Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation

TL;DR

This paper proves the convergence of a finite-volume scheme for aggregation-diffusion equations with saturation, extending previous methods to cases with free boundaries and less regular mobility and potentials.

Contribution

It establishes convergence results for the scheme under broader conditions, including cases with free boundaries and less regular data, using discrete compactness arguments.

Findings

Convergence proven for general entropies with smooth initial data and potentials.

Convergence shown for initial data with free boundaries under weaker regularity.

The scheme is validated for a wider class of aggregation-diffusion models.

Abstract

In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin--Simons compactness theorem due to Gallou\"et and Latch\'e. We use suitable discrete $H^1$ and $W^{-1,1}$ discrete norms. We provide two convergence results. A first result shows convergence with general entropies ($U$) (including singular and degenerate) if the initial datum does not have free boundaries, the mobility is Lipschitz, and the confinement ($V$) and aggregation ($K$) potentials are $W^{2,\infty}_0$. A second result shows convergence when the initial datum has free boundaries, mobility is just continuous, and $V$ and $K$ are $W^{1,\infty}$, but under the assumption that the entropy $U$ is $C^1$ and strictly convex.