Fundamental Solutions of Conservation Laws

In this paper we construct fundamental solutions of a scalar conservation law in one space dimension. These source-type solutions are well known for a convex case and hence our focus is on a general non-convex case which may have a finite number of inflection points. Signed fundamental solutions are constructed first and then under an extra hypothesis on nonnegativity of the flux two parameter family of fundamental solutions are constructed. This process is a natural generalization of N-waves. New N-waves constructed here consist of a series of increasing rarefaction waves and increasing shocks and then another series of decreasing rarefaction waves and decreasing shocks. This fundamental solution indicates why the famous one sided Oleinik estimate fails for a non-convex case and how it should be corrected. N-waves are also computed using the WENO and central type numerical schemes, which show the structure of the N-wave constructed in this paper.