The fundamental solution of a conservation law without convexity

Abstract

The nonnegative fundamental solution of a scalar conservation law is constructed when its flux may have a finite number of inflection points. The constructed solution can be either explicit and implicit depending on the flux. This fundamental solution consists of a series of rarefaction waves, contact discontinuities and a shock. These analytically constructed fundamental solutions are also compared with numerical approximations, which possess the structure of the analytically constructed fundamental solution.