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

TL;DR

This paper studies weighted ultrafast diffusion equations related to probability measure quantisation, establishing well-posedness, regularity, and long-term behavior using gradient flow techniques and the JKO scheme.

Contribution

It introduces a new framework for weak solutions of weighted ultrafast diffusion equations and proves key properties including existence, uniqueness, and exponential convergence.

Findings

Existence and uniqueness of weak solutions

Exponential convergence to steady state

Harnack inequalities and regularity estimates

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