2. Lifting a Non-Resonant Scalar Balance Law

The mathematical theory of scalar conservation laws has reached a state of completion: existence, uniqueness, regularity and stability with respect to initial data have been established in various settings (BV theory, compensated compactness, kinetic formulation, relaxation approximation, etc…). Here, we aim at presenting the special features holding when space-dependent, non-dissipative, source terms are added on the right-hand side (which complicates the picture), but under the simplifying assumption that stagnation points aren’t allowed (f′(u) ≠ 0).. This somehow tempers the effects of the source, for instance when it has compact support in ℝ, allowing for the derivation of peculiar, uniform in time, bounds in both amplitude and total variation which express the fact that convective waves exit the amplification area after some time. Such bounds lead to an improvement in Kuznetsov-type error estimates [24] for the so-called well-balanced schemes, obtained by approximating a 2 × 2 homogeneous Temple-class system by means of Godunov’s method.