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arxiv: 2606.02899 · v1 · pith:UMND4LZVnew · submitted 2026-06-01 · 🧮 math.AP

A coupled prediction-correction Hughes' model for congested crowd motion

Hamza Ennaji , Noureddine Igbida , Ghadir Jradi , Jos\'e Miguel Urbano This is my paper

Pith reviewed 2026-06-28 13:19 UTC · model grok-4.3

classification 🧮 math.AP
keywords crowd motionHughes modelconservation lawEikonal equationprediction-correctionwell-posednesspedestrian density
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The pith

A coupled prediction-correction variant of Hughes' model yields numerical evidence toward well-posedness of the classical version.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a macroscopic crowd model that couples a nonlinear conservation law for pedestrian density to an Eikonal equation, modifying Hughes' original formulation through a prediction step followed by a correction step. This structure adds anticipatory behavior and dynamic route adjustment while remaining close to the classical equations. The authors formulate the system, examine its properties, and run numerical simulations that display stable, unique-looking solutions. They conclude that the coupled system supplies a promising numerical route to proving well-posedness for the uncoupled Hughes model, an open question.

Core claim

The coupled prediction-correction Hughes model, obtained by inserting a prediction step for density evolution before solving the Eikonal equation in the correction step, produces well-behaved numerical solutions that suggest the original Hughes system may admit unique weak solutions.

What carries the argument

The coupled prediction-correction scheme that separates an anticipatory density prediction from an Eikonal-based route correction.

Load-bearing premise

Numerical behavior observed in the coupled system can be taken as reliable evidence for the mathematical well-posedness of the original uncoupled Hughes model.

What would settle it

An explicit example of a density and velocity field that satisfies the classical Hughes equations but violates uniqueness or existence, while the coupled prediction-correction system remains well-posed on the same data.

Figures

Figures reproduced from arXiv: 2606.02899 by Ghadir Jradi, Hamza Ennaji, Jos\'e Miguel Urbano, Noureddine Igbida.

Figure 1
Figure 1. Figure 1: Left: The graph of the multivalued function Sign+. Center & Right: Smooth approximations given respectively by the logistic function β 1 δ (r) = 1 1+exp(− r δ ) and the scaled arctangent β 2 δ (r) = 1 2 + 1 π arctan r δ  for different values of the scaling parameter δ. as ρ approaches the saturation threshold, generating repulsive corrections before complete saturation. The singular behavior near ρ = 1 en… view at source ↗
Figure 2
Figure 2. Figure 2: Discretization of the domain Ω [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: The Moreau-Yosida approximation of Sign+. Middle: The ho￾mographic approximation β 1 δ (p) = max(0,p) max(0,p)+δ . Right: The hyperbolic tangent approximation β 2 δ (p) = max 0,tanh p δ . Both regularizations β 1 δ and β 2 δ assign zero value for negative and null pressures. A standard choice in the literature is to use the sigmoid or the arctangent functions. However, as illustrated in [PITH_FULL… view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM) [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM). We observe that in the first row of both Figs. 4 and 5, the model (PCM) transports the population in a rigid way. Pedestrians remain highly concentr… view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM) [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM). As observed in the first row of Figs. 6 and 7, the constant velocity model (PCM) simply splits the population into two rigid groups moving directly … view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM) [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the crowd density ρ computed at different instants. First row: The constant velocity model governed by (PCM). Second row: The hard congestion Hughes’ model governed by (HC-HM). Third row: The soft congestion Hughes’ Model governed by (SC-HM). The first row of both Figs. 8 and 9 shows again that the constant velocity model (PCM) moves the initial shapes in a completely rigid manner. Since … view at source ↗
Figure 10
Figure 10. Figure 10: The crowd density ρ computed by the model (SC-HM) at different time steps. Top row: λ = 1.5. Middle row: λ = 3.75. Bottom row: λ = 7.25 [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The crowd density ρ computed by the model (HC-HM) at different time steps. Top row: λ = 1.5. Middle row: λ = 3.75. Bottom row: λ = 7.25 [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The crowd density ρ computed by the model (SC-HM) at different time steps. Top row: λ = 1.5. Middle row: λ = 3.75. Bottom row: λ = 7.25 [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The crowd density ρ computed by the model (HC-HM) at different time steps. Top row: λ = 1.5. Middle row: λ = 3.75. Bottom row: λ = 7.25 [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The crowd density ρ computed by the (SC-HM) model at different time steps. Top row: H1. Middle row: H2. Bottom row: H3 [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The crowd density ρ computed by the (HC-HM) model at different time steps. Top row: H1. Middle row: H2. Bottom row: H3. In the top rows (with cost H1) of both [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Evolution of the maximum pressure gradient norm. The curves in blue, green, and black track the maximum spatial norm of the pressure p, plotted on a logarithmic scale. The red line represents the theoretical saturation limit, i.e., |∇p|= 1. In Scenario 1, the density is given by (82), in Scenario 2 by (72), while in Scenario 3 it is given by (76). In all these examples, we make use of the classical Hughes… view at source ↗
read the original abstract

In this work, we introduce a new macroscopic model for crowd motion inspired by the celebrated Hughes' model \cite{Hughes2002, Hughes2003}, which couples a nonlinear conservation law for the pedestrian density with an Eikonal equation describing the shortest path to the target. Our approach can be viewed both as a modification of Hughes' original formulation and as a refinement of the prediction-correction framework proposed in the recent work \cite{ennaji2023prediction}. The resulting model incorporates anticipatory behavior and dynamic route adjustment, offering a more realistic representation of crowd dynamics in complex environments. We present the mathematical formulation of the model, discuss its well-posedness properties, and illustrate its qualitative behavior through numerical simulations. Ultimately, we show, at least from a numerical perspective, that this variant provides a promising avenue towards establishing the well-posedness of the classical Hughes' model, which has remained a challenging open problem for a long time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a coupled prediction-correction variant of Hughes' model for congested crowd motion, obtained by augmenting the classical conservation-law/Eikonal coupling with an explicit anticipatory correction step whose strength is controlled by an additional parameter. It formulates the model, discusses its well-posedness properties, and presents numerical simulations whose qualitative behavior is claimed to supply evidence, at least numerically, that the variant may open a route to proving well-posedness of the original uncoupled Hughes model.

Significance. The classical Hughes model remains a long-standing open problem in mathematical crowd dynamics. A rigorously justified limit argument showing that well-posedness of the coupled system implies well-posedness of the original would be a substantial contribution; the present numerical evidence alone does not yet establish that link.

major comments (2)
  1. [Abstract] Abstract (final sentence): the assertion that numerical simulations of the coupled system supply evidence for well-posedness of the classical Hughes model is not supported by any singular-limit analysis. No section derives the original Hughes equations as a limit of the coupled system when the correction parameter tends to zero, nor shows that well-posedness persists under that limit.
  2. [Well-posedness discussion] The well-posedness discussion for the new model (whatever section contains it) cannot be transferred to the classical model without an explicit convergence or stability estimate relating the two systems; the additional regularization introduced by the prediction-correction step may suppress precisely the singularities that render the uncoupled problem open.
minor comments (2)
  1. Clarify the precise functional setting (spaces, boundary conditions) in which the coupled system is stated to be well-posed.
  2. Add a short paragraph comparing the new parameter to existing regularizations in the literature on Hughes-type models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback. The comments correctly identify that our claims linking the numerical results to the classical model require careful qualification. We will revise the abstract accordingly and clarify the scope of our well-posedness discussion.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the assertion that numerical simulations of the coupled system supply evidence for well-posedness of the classical Hughes model is not supported by any singular-limit analysis. No section derives the original Hughes equations as a limit of the coupled system when the correction parameter tends to zero, nor shows that well-posedness persists under that limit.

    Authors: We agree that the manuscript lacks a singular-limit analysis relating the coupled system to the classical Hughes model as the correction parameter approaches zero. The numerical simulations are presented to demonstrate the well-posed behavior of the new model and to suggest it as a potential regularization approach. We will revise the abstract's final sentence to indicate that the variant provides a promising avenue for future study of the classical model's well-posedness, rather than claiming that the simulations supply evidence for it. revision: yes

  2. Referee: [Well-posedness discussion] The well-posedness discussion for the new model (whatever section contains it) cannot be transferred to the classical model without an explicit convergence or stability estimate relating the two systems; the additional regularization introduced by the prediction-correction step may suppress precisely the singularities that render the uncoupled problem open.

    Authors: We acknowledge the validity of this point. The well-posedness analysis in the manuscript applies specifically to the coupled prediction-correction model and does not include convergence estimates to the uncoupled case. The regularization effect of the correction step is indeed a key feature that may address the open issues in the classical model, but we do not claim that the results transfer directly. This observation reinforces the need for future work on the limit process. No changes are needed to the well-posedness section itself. revision: no

Circularity Check

0 steps flagged

Minor self-citation to authors' prior framework; central numerical suggestion remains independent of inputs.

full rationale

The paper introduces a coupled prediction-correction variant as a refinement of the framework in the authors' prior work <cite{ennaji2023prediction}> and presents numerical simulations as evidence that the variant offers a promising avenue toward well-posedness of the classical Hughes model. This suggestion is an independent observation drawn from the behavior of the new system; no derivation, limit argument, or fitted quantity reduces the claim to the self-citation or to any input by construction. The self-citation supports model construction but is not load-bearing for the well-posedness avenue.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the model is presented as a direct modification of existing equations.

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Works this paper leans on

48 extracted references · 40 canonical work pages

  1. [1]

    Adams and John J

    Robert A. Adams and John J. F. Fournier. Sobolev spaces, volume 140 of Pure Appl. Math., Academic Press. New York, NY: Academic Press, 2nd ed. edition, 2003

    2003

  2. [2]

    On the minimizing movement with the 1-W asserstein distance

    Martial Agueh, Guillaume Carlier, and Noureddine Igbid a. On the minimizing movement with the 1-W asserstein distance. ESAIM, Control Optim. Calc. Var. , 24(4):1415–1427, 2018. doi:10.1051/cocv/2017055

    work page doi:10.1051/cocv/2017055 2018

  3. [3]

    Quasilinear elli ptic-parabolic differential equations

    Hans Wilhelm Alt and Stephan Luckhaus. Quasilinear elli ptic-parabolic differential equations. Math. Z., 183:311–341, 1983. URL: https://eudml.org/doc/173321, doi:10.1007/BF01176474

    work page doi:10.1007/bf01176474 1983

  4. [4]

    Rosini, Giovanni Russo, Giu- lia Stivaletta, and Marie-Therese W olfram

    Debora Amadori, Boris Andreianov, Marco Di Francesco, S imone Fagioli, Théo Girard, Paola Goatin, Peter Markowich, Jan-Frederik Pietschmann, Massi miliano D. Rosini, Giovanni Russo, Giu- lia Stivaletta, and Marie-Therese W olfram. The mathematic al theory of Hughes’ model: A sur- vey of results. In Nicola Bellomo and Livio Gibelli, editors , Crowd Dynamic...

    work page doi:10.1007/978-3-031-46359-4_2 2023

  5. [5]

    Di Francesco

    Debora Amadori and M. Di Francesco. The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions. Acta Math. Sci. Ser. B (Engl. Ed.) , 32(1):259–280, 2012. doi:10.1016/S0252-9602(12)60016-2

    work page doi:10.1016/s0252-9602(12)60016-2 2012

  6. [6]

    Debora Amadori, Paola Goatin, and Massimiliano D. Rosin i. Existence results for Hughes’ model for pedestrian flows. J. Math. Anal. Appl. , 420(1):387–406, 2014. doi:10.1016/j.jmaa.2014.05.072

    work page doi:10.1016/j.jmaa.2014.05.072 2014

  7. [7]

    Time compactness tools for discretiz ed evolution equations and applications to degenerate parabolic PDEs

    Boris Andreianov. Time compactness tools for discretiz ed evolution equations and applications to degenerate parabolic PDEs. In Finite volumes for complex applications VI: Problems and pe rspectives. FVCA 6, international symposium, Prague, Czech Republich, Ju ne 6–10, 2011. Vol. 1 and 2. , pages 21–29. Berlin: Springer, 2011. doi:10.1007/978-3-642-20671-9_3

    work page doi:10.1007/978-3-642-20671-9_3 2011

  8. [8]

    From the microscale to collective crowd dynamics

    Nicola Bellomo, Abdelghani Bellouquid, and Damian Knop off. From the microscale to collective crowd dynamics. Multiscale Model. Simul. , 11(3):943–963, 2013. doi:10.1137/130904569

    work page doi:10.1137/130904569 2013

  9. [9]

    On the modeling of tr affic and crowds: a survey of models, speculations, and perspectives

    Nicola Bellomo and Christian Dogbe. On the modeling of tr affic and crowds: a survey of models, speculations, and perspectives. SIAM Rev. , 53(3):409–463, 2011. doi:10.1137/090746677

    work page doi:10.1137/090746677 2011

  10. [10]

    Blue and Jeffrey L

    Victor J. Blue and Jeffrey L. Adler. Emergent fundamenta l pedestrian flows from cellular automata microsimulation. Transportation Research Record , 1644(1):29–36, 1998. arXiv:https://doi.org/10.3141/1644-04, doi:10.3141/1644-04

    work page doi:10.3141/1644-04 1998

  11. [11]

    Blue and Jeffrey L

    Victor J. Blue and Jeffrey L. Adler. Modeling four-direc tional pedestrian flows. Transportation Research Record, 1710(1):20–27, 2000. arXiv:https://doi.org/10.3141/1710-03, doi:10.3141/1710-03

    work page doi:10.3141/1710-03 2000

  12. [12]

    Functional analysis, Sobolev spaces and partial differenti al equations

    Haim Brezis. Functional analysis, Sobolev spaces and partial differenti al equations . Universitext. New York, NY: Springer, 2011

    2011

  13. [13]

    Simulation of pedestrian dynamics us- ing a two-dimensional cellular automaton

    C Burstedde, K Klauck, A Schadschneider, and J Zittartz . Simulation of pedestrian dynamics us- ing a two-dimensional cellular automaton. Physica A: Statistical Mechanics and its Applications , 295(3):507–525, 2001. URL: https://www.sciencedirect.com/science/article/pii/S0378437101001418, doi:10.1016/S0378-4371(01)00141-8 . A PREDICTION-CORRECTION HUGHES’...

    work page doi:10.1016/s0378-4371(01)00141-8 2001

  14. [14]

    Elisabetta Carlini, Adriano Festa, and Francisco J. Si lva. The Hughes model for pedes- trian dynamics and congestion modelling. IF AC-PapersOnLine, 50(1):1655–1660, 2017. 20th IF AC W orld Congress. URL: https://www.sciencedirect.com/science/article/pii/S2405896317306614, doi:10.1016/j.ifacol.2017.08.333

    work page doi:10.1016/j.ifacol.2017.08.333 2017

  15. [15]

    Silva , and Marie-Therese W olfram

    Elisabetta Carlini, Adriano Festa, Francisco J. Silva , and Marie-Therese W olfram. A semi-Lagrangian scheme for a modified version of the Hughes’ model for Pedestr ian flow. Dyn. Games Appl. , 7(4):683– 705, 2017. doi:10.1007/s13235-016-0202-6

    work page doi:10.1007/s13235-016-0202-6 2017

  16. [16]

    Carrillo, Stephan Martin, and Marie-Therese W o lfram

    Jose A. Carrillo, Stephan Martin, and Marie-Therese W o lfram. An improved version of the Hughes model for pedestrian flow. Math. Models Methods Appl. Sci. , 26(4):671–697, 2016. doi:10.1142/S0218202516500147

    work page doi:10.1142/s0218202516500147 2016

  17. [17]

    A First–Order Primal–Dual Algo- rithm for Convex Problems with Applications to Imaging

    Antonin Chambolle and Thomas Pock. A first-order primal -dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. , 40(1):120–145, 2011. doi:10.1007/s10851-010-0251-1

    work page doi:10.1007/s10851-010-0251-1 2011

  18. [18]

    Multiscale modeling of pedestrian dynamics , volume 12 of MS&A, Model

    Emiliano Cristiani, Benedetto Piccoli, and Andrea Tos in. Multiscale modeling of pedestrian dynamics , volume 12 of MS&A, Model. Simul. Appl. Cham: Springer, 2014. doi:10.1007/978-3-319-06620-2

    work page doi:10.1007/978-3-319-06620-2 2014

  19. [19]

    Vision-based macroscopic pedestrian models

    Pierre Degond, Cécile Appert-Rolland, Julien Pettré, and Guy Theraulaz. Vision-based macroscopic pedestrian models. Kinet. Relat. Models , 6(4):809–839, 2013. doi:10.3934/krm.2013.6.809

    work page doi:10.3934/krm.2013.6.809 2013

  20. [20]

    Markowich, Jan-Frederik P ietschmann, and Marie-Therese W olfram

    Marco Di Francesco, Peter A. Markowich, Jan-Frederik P ietschmann, and Marie-Therese W olfram. On the Hughes’ model for pedestrian flow: the one-dimensional c ase. J. Differ. Equations , 250(3):1334– 1362, 2011. doi:10.1016/j.jde.2010.10.015

    work page doi:10.1016/j.jde.2010.10.015 2011

  21. [21]

    On the modelling of crowd dynamics by g eneralized kinetic models

    Christian Dogbe. On the modelling of crowd dynamics by g eneralized kinetic models. J. Math. Anal. Appl., 387(2):512–532, 2012. doi:10.1016/j.jmaa.2011.09.007

    work page doi:10.1016/j.jmaa.2011.09.007 2012

  22. [22]

    Nader El-Khatib, Paola Goatin, and Massimiliano D. Ros ini. On entropy weak solutions of Hughes’ model for pedestrian motion. Z. Angew. Math. Phys. , 64(2):223–251, 2013. doi:10.1007/s00033-012-0232-x

    work page doi:10.1007/s00033-012-0232-x 2013

  23. [23]

    Pre diction-correction pedestrian flow by means of minimum flow problem

    Hamza Ennaji, Noureddine Igbida, and Ghadir Jradi. Pre diction-correction pedestrian flow by means of minimum flow problem. Math. Models Methods Appl. Sci. , 34(3):385–416, 2024. doi:10.1142/S0218202524500052

    work page doi:10.1142/s0218202524500052 2024

  24. [24]

    Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations

    Hamza Ennaji, Noureddine Igbida, and Van Thanh Nguyen. Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. , 60(6):28, 2021. Id/No 238. doi:10.1007/s00526-021-02092-5

    work page doi:10.1007/s00526-021-02092-5 2021

  25. [25]

    A p rimal-dual algorithm for computing Finsler distances and applications

    Hamza Ennaji, Yvain Quéau, and Abderrahim Elmoataz. A p rimal-dual algorithm for computing Finsler distances and applications. Calcolo, 61(3):25, 2024. Id/No 53. doi:10.1007/s10092-024-00596-y

    work page doi:10.1007/s10092-024-00596-y 2024

  26. [26]

    Finite Volume Methods

    Robert Eymard, Thierry Gallouët, and Raphaèle Herbin. Finite volume methods. In P. G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis , volume 7, pages 713–1020. North-Holland, Amsterdam, 2000. doi:10.1016/S1570-8659(00)07005-8

    work page doi:10.1016/s1570-8659(00)07005-8 2000

  27. [27]

    A fully-discrete-state k inetic theory approach to modeling vehicular traffic

    Luisa Fermo and Andrea Tosin. A fully-discrete-state k inetic theory approach to modeling vehicular traffic. SIAM J. Appl. Math. , 73(4):1533–1556, 2013. doi:10.1137/120897110

    work page doi:10.1137/120897110 2013

  28. [28]

    Knopoff, Jie Liao, and W enbin Ya n

    Livio Gibelli, Damián A. Knopoff, Jie Liao, and W enbin Ya n. Macroscopic modeling of social crowds. Math. Models Methods Appl. Sci. , 34(6):1135–1151, 2024. doi:10.1142/S0218202524400098

    work page doi:10.1142/s0218202524400098 2024

  29. [29]

    Henderson

    Le Roy F. Henderson. On the fluid mechanics of human crowd motion. Transportation Research, 8:509–515, 1974. URL: https://api.semanticscholar.org/CorpusID:154000285

    1974

  30. [30]

    Optimal control of Hughes’ model for pedestrian flow via local attraction

    Roland Herzog, Jan-Frederik Pietschmann, and Max Wink ler. Optimal control of Hughes’ model for pedestrian flow via local attraction. Appl. Math. Optim. , 88(3):44, 2023. Id/No 87. doi:10.1007/s00245-023-10064-8

    work page doi:10.1007/s00245-023-10064-8 2023

  31. [31]

    W ong, Mengping Zhang, Chi-W ang Shu, an d William H.K

    Ling Huang, S.C. W ong, Mengping Zhang, Chi-W ang Shu, an d William H.K. Lam. Revisiting Hughes’ dynamic continuum model for pedestrian flow and the developm ent of an efficient solution algorithm. Transportation Research Part B: Methodological , 43(1):127–141, 2009. doi:10.1016/j.trb.2008.06.003

    work page doi:10.1016/j.trb.2008.06.003 2009

  32. [32]

    Roger L. Hughes. A continuum theory for the flow of pedest ri- ans. Transportation Research Part B: Methodological , 36(6):507–535,

  33. [33]

    URL: https://www.sciencedirect.com/science/article/pii/S0191261501000157, doi:10.1016/S0191-2615(01)00015-7

    work page doi:10.1016/s0191-2615(01)00015-7

  34. [34]

    The flow of human crowds

    Roger L Hughes. The flow of human crowds. Annu. Rev. Fluid Mech. , 35:169–182, 2003. doi:10.1146/annurev.fluid.35.101101.161136

    work page doi:10.1146/annurev 2003

  35. [35]

    Equivalent formulations for Monge -Kantorovich equation

    Noureddine Igbida. Equivalent formulations for Monge -Kantorovich equation. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods , 71(9):3805–3813, 2009. doi:10.1016/j.na.2009.02.039. 40 H. ENNAJI, N. IGBIDA, G. JRADI, AND J.M. URBANO

    work page doi:10.1016/j.na.2009.02.039 2009

  36. [36]

    Evolution Monge-Kantorovich equa tion

    Noureddine Igbida. Evolution Monge-Kantorovich equa tion. J. Differ. Equations , 255(7):1383–1407, 2013. doi:10.1016/j.jde.2013.04.020

    work page doi:10.1016/j.jde.2013.04.020 2013

  37. [37]

    A granular mo del for crowd motion and pedestrian flow

    Noureddine Igbida and José Miguel Urbano. A granular mo del for crowd motion and pedestrian flow. Journal of the London Mathematical Society , 111(5):e70184, 2025. URL: https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms.70184, doi:10.1112/jlms.70184

    work page doi:10.1112/jlms.70184 2025

  38. [38]

    Kruzhkov

    Stanislav N. Kruzhkov. Results concerning the nature o f the continuity of solutions of parabolic equa- tions and some of their applications. Mathematical Notes of the Academy of Sciences of the USSR , 6(1):517–523, 1969. doi:10.1007/BF01450257

    work page doi:10.1007/bf01450257 1969

  39. [39]

    Quelques méthodes de résolution des problèmes aux limites n on linéaires

    Jacques-Louis Lions. Quelques méthodes de résolution des problèmes aux limites n on linéaires. Dunod; Gauthier-Villars, Paris, 1969

    1969

  40. [40]

    On a nonlinear compactness lemma in Lp(0, T ; B)

    Emmanuel Maître. On a nonlinear compactness lemma in Lp(0, T ; B). International Journal of Math- ematics and Mathematical Sciences , 2003(27):1725–1730, 2003. doi:10.1155/S0161171203106175

    work page doi:10.1155/s0161171203106175 2003

  41. [41]

    Prise en compte de la congestion dans le s modèles de mouvements de foules

    Bertrand Maury. Prise en compte de la congestion dans le s modèles de mouvements de foules. In Actes des colloques EDP-Normandie, Caen 2010 – Rouen 2011 , pages 7–20. Fédération Normandie- Mathématiques, 2010

    2010

  42. [42]

    A macroscopic crowd mo- tion model of gradient flow type

    Bertrand Maury, Aude Roudneff-Chupin, and Filippo Sant ambrogio. A macroscopic crowd mo- tion model of gradient flow type. Math. Models Methods Appl. Sci. , 20(10):1787–1821, 2010. doi:10.1142/S0218202510004799

    work page doi:10.1142/s0218202510004799 2010

  43. [43]

    Handling conges- tion in crowd motion modeling

    Bertrand Maury, Aude Roudneff-Chupin, Filippo Santamb rogio, and Juliette Venel. Handling conges- tion in crowd motion modeling. Netw. Heterog. Media, 6(3):485–519, 2011. doi:10.3934/nhm.2011.6.485

    work page doi:10.3934/nhm.2011.6.485 2011

  44. [44]

    A model of the motion of crowds

    Bertrand Maury and Juliette Venel. A model of the motion of crowds. ESAIM, Proc. , 18:143–152,

  45. [45]

    doi:10.1051/proc:071812

    work page doi:10.1051/proc:071812

  46. [46]

    A mathematical fram ework for a crowd motion model

    Bertrand Maury and Juliette Venel. A mathematical fram ework for a crowd motion model. C. R. Math. Acad. Sci. Paris , 346(23-24):1245–1250, 2008. doi:10.1016/j.crma.2008.10.014

    work page doi:10.1016/j.crma.2008.10.014 2008

  47. [47]

    Duality theory and o ptimal transport for sand piles growing in a silos

    Luigi De Pascale and Chloé Jimenez. Duality theory and o ptimal transport for sand piles growing in a silos. Advances in Differential Equations , 20(9/10):859 – 886, 2015. doi:10.57262/ade/1435064516

    work page doi:10.57262/ade/1435064516 2015

  48. [48]

    Méthodes de volumes finis pour les fluide s compressibles

    Nicolas Seguin. Méthodes de volumes finis pour les fluide s compressibles. Université Pierre et Marie Curie-Paris 6 , 2010. URL: https://seguin.perso.math.cnrs.fr/DOCS/cours.pdf

    2010