Calculus of Variations and Geometric Measure Theory

M. Iacobelli - F. S. Patacchini - F. Santambrogio

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

created by santambro on 22 Aug 2018
modified on 23 Oct 2018


Accepted Paper

Inserted: 22 aug 2018
Last Updated: 23 oct 2018

Journal: Arch. Rati. Mech. An.
Year: 2018


n 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 $H^1$ estimates, $L^1$ weighted contractivity, instantaneous regularisation, Harnack inequalities, and exponential convergence to a steady state.