Contemporary Mathematics Prototypes for Nonstrict Hyperbolicity in Conservation Laws

Many conservation law systems that arise in continuum mechanics fail to be strictly hyperbolic; this situation may even arise generically in some dimensions. By examining simple systems in a single space variable, where the mathematical theory of weak solutions has been developed, one may try to gain insight into how more realistic systems will behave. However, even in simple systems, there is no standard model for the breakdown of strict hyperbolicity. This paper surveys a number of nonstrictly hyperbolic models. Some may be good prototypes for multidimensional problems in continuum mechanics; some appear frequently in models for multiphase ow; some are associated with change of type (loss of hyperbolicity). All have complicated Riemann solutions; for some, existence of solutions to the Cauchy problem has been proved, and interesting qualitative behavior has been observed.