A TWO-DIMENSIONAL RIEMANN PROBLEM FOR SCALAR CONSERVATION LAWS

This paper provides an introduction to current research in self-similar problems for multidimensional conservation laws. This is presently an active area. To set the stage, we begin by solving a classic problem. We prove that the solution of the scalar equation u(t) f (u)(x) + g(u)(y) = 0 with Riemann initial data of the form u(0, x, y) = u(0)(theta) (0 <= theta <= 2 pi) remains smooth outside a circle with center at the origin in the self-similar plane. Based on this approach to Riemann problems, and on recent research by a number of contributors, we give a list of open problems.