On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation
type: Item
instance of: scholarly article
title: On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation
zbMATH Open document ID: 1133.35028
published in: Journal of Differential Equations
publication date: 2007-03-12T00:00:00Z
review text: The authors prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation \[ \partial_t u-\partial_{txx}^{3}u+4u\partial_x u=3\partial_{x}u\partial_{xx}^{2}u+u\partial_{xxx}^{3}u. \] In a recent paper, the same authors proved for this equation the existence and uniqueness of \(L^1\cap BV\) weak solutions satisfying an infinite family of Kružkov-type entropy inequalities. The purpose of this paper is to replace the Kružkov-type entropy inequalities by an Oleǐnik-type estimate and to prove the uniquess via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Degasperis-Procesi equation is admissible only if it jumps down in value.
reviewed by: Meng Wang
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 5132871
zbMATH Keywords: shallow water equation
zbMATH Keywords: integrable equation
zbMATH Keywords: entropy condition
zbMATH Keywords: Kružkov-type entropy inequalities
zbMATH Keywords: Oleǐnik-type estimate
zbMATH Keywords: nonlocal adjoint problem
author: Giuseppe Maria Coclite
author: Kenneth H. Karlsen
MaRDI profile type: Publication
full work available at URL: j.jde.2006.11.008
OpenAlex ID: W2105014493
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DOI: 10.1016/J.JDE.2006.11.008
description: scientific article; zbMATH DE number 5132871