Inserted: 13 aug 2016
Last Updated: 7 oct 2019
Journal: Arch. Ration. Mech. Anal.
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.