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Nonlinear systems coupled through multi-marginal transport problems

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

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: maxime.laborde@mcgill.ca
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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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