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Publication 18-CNA-017

Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour

Mikaela Iacobelli
Department of Mathematical Sciences
Durham University
Durham, Lower Mountjoy, DH1 3LE, UK
mikaela.iacobelli@durham.ac.uk

Francesco S. Patacchini
Deparment of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15203, USA
fpatacch@math.cmu.edu

Filippo Santambrogio
Laboratoire de Mathématiques d’Orsay
Univ. Paris-Sud, CNRS, Université Paris-Saclay
91405 Orsay, France
filippo.santambrogio@math.u-psud.fr

Abstract: 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.

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