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 partial derivative(t)u + div(x) F(u) = 0 in (0, T) x R-d. 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.

Automated summary

We study weak solutions to the scalar conservation law with finite entropy production. Under suitable nonlinearity assumption, we prove that the set of non Lebesgue points of the solution has Hausdorff dimension at most d. We introduce the notion of Lagrangian representation for this class of solutions, which provides a new interpretation of the entropy dissipation measure.