**Rinaldo M. Colombo** and **Francesca Marcellini**

*Smooth and discontinuous junctions in the p-system and in the 3 × 3 Euler system*

**Pages:** 55-69

**Received:** 23 September 2010

**Accepted:** 5 November 2010

**Mathematics Subject Classification (2000):** 35L65, 76N10.

**Keywords:** Conservation laws at junctions, nozzle flow, coupling conditions at junctions.

**Abstract:**
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.

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