Smooth and discontinuous junctions in the p-system and in the 3x3 Euler system

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a = a(x). We analyze the mathematical problem related to a junction, i.e., a sharp discontinuity in the pipe’s geometry, we consider the case of a picewise constant pipe’s section and then, the smooth case. In particular, through a limit procedure, we prove the well posedness of the smooth case from the discontinuous one and also the opposite case for the full 3×3 Euler system. Then, all the basic analytical properties of the equations governing a fluid flowing in a duct with varying section are extended to the Euler system. In both cases of the p-system and the Euler system, a key assumption is the boundedness of the total variation of the pipe’s section. We provide explicit examples to show that this bound is necessary.