Energy Dissipated Across Shocks in Weak Solutions of Conservation Laws

Manipulation of conservation laws through multiplication by mappings of the unknown function result in singular terms that are densities supported on the discontinuity locus. We derive an expression for these singular terms which have been interpreted as the energy dissipated at the shocks. As an application, the expression for these singular terms is obtained from the transport equation governing the propagation of small amplitude high frequency waves in hyperbolic conservation laws. Analogous formulas are obtained for rich systems in several space dimensions.