  Given a first-order nonlinear hyperbolic system of conservation laws endowed
with a convex entropy-entropy flux pair, we can consider the class of weak
solutions containing shock waves depending upon some small scale parameters. In
this Note, we define and derive several classes of entropy-dissipating
augmented models, as we call them, which involve (possibly nonlinear) second-
and third-order augmentation terms. Such terms typically arise in continuum
physics and model the viscosity and capillarity effects in a fluid, for
instance. By introducing a new notion of positive entropy production that
concerns general functions (rather than solutions) we can easily check the
entropy-dissipating property for a broad class of augmented models. The weak
solutions associated with the zero diffusion/dispersion limit may contain
(nonclassical undercompressive) shocks whose selection is determined from these
diffusive and dispersive effects (for instance by using traveling wave
solutions), and having a classification of the models, as we propose, is
essential for developing a general theory.
