Riemann Invariants, Entropy, and Uniqueness

There are better results known for pairs of conservation laws than for systems with more than two equations. We have already seen an example of this in the last chapter ; namely, the interaction estimates are stronger when n = 2 than when n> 2. This was due to the existence of a distinguished coordinate system called Riemann invariants, which in general exists only for two equations. We shall study the implications one can draw using these coordinates. It turns out that the equations take a particularly nice form when written in terms of the Riemann invariants, and using this we can prove that for genuinely nonlinear systems, global classical solutions generally do not exist. (We only know this now for a single conservation law ; see Chapter 15, §B.)