Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour

  • Mikaela IacobelliEmail author
  • Francesco S. Patacchini
  • Filippo Santambrogio
Open Access


In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H1 estimates, L1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.



MI thanks Matteo Bonforte for his comments to a preliminary version of this manuscript and José A. Carrillo for interesting discussions on this topic during a visit at Imperial College in 2015. FSP thanks the CNA at Carnegie Mellon University for their kind support. FS thanks Imperial College for their warm hospitality in 2017 when this work started, via a CNRS-Imperial fellowship.

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  • Mikaela Iacobelli
    • 1
    Email author
  • Francesco S. Patacchini
    • 2
  • Filippo Santambrogio
    • 3
  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUK
  2. 2.Deparment of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  3. 3.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

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