  This article is concerned with the study of weak solutions of a linear
transport equation on a bounded domain with coupled boundary data for general
non smooth space and time dependent velocity fields. The existence of
solutions, its uniqueness and the continuous dependence of the solution on the
initial and boundary data as well on the velocity is proven. The results are
based on the renormalization property. At the end, the theory is shown to be
applicable to the continuity equation on a network.
