  This paper is concerned with a set of novel coupling conditions for the
$3\times 3$ one-dimensional Euler system with source terms at a junction of
pipes with possibly different cross-sectional areas. Beside conservation of
mass, we require the equality of the total enthalpy at the junction and that
the specific entropy for pipes with outgoing flow equals the convex combination
of all entropies that belong to pipes with incoming flow. Previously used
coupling conditions include equality of pressure or dynamic pressure. They are
restricted to the special case of a junction having only one pipe with outgoing
flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679--702, 2015]
showed that such pressure-based coupling conditions can produce non-physical
solutions for isothermal flows through the production of mechanical energy. Our
new coupling conditions ensure energy as well as entropy conservation and also
apply to junctions connecting an arbitrary number of pipes with flexible flow
directions. We prove the existence and uniqueness of solutions to the
generalised Riemann problem at a junction in the neighbourhood of constant
stationary states which belong to the subsonic region. This provides the basis
for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for
initial data with sufficiently small total variation.

Analysis of PDEs