  We consider relaxation systems of transport equations with heterogeneous
source terms and with boundary conditions, which limits are scalar conservation
laws. Classical bounds fail in this context and in particular BV estimates.
They are the most standard and simplest way to prove compactness and
convergence. We provide a novel and simple method to obtain partial BV
regularity and strong compactness in this framework. The standard notion of
entropy is not convenient either and we also indicate another, but closely
related, notion. We give two examples motivated by renal flows which consist of
2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more
general.

Analysis of PDEs