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

Iacobelli, Mikaela; Patacchini, Francesco; Santambrogio, Filippo

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.

Citation

Iacobelli, M., Patacchini, F., & Santambrogio, F. (2019). Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour. Archive for Rational Mechanics and Analysis, 232(3), 1165-1206. https://doi.org/10.1007/s00205-018-01341-w

• Journal Article Type: Article
• Acceptance Date: Nov 23, 2018
• Online Publication Date: Dec 18, 2018
• Publication Date: Jun 30, 2019
• Deposit Date: Nov 5, 2018
• Publicly Available Date: May 10, 2019
• Journal: Archive for Rational Mechanics and Analysis
• Print ISSN: 0003-9527
• Electronic ISSN: 1432-0673
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 232
• Issue: 3
• Pages: 1165-1206
• DOI: https://doi.org/10.1007/s00205-018-01341-w
• Public URL: https://durham-repository.worktribe.com/output/1309756
• Related Public URLs: https://arxiv.org/abs/1808.07743v2