Submitted Paper

Inserted: 4 nov 2026
Last Updated: 4 nov 2026

Year: 2026

Abstract:

In this paper we consider balance laws with measure-valued sources modeling commuters traveling along a road from residential to workplace locations. We introduce a notion of $(m_I, m_F)$-entropy solution, prove a compactness result for this class, and establish existence of solutions with prescribed spatial marginals for the sources representing the lodging and workplace distributions. Moreover, adopting a Lagrangian viewpoint, we describe commuting strategies as measures on Lipschitz curves with assigned entry and exit times, compatible with a given transport plan. Using the compactness result and a semi-discrete micro-macro scheme with source, based on the celebrated Follow-the-Leader model, we prove existence of admissible commuting strategies. Finally we prove the existence of optimal solutions for classical traffic cost functionals.