Scalar conservation laws with discontinuous flux

In order to obtain uniqueness for solutions of scalar conservation laws with discontinuous flux, Kruzhkov's entropy conditions are not enough and additional dissipation conditions have to be imposed on the discontinuity set of the flux. Understanding these conditions requires to study the structure of solutions on the discontinuity set. I will show that under quite general assumptions on the flux, solutions admit traces on the discontinuity set of the ux. This allows to show that any pair of solutions satises a Kato type inequality with an explicit reminder term concentrated on the discontinuities of the flux. Applications to uniqueness is then discussed.