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Balance laws with measure valued source for commuting dynamics

  • Fabio Ancona
  • Annalisa Cesaroni
  • Elio Marconi
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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 (mI, mF)-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.

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Cite

BibTeXRIS
BibTeX
@misc{ancona2026balance,
  title = {{Balance laws with measure valued source for commuting dynamics}},
  author = {Fabio Ancona and Annalisa Cesaroni and Elio Marconi},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04503},
  eprint = {2611.04503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04503},
}
RIS
TY  - UNPB
TI  - Balance laws with measure valued source for commuting dynamics
AU  - Fabio Ancona
AU  - Annalisa Cesaroni
AU  - Elio Marconi
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04503
UR  - https://arxiv.org/abs/2611.04503
AB  - 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.
ER  -