Porous medium equation and cross-diffusion systems as limit of nonlocal interaction

TL;DR

This paper demonstrates how nonlocal interaction equations and systems converge to the quadratic porous medium equation and cross-diffusion systems using gradient flow techniques, even at low regularity.

Contribution

It introduces a novel discretisation scheme that handles nonlocal to local PDE limits without requiring energy convexity or gradient flow structure.

Findings

Proves convergence of nonlocal models to local PDEs in the limit of localized interactions.

Develops bounds on energy, moments, and entropy for the solutions.

Extends results to weak solutions with low regularity.

Abstract

This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of the quadratic porous medium equation, resp. cross-diffusion system, in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use gradient flow techniques to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme, which can be slightly modified in order to extend our result to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy.