On the structure of weak solutions to scalar conservation laws with finite entropy production

Abstract

We consider weak solutions with finite entropy production to the scalar conservation law $$\begin{aligned} \partial _t u+\mathrm {div}_x F(u)=0 \quad \text{ in } (0,T)\times \mathbb {R}^d. \end{aligned}$$ Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.