Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.