Calculus of Variations and Geometric Measure Theory

S. Bianchini - E. Marconi

On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension

created by marconi on 13 Aug 2016
modified on 07 Oct 2019


Published Paper

Inserted: 13 aug 2016
Last Updated: 7 oct 2019

Journal: Arch. Ration. Mech. Anal.
Year: 2017


We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics.
In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f''=0$.

We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.