$L^2$-stability $\mathrm{\&}$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes

In this paper, we study stability properties of solutions to scalar conservation laws with a class of non-convex fluxes. Using the theory of $a$-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient $a$ in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2\times 2$ system setting by Chen, Golding, Krupa, $\mathrm{\&}$ Vasseur.