Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

TL;DR

This paper establishes the existence of weak solutions for a class of nonlinear mobility aggregation-diffusion PDEs using a particle approximation method, addressing degenerate diffusion and nonlinear mobility complexities.

Contribution

It introduces a deterministic particle approximation scheme for aggregation-diffusion equations with nonlinear mobility, including degenerate cases, and proves existence of weak solutions.

Findings

Existence of weak solutions for nonlinear mobility aggregation-diffusion PDEs.

The particle approximation scheme effectively handles degenerate diffusion.

Numerical simulations support the theoretical results.

Abstract

We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to nonnegative initial data in $L^{\infty} \cap BV$ away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function. As a consequence, our result applies also to strongly degenerate diffusion equations. The conclusions are complemented with some numerical simulations.