Barbara Lee Keyfitz
Fields Institute and University of Houston

Can Self-Similar Problems Tell us Anything about Multi-Dimensional Conservation Laws?

Riemann problems (self-similar problems in space-time) have played an important part in developing the theory of hyperbolic conservation laws in one space dimension. It is unlikely that they will have the same role in multidimensional problems. However, they have some interest. First of all, in one space dimension, the novelty lies in the way linear discontinuities become shocks or rarefactions for a nonlinear equation. After that, interactions among different wave families can be quite complicated, but the full story does not appear until one studies the interaction of solutions of two or more Riemann problems. In multidimensional Riemann problems, by contrast, complicated nonlinear wave interactions appear already with quite simple Riemann data. Thus, self-similar problems are relatively more complicated in two or more space dimensions, but it is possible that they will reward study. During the past decade, several groups have begun to analyse two-dimensional Riemann problems for some model systems of conservation laws. This talk will outline some successes, and describe some tantalizing paradoxes whose resolution seems still out of reach.