \] where \(m = u - u_{xx}\). Based on the bilinear estimates, the local-posedness of the initial value problem associated with the VDP equation is proved. They construct the energy identity to prove the global existence theorem of the VDP equation. It is shown that there exists a unique global solution of the VDP equation. In addition, the solution is smooth on \((0,+ \infty) \times \mathbb{R}\). This is an interesting paper which is concerned with the effect of viscosity of the VDP equation.
reviewed by: Lokenath Debnath
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 5629200
zbMATH Keywords: energy identity
zbMATH Keywords: viscous Degasperis-Procesi equation
zbMATH Keywords: Picard contraction principle
zbMATH Keywords: bilinear estimate
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.07.031
OpenAlex ID: W2078293116
cites work: Global conservative solutions of the Camassa-Holm equation
cites work: Acoustic scattering and the extended Korteweg-de Vries hierarchy
cites work: An integrable shallow water equation with peaked solitons
cites work: Q4311808
cites work: Q4210480
cites work: Wellposedness for a parabolic-elliptic system
cites work: On the well-posedness of the Degasperis-Procesi equation
cites work: Finite propagation speed for the Camassa–Holm equation
cites work: On the scattering problem for the Camassa-Holm equation
cites work: Wave breaking for nonlinear nonlocal shallow water equations
cites work: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation
cites work: Stability of peakons
cites work: Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves
cites work: Q4508328
cites work: Global weak solutions and blow-up structure for the Degasperis-Procesi equation
cites work: Symplectic structures, their Bäcklund transformations and hereditary symmetries
cites work: Global existence and blow-up phenomena for the peakon $b$-family of equations
cites work: Infinite propagation speed for the Degasperis-Procesi equation
cites work: Compactly Supported Solutions of the Camassa-Holm Equation
cites work: Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs
cites work: Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE
cites work: Q4782579
cites work: Traveling wave solutions of the Degasperis-Procesi equation
cites work: Global existence and blow-up phenomena for the Degasperis-Procesi equation
cites work: On the Blow-up Phenomena for the Degasperis-Procesi Equation
cites work: Formation and dynamics of shock waves in the Degasperis-Procesi equation
cites work: Multi-peakon solutions of the Degasperis–Procesi equation
cites work: Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit
cites work: A Note on the Degasperis-Procesi Equation
cites work: On the Cauchy problem for the Camassa-Holm equation
cites work: Q3142876
cites work: On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation
cites work: On the weak solutions to a shallow water equation.
cites work: On the Cauchy problem for an integrable equation with peakon solutions
DOI: 10.1016/J.JMAA.2009.07.031
description: scientific article; zbMATH DE number 5629200