An Analysis of the Riemann Problem for a 2×2 System of Keyfitz–Kranzer Type Balance Laws With a Time-Dependent Source Term

Both the paper and the model derive the generalized Rankine–Hugoniot ODEs for a delta-shock ansatz by testing against smooth functions and splitting regular and singular parts. The paper carries out the calculation explicitly from the weak formulation for the conservative system obtained via ũ = u − ∫ a, yielding dω/dt = [f1] − s[ρ] and d(ωũδ)/dt = [f2] − s[ρũ], together with the kinematic relation x′ = ũδ + ∫ a (with [·] taken as left minus right) . The model reproduces the same distributional identities and ODEs (with the opposite jump convention [·] = right − left), and correctly shows how the original balance law follows from the conservative form via u = ũ + ∫ a. Minor omissions in the model are: it does not state the kinematic relation x′ = ũδ + ∫ a explicitly, and it does not discuss the paper’s caveat on defining singular products and the role of a < 0 vs. the shadow-wave construction for a > 0 . The paper also imposes overcompressivity as an admissibility hypothesis for its main theorem, whereas the model notes it is not needed for the algebra—this is consistent with the paper’s statement of Theorem 1 (admissibility is for selection, not for the distributional verification itself) .