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Nonlinear systems coupled through multi-marginal transport problems

Part of: Parabolic equations and systems Existence theories Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  29 April 2019

M. LABORDE Open the ORCID record for M. LABORDE [Opens in a new window]
M. LABORDE*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Canada e-mail: [email protected]
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Abstract

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

Keywords

Wasserstein gradient flowsmulti-marginal optimal transporturban planning

MSC classification

Primary: 35K40: Second-order parabolic systems 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 3 , June 2020 , pp. 450 - 469
Copyright
© Cambridge University Press 2019

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References

Agueh, M. (2005) Existence of solutions to degenerate parabolic equations via the Monge- Kantorovich theory. Adv. Differ. Equations 10(3), 309–360.Google Scholar
Agueh, M. & Carlier, G. (2011) Barycenters in theWasserstein space. SIAM J. Math. Anal. 43(2), 904–924.CrossRefGoogle Scholar
Ambrosio, L., Gigli, N. & Savaré, G. (2005) Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel.Google Scholar
Baillon, J.-B. & Carlier, G. (2012) From discrete to continuous Wardrop equilibria. Netw. Heterog. Media 7(2), 219–241.CrossRefGoogle Scholar
Brenier, Y. (1991) Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375–417.CrossRefGoogle Scholar
Buttazzo, G. & Santambrogio, F. (2005) A model for the optimal planning of an urban area. SIAM J. Math. Anal. 37(2), 514–530.CrossRefGoogle Scholar
Buttazzo, G. & Santambrogio, F. (2009) A mass transportation model for the optimal planning of an urban region. SIAM Rev. 51(3), 593–610.CrossRefGoogle Scholar
Carlier, G. (2003) On a class of multidimensional optimal transportation problems. J. Convex Anal. 10(2), 517–529.Google Scholar
Carlier, G. & Ekeland, I. (2004) The structure of cities. J. Global Optim. 29(4), 371–376.CrossRefGoogle Scholar
Carlier, G. & Ekeland, I. (2007) Equilibrium structure of a bidimensional asymmetric city. Nonlinear Anal. Real World Appl. 8(3), 725–748.CrossRefGoogle Scholar
Carlier, G. & Laborde, M. (2017) A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts. Nonlinear Anal. Theory Methods Appl. 150, 1–18.CrossRefGoogle Scholar
Carlier, G. & Santambrogio, F. (2005) A variational model for urban planning with traffic congestion. ESAIM Control Optim. Calc. Var. 11(4), 595–613.CrossRefGoogle Scholar
De Giorgi, E. (1993) New problems on minimizing movements. In: Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Vol. 29, Masson, Paris, pp. 81–98.Google Scholar
Di Francesco, M. & Fagioli, S. (2013) Measure solutions for non-local interaction PDEs with two species. Nonlinearity 26(10), 2777–2808.CrossRefGoogle Scholar
Di Marino, S., Gerolin, A. & Nenna, L. (2017) Optimal transportation theory with repulsive costs. In: Topological Optimization and Optimal Transport, Radon Ser. Comput. Appl. Math., Vol. 17, De Gruyter, Berlin, pp. 204–256.Google Scholar
Fujita, M. (1989) Urban Economic Theory: Land Use and City Size, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Jordan, R., Kinderlehrer, D. & Otto, F. (1998) The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17.CrossRefGoogle Scholar
Kinderlehrer, D., Monsaingeon, L. & Xu, X. (2017) A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations. ESAIM Control Optim. Calc. Var. 23(1), 137–164.CrossRefGoogle Scholar
Laborde, M. (2017) On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows. In: Topological Optimization and Optimal Transport, Radon Ser. Comput. Appl. Math., Vol. 17, De Gruyter, Berlin, pp. 304–332.Google Scholar
Lucas, R. E. & Rossi-Hansberg, E. (2002) On the internal structure of cities. Econometrica 70(4), 1445–1476.CrossRefGoogle Scholar
Mccann, R. J. (1997) A convexity principle for interacting gases. Adv. Math. 128(1), 153–179.CrossRefGoogle Scholar
Pass, B. (2015) Multi-marginal optimal transport: theory and applications. ESAIM Math. Model. Numer. Anal. 49(6), 1771–1790.CrossRefGoogle Scholar
Rossi, R. & Savaré, G. (2003) Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(2), 395–431.Google Scholar
Santambrogio, F. (2007) Transport and concentration problems with interaction effects. J. Global Optim. 38(1), 129–141.CrossRefGoogle Scholar
Santambrogio, F.Variational problems in transport theory with mass concentration, Tesi. Scuola Normale Superiore di Pisa (Nuova Series)], Vol. 4, Edizioni della Normale, Pisa, 2007.Google Scholar
Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser Verlag, Basel.CrossRefGoogle Scholar
Villani, C. (2003) Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58. American Mathematical Society, Providence, RI.Google Scholar
Villani, C. (2009) Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin. Old and new.CrossRefGoogle Scholar