Backward-forward characterization of attainable set for conservation laws with spatially discontinuous flux

Abstract

Consider a scalar conservation law with a spatially discontinuous flux at a single point $x=0$, and assume that the flux is uniformly convex when $x\ne 0$. Given an interface connection $(A,B)$, we define a backward solution operator consistent with the concept of AB-entropy solution [4], [16], [20]. We then analyze the family $A^{\left[AB\right]}\left(T\right)$ of profiles that can be attained at time $T>0$ by AB-entropy solutions with $L^{\infty }$-initial data. We provide a characterization of $A^{\left[AB\right]}\left(T\right)$ as fixed points of the backward-forward solution operator. As an intermediate step we establish for the first time a full characterization of $A^{\left[AB\right]}\left(T\right)$ in terms of unilateral constraints and Oleı̌nik-type estimates, valid for all connections. Building on such a characterization we derive uniform BV bounds on the flux of AB-entropy solutions, which in turn yield the $L_{\mathrm{loc}}^1$-Lipschitz continuity in time of these solutions.