Convergence rate to the nonlinear waves for viscous conservation laws on the half line
type: Item
instance of: scholarly article
title: Convergence rate to the nonlinear waves for viscous conservation laws on the half line
zbMATH Open document ID: 1204.35112
author: Itsuko Hashimoto
author: Yoshihiro Ueda
author: Shuichi Kawashima
published in: Methods and Applications of Analysis
publication date: 2010-07-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.maa/1273002799
review text: The authors in this study investigate the convergence rate of solutions to the initial-boundary value problem for scalar viscous conservation laws on the half line. Especially, they deal with the case where the Riemann problem for the corresponding hyperbolic equation admits transonic rarefaction waves. In this case, it is known that the solution tends toward a linear superposition of the stationary solution and the rarefaction wave. We show that the convergence rate is \((1+t)^{-\frac 1 2 (1-\frac 1 p)}\log ^2(2+t)\) in \(L^p\) norm \((1 \leq p < \infty)\) and \((1 + t)^{-\frac 1 2 +\varepsilon} \) in \(L^\infty\) norm if the initial perturbation from the corresponding superposition is located in \(H ^1 \cap L^1\). In this article, the proof is given by a combination of the weighted \(L^p\) energy method and the \(L^1\) estimate.
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 5761643
zbMATH Keywords: weighted energy method
zbMATH Keywords: nonlinear waves
zbMATH Keywords: Riemann problem
zbMATH Keywords: transonic rarefaction waves
reviewed by: Ömer Kavaklioglu
MaRDI profile type: Publication
DOI: 10.4310/MAA.2009.V16.N3.A7
description: scientific article; zbMATH DE number 5761643