In Definition 2.15 we gave the most practical version of the entropy condition. It limits admissible shocks to those obtained by the intersection of “forward-moving” characteristics. These are therefore discontinuities that “cannot be avoided” or replaced by a rarefaction wave. In this Appendix we give some further insight into this concept of an “entropy satisfying” weak solution to (2.1), (2.2).

Our starting point is the physical notion of a “vanishing viscosity solution.” In general terms, an equation leading to discontinuous solutions [such as (2.1)] is supplemented by “dissipative terms” (also referred to as “viscous terms”). In analogy to the physical situation, such terms have a “smoothing effect” on solutions with large gradients, thus replacing discontinuities by “transition zones” where the solution varies smoothly, albeit rapidly. As the viscous effects are diminished, those transition zones shrink to surfaces of zero width, across which the solution has a sharp jump. Mathematically speaking, the additional viscous terms are often represented by second-order derivatives with a small (“vanishing”) coefficient.

To illustrate the situation, we consider the “moving step” problem for Burgers' equation (Example 2.12).