The rectifiability of entropy measures in one space dimension.
type: Item
instance of: scholarly article
title: The rectifiability of entropy measures in one space dimension.
zbMATH Open document ID: 1037.35044
DOI: 10.1016/S0021-7824(03)00061-8
author: Camillo De Lellis
author: Tristan Rivière
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-12-03T00:00:00Z
review text: This paper deals with entropy solutions to one-dimensional scalar conservation laws \(\partial_t u+\partial_x(f(u))= 0\) and for arbitrary measurable bounded data. The flux function \(f\in C^2\) and the set \(\{x: f''(x)= 0\}\) are assumed to be locally finite. The singular set (shock waves) of the entropy solutions is contained in a countable union of \(C^1\) curves, and \({\mathcal H}^1\) amost everywhere along these curves the solution \(u\) possesses left and right approximate limits \(u_{\pm}\). Here the symbol \({\mathcal H}^1\) denotes the one-dimensional Hausdorff measure. Moreover, it is proved that for any smooth entropy-entropy flux pair \((\eta,q)\) the entropy production is concentrated on the shock waves and can be computed in terms of \(u_{\pm}\). The solution \(u\) is approximately continuous \({\mathcal H}^1\) almost everywhere outside the above-mentioned union of curves. On the other hand, it is well known that for strictly convex fluxes the entropy solution is continuous outside the shocks.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 2012740
zbMATH Keywords: conservation laws
zbMATH Keywords: entropy solutions
zbMATH Keywords: measurable bounded data
zbMATH Keywords: concentration
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
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full work available at URL: s0021-7824(03)00061-8
OpenAlex ID: W1990492870
description: scientific article; zbMATH DE number 2012740