An integrable example of gradient flows based on optimal transport of differential forms

TL;DR

This paper introduces a fully integrable gradient flow system based on optimal transport of differential forms, providing new insights into parabolic equations and their solutions in Euclidean space.

Contribution

It presents a novel integrable example of gradient flows for differential forms, extending optimal transport theory to complex geometric structures.

Findings

The system is fully integrable despite its complexity.

Global existence and weak-strong uniqueness are established.

The system is a Euclidean analogue of the heat equation for curves.

Abstract

Optimal transport theory has been a powerful tool for the analysis of parabolic equationsviewed as gradient flows of volume forms according to suitable transportation metrics.In this paper, we present an example of gradient flows for closed (d-1)-forms in theEuclidean space R^d. In spite of its apparent complexity, the resulting verydegenerate parabolic system is fully integrable and can be viewed as the Eulerianversion of the heat equation for curves in the Euclidean space.We analyze this system in terms of "relative entropy" and "dissipative solutions" and provide global existence and weak-strong uniqueness results.