Concentration estimates for entropy measures

Camillo De Lellis and Tristan Rivière

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Submission date: 06. May. 2003 (revised version: May 2003)
Pages: 22
published in: Journal de mathématiques pures et appliquées, 82 (2003) 10, p. 1343-1367 
DOI number (of the published article): 10.1016/S0021-7824(03)00061-8
Bibtex
with the following different title: The rectifiability of entropy measures in one space dimension
MSC-Numbers: 35D10, 35L65, 35L67, 28A75
Keywords and phrases: conservation laws, entropy solutions, shocks, concentration
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Abstract:
We show that entropy solutions to 1 dimensional scalar conservation laws for totally nonlinear fluxes and for arbitrary measurable bounded data have a structure similar to the one of BV maps without being always BV. The singular set -shock waves- of such solutions is contained in a countable union of curves and almost everywhere along these curves the solution has left and right approximate limits. The entropy production is concentrated on the shock waves and can be explicitly computed in terms of the approximate limits. The solution is approximately continuous almost everywhere outside this union of curves.