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

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.