Calculus of Variations and Geometric Measure Theory

C. De Lellis - T. Rivière

Concentration estimates for entropy measures

created on 06 May 2003
modified by delellis on 03 May 2011


Published Paper

Inserted: 6 may 2003
Last Updated: 3 may 2011

Journal: Journal de Mathématiques Pures et Appliquées
Volume: 82
Pages: 1343-1367
Year: 2003


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 $C^1$ curves and $\haus^1$ 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 $\haus^1$ almost everywhere outside this union of curves.

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Keywords: conservation laws, entropy solutions, concentration, structure