  We consider networks for isentropic gas and prove existence of weak solutions
for a large class of coupling conditions. First, we construct approximate
solutions by a vector-valued BGK model with a kinetic coupling function.
Introducing so-called kinetic invariant domains and using the method of
compensated compactness justifies the relaxation towards the isentropic gas
equations. We will prove that certain entropy flux inequalities for the kinetic
coupling function remain true for the traces of the macroscopic solution. These
inequalities define the macroscopic coupling condition. Our techniques are also
applicable to networks with arbitrary many junctions which may possibly contain
circles. We give several examples for coupling functions and prove
corresponding entropy flux inequalities. We prove also new existence results
for solid wall boundary conditions and pipelines with discontinuous
cross-sectional area.
