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MiS Preprint

Concentration estimates for entropy measures

Camillo De Lellis and Tristan Rivière


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 $\mathcal{H}^1$ 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 $\mathcal{H}^1$ almost everywhere outside this union of curves.

MSC Codes:
35D10, 35L65, 35L67, 28A75
conservation laws, entropy solutions, shocks, concentration

Related publications

2003 Repository Open Access
Camillo De Lellis and Tristan Riviére

The rectifiability of entropy measures in one space dimension

In: Journal de mathématiques pures et appliquées, 82 (2003) 10, pp. 1343-1367