# Weak Measure-Valued Solutions of a Nonlinear Hyperbolic Conservation Law

## TL;DR

This paper develops a framework for analyzing measure-valued solutions to a nonlinear hyperbolic conservation law modeling semi-conductor manufacturing, addressing discontinuities and establishing existence, uniqueness, and continuity properties.

## Contribution

It introduces a novel weak solution concept for measure-valued states with discontinuous velocities, extending the analysis of hyperbolic conservation laws in a finite domain.

## Key findings

- Established existence and uniqueness of solutions.
- Proved continuity of the flow with respect to time.
- Developed regularity results using semi-norms related to the flat norm.

## Abstract

We revisit a well-established model for highly re-entrant semi-conductor 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 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 semi-norms that are modifications of the flat norm. Generally. the solution is not continuous with respect to any norm on the space of measures.