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

We consider weak solutions with finite entropy production to the scalar conservation law ∂tu+divxF(u)=0in(0,T)×Rd.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