  Consider a balance law where the flux depends explicitly on the space
variable. At jump discontinuities, modeling considerations may impose the
defect in the conservation of some quantities, thus leading to non conservative
products. Below, we deduce the evolution in the smooth case from the jump
conditions at discontinuities. Moreover, the resulting framework enjoys well
posedness and solutions are uniquely characterized. These results apply, for
instance, to the flow of water in a canal with varying width and depth, as well
as to the inviscid Euler equations in pipes with varying geometry.
