Nonlocal approximation of nonlinear diffusion equations
Abstract
We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of -Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.
Cite
@article{arxiv.2302.08248,
title = {Nonlocal approximation of nonlinear diffusion equations},
author = {José Antonio Carrillo and Antonio Esposito and Jeremy Sheung-Him Wu},
journal= {arXiv preprint arXiv:2302.08248},
year = {2023}
}
Comments
39 pages, revised with more precise scaling for modulus of convexity in Section 6