Structure of fundamental solutions of a conservation law without convexity
Yong-Jung Kim and Young-Ran Lee
Abstract: This note is devoted to reveal the structure of signed fundamental solutions of a conservation law without the convexity assumption. It is assumed that the flux is in C1(R) and has a finite number of inflection points. Fundamental solutions of the case have been constructed in [Preprint 2007-022] employing convex and concave envelopes. This construction provides useful information on the structure of a fundamental solution in terms of the envelopes and we clarify the dynamics of a fundamental solution using this structural information. We classify the types of shocks, rarefaction waves and their interactions to reach the final stage of its asymptotics. A complete example of such a dynamics is given with a full characteristic map.