Open Questions in the Theory of One Dimensional Hyperbolic Conservation Laws

We remark that hyperbolic conservation laws are a class of nonlinear evolution equations. As such, it would be natural to study them also from the point of view of dynamical systems. In particular, this would mean looking at periodic orbits, bifurcations, attractors, chaotic dynamics, etc. . . At present, however, very little of this is seen, within hyperbolic theory. Apparently, the main reason is that the known existence-uniqueness results are mainly restricted to solutions with small total variation. For such solutions, the asymptotic behavior is, in a sense, trivial. Indeed, as proved by T. P. Liu [52, 53], as t → +∞ every solution with small total variation approaches the solution to a corresponding Riemann problem.