Propagation of Singularities in Nonconvex Hamilton--Jacobi Problems: Local Structure in Two Dimensions

Source abstract

The evolution of the singular set of points in solutions to Hamilton--Jacobi equations is an active area of research. Recent work by L. C. Evans has shed some light on the singular structure in nonconvex problems, including envelope shocks, from which characteristics leave tangentially. The present paper contributes to this area by considering Riemann problems, for which the data and viscosity solutions are self-similar. Riemann problems explore the local structure of solutions in the vicinity of singularities. A framework is developed for constructing solutions to these problems, including a device for establishing viscosity admissibility of envelope shocks using a notion of duality. A nonconvex example is presented in which a point singularity in the data, with three reachable gradients, evolves into two envelope shocks and two point singularities, one of which has four reachable gradients.