Periodic Structure in Two-Dimensional Riemann Problems for Hamilton-Jacobi Equations

This paper investigates the structure of two-dimensional Riemann problems for Hamilton-Jacobi equations. We show that it is possible for the viscosity solution to contain closed characteristic orbits , enclosing furthermore a periodic sonic structure, which in turn encloses a parabolic structure. The existence of such examples elucidates the diiculties encountered in designing construction methods for viscosity solutions to Riemann problems in dimension 2. This investigation was prompted by the discovery of numerical evidence of examples displaying an even richer internal structure. Here we establishe the existence of Riemann problems with viscosity solutions of considerable complexity.