Source-solutions and asymptotic behavior in conservation laws
type: Item
instance of: scholarly article
title: Source-solutions and asymptotic behavior in conservation laws
zbMATH Open document ID: 0545.35057
DOI: 10.1016/0022-0396(84)90096-2
author: Tai-Ping Liu
author: Michel Pierre
published in: Journal of Differential Equations
publication date: 1984-01-01T00:00:00Z
review text: The authors consider a scalar conservation law \(u_ t+\phi(u)_ x=0\) in a single space variable. The initial data is taken to be a finite measure \(\mu\) ; hence the initial condition is stated in the weak form \(\lim_{t\to 0}<\psi(\cdot),u(\cdot,t)>=<\psi(\cdot),d\mu>,\) for all bounded continuous \(\psi\). The motivation for this study comes from the fact that the case where \(\mu\) is the Dirac measure is relevant to the study of the asymptotic behavior of u when \(u(\cdot,0)\) is taken to be an integrable function. The results are different depending on whether u is non-negative, or not, and whether \(\phi\) is odd or convex. In fact, if u is negative somewhere, the problem is not well posed in general, for data which are measures. These differences do not occur if the data is a bounded integrable function. This is an interesting paper having lots of new ideas and it opens up a new area for study.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 3867838
zbMATH Keywords: scalar conservation law
zbMATH Keywords: weak form
zbMATH Keywords: Dirac measure
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: not well posed
zbMATH Keywords: bounded integrable function
MaRDI profile type: Publication
full work available at URL: 0022-0396(84)90096-2
OpenAlex ID: W2000690999
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description: scientific article; zbMATH DE number 3867838