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...
moreWe 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. • a continuum description at the level of the interaction of sub-populations of the system, known as the macroscopic level. Each level has an associated class of mathematical equations which provide an appropriate model. Usually their structures at various levels are completely different, see e.g. [9]. We defer the reader to [32-34] for an overview of the various research directions in the field of crowd dynamics. One of the most original and mathematically challenging models is the one proposed by Roger L. Hughes [41], which is part of the third approach above. Hughes' model describes evacuation scenarios, in which a crowd wants to exit a given domain D ⊂ R 2 , with one or several exits, as fast as possible. The crowd population is described through a density. The driving force towards the exits is the gradient of a potential , which satisfies an eikonal equation coupled with the density. The potential represents the expected travel time towards an exit and accounts for the best strategy to minimise the exit time. The resulting model is a nonlinear conservation law for coupled with the gradient of the potential ; the latter depends on non-locally in space. Even in the simplest case of space dimension one, no more than Lipschitz continuity can be expected for. Moreover, can change its sign just once from positive to negative. In this case, a turning curve = () may be defined in the one-dimensional domain, at which reaches its maximum. As a result, the flux of the conservation law for is possibly discontinuous along = () and, on the other hand, depends non-locally on. Furthermore, is not expected to satisfy in general the Lax entropy inequalities (see [43]). Therefore, the possible appearance of non-classical shocks along the turning curve has to be taken into account. Non-classical shocks correspond to pedestrians changing direction during the evacuation. Because of these various difficulties, the existence and uniqueness analysis for the Hughes' model appears to be challenging. This motivates the first approach to the problem developed in [27], in which a smoothened version of the eikonal equation is considered, with an extra Laplacian term. Despite it covers only the one-dimensional case and despite it deals only with an approximated version of the model rather than the actual model, the result in [27] remains until now the only result in a large data setting which holds for a fairly general class of density-potential coupling. As we will detail later on, the first existence results for the actual Hughes model appeared only very recently, and only for specific couplings. Let us also mention the existence and uniqueness result proved in [19, Theorem 2.6] for a two dimensional regularized version of the Hughes model, which holds for large data but with a potential depending only on the given domain D, see also [36]. Parallel to [27] or shortly after it, some researchers started to study the Riemann problem for the model in the spirit of scalar conservation laws and to develop proper numerical schemes, see [2, 3, 7, 30] for the Riemann data part and [11, 12, 16, 17, 35, 40, 51] for the numerics. Both in [2] and [30], the authors study the Riemann problems for the Hughes' model in detail. This study is strictly related to the effectiveness of the Wave-Front Tracking (WFT) strategy [21] for the Hughes' model. The WFT algorithm was then first exploited in [35], but only for numerical purposes, and then in [3] to prove the first existence result for the Hughes' model, but under very restrictive assumptions that rule out non-classical shocks. A simpler proof of an analogous existence result was then obtained in [25] by means of a Deterministic Particle Approximation (DPA) and the results proved in [28], see also [23, 24, 26, 29]. The first existence result accounting for the possible presence of non-classical shocks was recently obtained in [7]. The authors obtain this result by exploiting the properties of the linear cost introduced in [30] (the key fact here is that linear costs yield a uniform Lipschitz bound on), combined with the DPA adapted to the Hughes' model in [25]. Despite being only valid for linear costs and in one space dimension, this result has the merit of being the first existence result on the Hughes model for large data and in presence of non-classical shocks. This result is reproved in [6] via a non-constructive Schauder fixed-point approach allowing for a wide variety of generalizations of the one-dimensional Hughes' model (different ways to compute the turning curve from the density , different exit conditions). Concerning uniqueness, only very partial results are available for the one-dimensional Hughes' model; they require BV regularity of the density and the highly restrictive assumption of zero density traces (, () ±) at the turning curve (see [7, Theorem 4], see also [3, 25] for particular cases). Apart from the regularised version proposed in [27], other variants of the Hughes model have been proposed: a first one in [14, 37] obtained a similar model with a time derivative in the eikonal equation, justified through an optimal control problem, and a second one in [18] trying to remove the global awareness of the pedestrians in the model, which seems unrealistic in some situations. Further variants with more flexible boundary conditions for the density, with memory or relaxation effects in the dynamics of , are proposed and studied in [6]. The chapter is structured as follows. In Section 2 we derive Hughes' model in the way it was done in the original paper [41] by Roger L. Hughes, plus some additional considerations by the authors of this survey. We also provide a rephrasement of the model in the special case of one space dimension. In Section 3 we detail the local-in-time solution of the Riemann problem. In Section 4 we collect the existence result provided for the model, from the ones holding only for small data or symmetric data provided in [3, 26], to the main one provided in [7] for the case of linear cost. In Section 5 we describe the construction of the Wave-Front Tracking (WFT) algorithm used to prove the existence results in [3]. In Section 6 we introduce the Deterministic Particle Approximation (DPA) of the model leading to the results in [26] and [7]. In Section 7 we describe in detail the main existence result of [7]. In Section 8 we briefly describe the fixed-point approach of [6], with a second proof of this main existence result and several extensions. Section 9 is devoted to numerical simulations, both using the WFT algorithm and the DPA scheme. Finally, in Section 10 we summarise the