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\neq 0. Given an interface connection (A,B), we define a backward solution operator consistent with the concept of AB-entropy solution [4,13,16]. We then analyze the family A^{[AB]}(T) 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^{[AB]}(T) as fixed points of the backward-forward solution operator. As an intermediate step we establish a full characterization of A^{[AB]}(T) in terms of unilateral constraints and Ole\v{\i}nik-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^1_{loc}-Lipschitz continuity in time of these solutions.