Continuum of coupled Wasserstein gradient flows

TL;DR

This paper develops a mathematical framework for analyzing an infinite system of coupled drift-diffusion PDEs using Wasserstein gradient flows, establishing existence, stability, and convergence results, and demonstrating the approach with numerical examples.

Contribution

It introduces a novel interpretation of coupled PDEs as gradient flows in the space of couplings with a fibered Wasserstein metric, proving convergence and stability in the infinite-phase limit.

Findings

Existence of weak solutions for the coupled PDE system

Unconditional convergence of the entropic relaxation scheme

Numerical illustrations confirming theoretical results

Abstract

We study a system of drift-diffusion PDEs for a potentially infinite number of incompressible phases, subject to a joint pointwise volume constraint. Our analysis is based on the interpretation as a collection of coupled Wasserstein gradient flows or, equivalently, as a gradient flow in the space of couplings under a `fibered' Wasserstein distance. We prove existence of weak solutions, long-time asymptotics, and stability with respect to the mass distribution of the phases, including the discrete to continuous limit. A key step is to establish convergence of the product of pressure gradient and density, jointly over the infinite number of phases. The underlying energy functional is the objective of entropy regularized optimal transport, which allows us to interpret the model as the relaxation of the classical Angenent-Haker-Tannenbaum (AHT) scheme to the entropic setting. However, in contrast to the AHT scheme's lack of convergence guarantees, the relaxed scheme is unconditionally convergent. We conclude with numerical illustrations of the main results.