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

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.