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Nonlocal approximation of nonlinear diffusion equations

Authors :
Carrillo, José Antonio
Esposito, Antonio
Wu, Jeremy Sheung-Him
Publication Year :
2023

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 $2$-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.<br />Comment: 39 pages, revised with more precise scaling for modulus of convexity in Section 6

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2302.08248
Document Type :
Working Paper