Inserted: 22 aug 2018
Last Updated: 22 aug 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.