Structure of entropy solutions to general scalar conservation laws in one space dimension

In this paper, we show that the entropy solution of a scalar conservation law is \begin{itemize} \item continuous outside a $1$-rectifiable set $\Xi$, \item up to a $\mathcal H^1$ negligible set, for each point $(\bar t,\bar x) \in \Xi$ there exists two regions where $u$ is left/right continuous in $(\bar t,\bar x)$. \end{itemize} We provide examples showing that these estimates are nearly optimal. In order to achieve these regularity results, we extend the wave representation of the wavefront approximate solutions to entropy solution. This representation can the interpreted as some sort of Lagrangian representation of the solution to the nonlinear scalar PDE, and implies a fine structure on the level sets of the entropy solution.