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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

Francesco S. Patacchini
Deparment of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15203, USA

Filippo Santambrogio
Laboratoire de Mathématiques d’Orsay
Univ. Paris-Sud, CNRS, Université Paris-Saclay
91405 Orsay, France

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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