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

We provide an overview of the results on Hughes’ model for pedestrian movements available in the literature. The model consists of a nonlinear conservation law coupled with an eikonal equation. The main difficulty in developing a proper mathematical theory lies in the lack of regularity of the flux in the conservation law, which yields the possibility of non-classical shocks that are generated non-locally by the whole distribution of pedestrians. This is a possible reason behind the availability of existence results only on one-dimensional spatial domains, despite the model having a more natural setting in two spatial dimensions. 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.