Existence of solutions to hyperbolic conservation laws with a source

Existence of solutions to three different systems of Eqs. (1.1), (1.2) and (1.3) coming from physically relevant models is shown, each needing a different proof which are given in Sects. 2, 3 and 4. The unifying theme is the presence of source terms and the general method of proof is vanishing viscosity together with compensated compactness, For system (1.2) entropy-entropy flux pairs of Lax type are constructed and estimates from singular perturbation theory of ODEs are used. For (1.1) and (1.3) weak entropy entropy flux pairs are constructed following the compensated compactness framework set up by Diperna [4].