Solutions to a set of initial value problems that arise in deformation of elastic-plastic solids, two phase flow, and other physical situations are discussed in this mathematical research paper. Even if the functions (convection and diffusion matrices) that define the equations remain continuously differentiable, solutions that arise are discontinuous at a point that moves (a shock wave). Furthermore, multiple solutions potentially arise, although the viscous profile criterion provides a way of identifying an essentially unique solution.

A good overview of this area is provided in the paper. Results that had been obtained by others using rather specific techniques are revisited in the light of general theory (center manifold reduction) and put into perspective, and useful links are made with work in related areas.

While there is little that is new, the paper provides a useful introduction to the contemporary underpinning mathematics of a family of problems with real applications. Questions of uniqueness within specific conditions are resolved. There is no link to any computation.

Although the paper is fairly dense in its use of space, it is pleasant to the eye, straightforward to follow, and well argued. Equations are clear and well linked. The mathematical terminology used is minimized, diagrams are nicely presented, and there is a good set of pertinent up-to-date references.