Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to \(N\)-waves
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to \(N\)-waves
zbMATH Open document ID: 1026.35066
DOI: 10.1016/S0022-0396(03)00058-5
published in: Journal of Differential Equations
publication date: 2003-07-30T00:00:00Z
review text: The author of this interesting paper develops a new technique to obtain better asymptotic estimates for scalar conservation laws, which are hyperbolic partial differential equations of first order, \( u_t+f(u)_x=0\). The Cauchy problem, \(u(x,0)=u_0(x)\), \(x\in \mathbb R\), \(t>0\) is considered with integrable initial value \(u_0\) defined on a compact support, \(u_0\in L^1(\mathbb R)\), supp \(u_0\subset [-L,L]\), \(L\in \mathbb R \). It is assumed that the flux \(f(u)\) is convex, \(f(0)=f^{\prime }(0)=0\), \(f^{\prime\prime }\geq 0\), and has an algebraic growth of order \(\gamma >1\) near the zero state \(u=0\) in the sense that \(\lim\limits_{u\to 0}uf^{\prime }(u)/f(u)=\gamma \), \(\gamma >1\). It is shown that under suitable conditions on the initial value, the solution of the considered equation converges to an \(N-\)wave in \(L^1\) norm with optimal convergence order of \(O(1/t)\). The technique applied here is to enclose the solution with two rarefaction waves. A uniform convergence order in the sense of graphs is shown as well. An interesting numerical example is given to illustrate the theory and considered phenomenon.
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 1957263
zbMATH Keywords: asymptotics
zbMATH Keywords: characteristics
zbMATH Keywords: rarefaction waves
zbMATH Keywords: similarity
author: Yong-Jung Kim
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957263