Nonlinear systems coupled through multi-marginal transport problems

TL;DR

This paper introduces a dynamical urban planning model involving nonlinear systems coupled via multi-marginal optimal transport, utilizing Wasserstein gradient flows and establishing uniqueness in one dimension.

Contribution

It presents a novel framework for solving coupled nonlinear equations in urban planning using multi-marginal optimal transport and Wasserstein gradient flows, including a uniqueness result.

Findings

Effective framework for solving coupled nonlinear systems

Application of Wasserstein gradient flows to complex transport problems

Uniqueness of solutions in one-dimensional cases

Abstract

In this paper, we introduce a dynamical urban planning model. This leads us to study a system of nonlinear equations coupled through multi-marginal optimal transport problems. A simple case consists in solving two equations coupled through the solution to the Monge-Amp{\`e}re equation. We show that the Wasserstein gradient flow theory provides a very good framework to solve this highly nonlinear system. At the end, an uniqueness result is presented in dimension one based on convexity arguments.