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

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≠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[AB](T) of profiles that can be attained at time T>0 by AB -entropy solutions with L∞-initial data. We provide a characterization of A[AB](T) as fixed points of the backward-forward solution operator. As an intermediate step we establish for the first time a full characterization of A[AB](T) 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 Lloc1-Lipschitz continuity in time of these solutions.