Existence of Solutions to Hyperbolic Conservation Laws with a Source

Abstract:

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].