Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed 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 the Euclidean space R^d. In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed as the Eulerian version 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.