Weak measure-valued solutions of a nonlinear hyperbolic conservation law

We revisit a well-established model for highly re-entrant semiconductor manufacturing systems, and analyze it in the setting of states, in- and outfluxes being Borel measures. This is motivated by the lack of optimal solutions in the L1-setting for transitions from a smaller to a larger equilibrium with zero backlog. Key innovations involve dealing with discontinuous velocities in the presence of point masses, and a finite domain with in- and outfluxes. Taking a Lagrangian point of view, we establish existence and uniqueness of solutions and formulate a notion of weak solution. We prove that the Lagrangian solutions are weak solutions and vice versa. We prove continuity of the flow with respect to time (and almost also with respect to the initial state). Due to generally discontinuous velocities, these delicate regularity results hold only with respect to carefully crafted seminorms that are modifications of the flat norm. Generally, the solution cannot be continuous with respect to any norm on the space of measures.