Global weak solutions to the weakly dissipative Degasperis-Procesi equation
type: Item
instance of: scholarly article
title: Global weak solutions to the weakly dissipative Degasperis-Procesi equation
zbMATH Open document ID: 1222.35064
author: YunXi Guo
author: Shaoyong Lai
author: Ying Wang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-07-19T00:00:00Z
review text: The authors consider the Degasperis-Procesi equation with a dissipative term, \[ (1 - \partial_x^2) u_t + 4 u u_x - 3 u_x u_{xx} - u u_{xxx} + \lambda (1 - \partial_x^2) u = 0, \] where \(\lambda > 0\) is the dissipation constant. Wave breaking and decay of the solutions were analyzed for this equation by \textit{S. Wu} and \textit{Z. Yin} [SIAM J. Math. Anal. 40, No. 2, 475--490 (2008; Zbl 1216.35126)]. Local well-posedness, blow-up scenario, global existence of strong solutions were investigated by \textit{Z. Guo} [J. Differ. Equations 246, No. 11, 4332--4344 (2009; Zbl 1170.35083)]. In the present article, the authors consider the global weak solutions to the dissipative Degasperis-Procesi equation in the energy space \(H^1\). The technique is inspired by similar works of \textit{A. Constantin} and \textit{L. Molinet} [Commun. Math. Phys. 211, No. 1, 45--61 (2000; Zbl 1002.35101)] and \textit{J. Escher, Y. Liu} and \textit{Z. Yin} [J. Funct. Anal. 241, No. 2, 457--485 (2006; Zbl 1126.35053)] in the context of the Camassa-Holm and conservative Degasperis-Procesi equations, respectively. After details of analysis, the authors also compare their results with the outcomes of the recent work of \textit{Z. Yin, S. Lai} and \textit{Y. Guo} [Comput. Math. Appl. 60, No. 9, 2645--2652 (2010; Zbl 1205.76049)] for a model shallow-water equation.
Mathematics Subject Classification ID: 35G25
zbMATH DE Number: 5925791
zbMATH Keywords: well-posedness
zbMATH Keywords: global weak solutions
zbMATH Keywords: Degasperis-Procesi equation
zbMATH Keywords: Camassa-Holm equation
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
full work available at URL: j.na.2011.04.051
OpenAlex ID: W2006203686
cites work: Q4508328
cites work: On Solutions to a Two-Component Generalized Camassa-Holm Equation
cites work: Existence of permanent and breaking waves for a shallow water equation: a geometric approach
cites work: On the inverse spectral problem for the Camassa-Holm equation
cites work: Global weak solutions for a shallow water equation
cites work: On the scattering problem for the Camassa-Holm equation
cites work: Q4253188
cites work: Wave breaking for a periodic shallow water equation.
cites work: Wave breaking for a shallow water equation
cites work: Persistence properties and unique continuation of solutions of the Camassa-Holm equation
cites work: On the Cauchy problem for the Camassa-Holm equation
cites work: Global solutions and blow-up phenomena to a shallow water equation
cites work: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation
cites work: Wave Breaking and Persistence Properties for the Dispersive Rod Equation
cites work: Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation
cites work: Stability of solitary waves for a rod equation
cites work: On the Cauchy problem for an integrable equation with peakon solutions
cites work: Stability of peakons for the Degasperis-Procesi equation
cites work: On the well-posedness of the Degasperis-Procesi equation
cites work: Multi-peakon solutions of the Degasperis–Procesi equation
cites work: Periodic and solitary-wave solutions of the Degasperis-Procesi equation
cites work: Global weak solutions and blow-up structure for the Degasperis-Procesi equation
cites work: Infinite propagation speed for the Degasperis-Procesi equation
cites work: Blow-up phenomenon for the integrable Degasperis-Procesi equation
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 solutions to the Degasperis-Procesi equation
cites work: On a two-component Degasperis-Procesi shallow water system
cites work: Wave breaking and propagation speed for a class of nonlocal dispersive \(\theta \)-equations
cites work: On solutions to the Holm–Staleyb-family of equations
cites work: Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation
cites work: Blow-Up and Decay of the Solution of the Weakly Dissipative Degasperis–Procesi Equation
cites work: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation
cites work: Q5536399
cites work: Q4865486
cites work: Global existence of weak solutions for a shallow water equation
DOI: 10.1016/J.NA.2011.04.051
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description: scientific article; zbMATH DE number 5925791