Perestroikas of Shocks and Singularities of Minimum Functions

Abstract

The shock discontinuities, generically present in inviscid solutions of the forced Burgers equation, and their bifurcations happening in the course of time (perestroikas) are classified in two and three dimensions -- the one-dimensional case is well known. This classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time, the ones permitted by the convexity of the Hamiltonian of the Burgers dynamics. Topological restrictions on the admissible perestroikas of shocks are obtained. The resulting classification can be extended to the so-called viscosity solutions of a Hamilton--Jacobi equation, provided the Hamiltonian is convex.