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
Department 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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