Deterministic particle approximation of aggregation diffusion equations with nonlinear mobility

TL;DR

This paper introduces a deterministic particle approximation method for aggregation-diffusion equations with nonlinear mobility, proving strong convergence to weak solutions without requiring BV or vacuum assumptions.

Contribution

It extends previous techniques to establish convergence of particle schemes for aggregation-diffusion PDEs with nonlinear mobility on unbounded domains.

Findings

Proves strong $L^1$-convergence of the particle scheme.

Handles equations with Lipschitz nonincreasing mobility functions.

Provides well-posedness results without BV or vacuum assumptions.

Abstract

We consider a class of aggregation-diffusion equations on unbounded one dimensional domains with Lipschitz nonincreasing mobility function. We show strong $L^1$-convergence of a suitable deterministic particle approximation to weak solutions of a class aggregation-diffusion PDEs (coinciding with the classical ones in the no vacuum regions) for any bounded initial data of finite energy. In order to prove well-posedness and convergence of the scheme with no BV or no vacuum assumptions and overcome the issues posed in this setting by the presence of a mobility function, we improve and strengthen the techniques introduced in arXiv:2012.01966(2).