  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.
