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arXiv · 2305.10076

The mathematical theory of Hughes' model: a survey of results

Debora Amadori·Boris Andreianov·Marco Di Francesco·Simone Fagioli·Théo Girard·Paola Goatin·Peter Markowich·Jan F. Pietschmann·Massimiliano D. Rosini·Giovanni Russo·Graziano Stivaletta·Marie-Therese Wolfram
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Abstract

We provide an overview of the results on Hughes' model for pedestrian movements available in the literature. After the first successful approaches to solving a regularised version of the model, researchers focused on the structure of the Riemann problem, which led to local-in-time existence results for Riemann-type data and paved the way for a WFT (Wave-Front Tracking) approach to the solution semigroup. In parallel, a DPA (Deterministic Particles Approximation) approach was developed in the spirit of follow-the-leader approximation results for scalar conservation laws. Beyond having proved to be powerful analytical tools, the WFT and the DPA approaches also led to interesting numerical results. However, only existence theorems on very specific classes of initial data (essentially ruling out non-classical shocks) have been available until very recently. A proper existence result using a DPA approach was proven not long ago in the case of a linear coupling with the density in the eikonal equation. Shortly after, a similar result was proven via a fixed point approach. We provide a detailed statement of the aforementioned results and sketch the main proofs. We also provide a brief overview of results that are related to Hughes' model, such as the derivation of a dynamic version of the model via a mean-field game strategy, an alternative optimal control approach, and a localized version of the model. We also present the main numerical results within the WFT and DPA frameworks.

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Debora Amadori, Boris Andreianov, Marco Di Francesco, Simone Fagioli, Théo Girard, Paola Goatin, Peter Markowich, Jan F. Pietschmann, Massimiliano D. Rosini, Giovanni Russo, Graziano Stivaletta, Marie-Therese Wolfram. 2023-05-17. The mathematical theory of Hughes' model: a survey of results. https://arxiv.org/abs/2305.10076

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