Well Posedness and Characterization of Solutions to Non Conservative Products in Non Homogeneous Fluid Dynamics Equations

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.