A characterization of the reachable profiles of entropy solutions for the elementary wave interaction problem of convex scalar conservation laws

In this paper, we analyze and characterize the set ${\mathcal{A}}_T$ which consists of all possible profiles at a fixed time of the entropy solution of the elementary wave interaction problem in a bounded domain for a convex scalar conservation law. The elementary wave interaction problem is the initial and boundary value problem for a scalar conservation law, where the flux is a strictly convex function, and the initial and boundary data are constant functions. In the first main result of the article, we state and prove that ${\mathcal{A}}_T$ is a subset of the set of piecewise functions that are constant on each subdomain, or there is a subdomain where the function is strictly increasing. We prove the result by applying the method of characteristics in three steps: the Riemann problem solution, the entropy solution of the interaction of two Riemann problems, and restriction of the entropy solution to the spatial bounded domain. Moreover, we characterize the strictly increasing part of the solution's profile regarding the flux function. In the second result, which is stated as an application of the first result, we introduce the conditions for ill-posedness and local flux identification from the knowledge of the entropy solution's profile.