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

  • Yann BrenierEmail author
  • Xianglong Duan


Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed as gradient flows of volume forms (or, in other words, 0-currents) according to suitable transportation metrics. In this paper, we present an example of gradient flow for closed \((d-1)\)-differential forms, or, more appropriately, to closed 1-currents, which can be identified to divergence-free vector fields, in the Euclidean space \(\mathbb {R}^d\). In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed, in a suitable sense, as an Eulerian version of the heat equation for loops 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.


Optimal transportation Gradient flow Differential forms Dissipative solutions Relative entropy 

Mathematics Subject Classification

49Q20 58A10 35K55 



This work has been partly supported by the ANR contract ISOTACE. The first author is grateful to the Erwin Schrödinger Institute for its hospitality during the first stage of this work. He also thanks the INRIA team MOKAPLAN where the work was partly completed.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS UMR 7640Ecole PolytechniquePalaiseauFrance

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