A modest proposal for MFG with density constraints

Previous Article
Liquidity generated by heterogeneous beliefs and costly estimations
NHM Home
This Issue
Next Article
New numerical methods for mean field games with quadratic costs

June 2012, 7(2): 337-347. doi: 10.3934/nhm.2012.7.337

A modest proposal for MFG with density constraints

Filippo Santambrogio ^1,

1.	Laboratoire de Mathématiques d'Orsay, Faculté de Sciences, Université Paris-Sud, 91405 Orsay cedex, France

Received November 2011 Revised March 2012 Published June 2012

Abstract
Related Papers

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.

Keywords: density constraints, Crowd motion, pressure., MFG system.

Mathematics Subject Classification: Primary: 91A23; Secondary: 49K15, 49K20, 49L20, 35F25. .

Citation: Filippo Santambrogio. A modest proposal for MFG with density constraints. Networks and Heterogeneous Media, 2012, 7 (2) : 337-347. doi: 10.3934/nhm.2012.7.337

References:

[1]	L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.
[2]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[4]	G. Buttazzo, C. Jimenez and E. Oudet, An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X.
[5]	G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.
[6]	E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98.
[7]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[8]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.
[9]	J.-M. Lasry and P.-L. Lions, Mean-field games, Japan. J. Math, 2 (2007), 229-260.
[10]	B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.
[11]	B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Net. Het. Media, 6 (2011), 485-519.
[12]	B. Maury and J. Venel, "Handling of Contacts in Crowd Motion Simulations," Traffic and Granular Flow, Springer, 2007.
[13]	R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.
[14]	F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[15]	S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.
[16]	C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58, AMS, Providence, RI, 2003.
[17]	C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

show all references

References:

[1]	L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.
[2]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[4]	G. Buttazzo, C. Jimenez and E. Oudet, An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X.
[5]	G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.
[6]	E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98.
[7]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[8]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.
[9]	J.-M. Lasry and P.-L. Lions, Mean-field games, Japan. J. Math, 2 (2007), 229-260.
[10]	B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.
[11]	B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Net. Het. Media, 6 (2011), 485-519.
[12]	B. Maury and J. Venel, "Handling of Contacts in Crowd Motion Simulations," Traffic and Granular Flow, Springer, 2007.
[13]	R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.
[14]	F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[15]	S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.
[16]	C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58, AMS, Providence, RI, 2003.
[17]	C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

[1]	Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling. Networks and Heterogeneous Media, 2011, 6 (3) : 485-519. doi: 10.3934/nhm.2011.6.485
[2]	Tan H. Cao, Boris S. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 267-306. doi: 10.3934/dcdsb.2017014
[3]	Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic and Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025
[4]	Alessandro Bertuzzi, Antonio Fasano, Alberto Gandolfi, Carmela Sinisgalli. Interstitial Pressure And Fluid Motion In Tumor Cords. Mathematical Biosciences & Engineering, 2005, 2 (3) : 445-460. doi: 10.3934/mbe.2005.2.445
[5]	Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[6]	Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078
[7]	Félicien BOURDIN. Splitting scheme for a macroscopic crowd motion model with congestion for a two-typed population. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022026
[8]	Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[9]	Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365
[10]	Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150
[11]	Gui-Qiang G. Chen, Qin Wang, Shengguo Zhu. Global solutions of a two-dimensional Riemann problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2475-2503. doi: 10.3934/cpaa.2021014
[12]	Rhouma Mlayeh. Stability analysis of pressure and penetration rate in rotary drilling system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1547-1560. doi: 10.3934/dcdss.2022007
[13]	Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic and Related Models, 2014, 7 (1) : 195-203. doi: 10.3934/krm.2014.7.195
[14]	Yuanshi Wang, Donald L. DeAngelis. A mutualism-parasitism system modeling host and parasite with mutualism at low density. Mathematical Biosciences & Engineering, 2012, 9 (2) : 431-444. doi: 10.3934/mbe.2012.9.431
[15]	Roger Lui, Hirokazu Ninomiya. Traveling wave solutions for a bacteria system with density-suppressed motility. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 931-940. doi: 10.3934/dcdsb.2018213
[16]	Banavara N. Shashikanth. Kirchhoff's equations of motion via a constrained Zakharov system. Journal of Geometric Mechanics, 2016, 8 (4) : 461-485. doi: 10.3934/jgm.2016016
[17]	Yinzheng Sun, Qin Wang, Kyungwoo Song. Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4899-4920. doi: 10.3934/cpaa.2020217
[18]	Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609
[19]	Yu Liu, Ting Zhang. On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021307
[20]	Boris P. Andreianov, Carlotta Donadello, Ulrich Razafison, Julien Y. Rolland, Massimiliano D. Rosini. Solutions of the Aw-Rascle-Zhang system with point constraints. Networks and Heterogeneous Media, 2016, 11 (1) : 29-47. doi: 10.3934/nhm.2016.11.29

2021 Impact Factor: 1.41

Tools

Metrics

Other articles
by authors

on AIMS
Filippo Santambrogio
on Google Scholar
Filippo Santambrogio

[Back to Top]