Aggregation-Diffusion Equations for Collective Behaviour in the Sciences

TL;DR

This survey frames aggregation-diffusion equations as gradient flows in the $2$-Wasserstein space, unifying microscopic many-particle models and macroscopic PDEs through the free-energy functional $\mathcal{F}[\rho]$ and the JKO scheme. It surveys analytical results on minimisers, steady states, symmetry, and regime-dependent phenomena under degenerate and fast diffusion, highlighting critical exponents and concentration effects governed by Hardy–Littlewood–Sobolev-type inequalities and their reversals. The article also reviews numerical schemes that preserve mass, positivity, and energy dissipation, and connects theory to applications in mathematical biology (notably cell adhesion and chemotaxis) and broader areas such as optimisation and machine learning. Together, these threads illuminate how diffusion, external drift, and nonlocal interactions shape pattern formation, long-time behavior, and computational approaches across disciplines.

Abstract

This is a survey article based on the content of the plenary lecture given by José A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.

Figures (6)

• Figure 1: Interaction forces of an agent-based model for cell adhesion, see CARRILLO201875carrillo2019population.
• Figure 1: Numerical approximation of a geodesic curve using the $2$-Wasserstein distance between the characteristic sets of Pac-Man and the Ghost (suitably normalised). https://figshare.com/articles/media/Wasserstein_Geodesic_between_PacMan_and_Ghost/7665377.
• Figure 1: In vitro experiments from katsunuma2016synergistic compared to in silico experiments from carrillo2019population. The numerical schemes are based on finite volumes, as in BCH2020BCH2023. https://figshare.com/articles/media/Front_propagation_and_intermingling_of_cell_types_experiments_versus_mathematical_model_simulations_/7707890.
• Figure 1: Sketch of the main results known for the fast diffusion range. The dark grey region corresponds to the area in which global minimisers of the free energy are integrable. The arrowed red line is the region for which we know the exact value of $m$ separating integrable from partially mass-concentrated global minimisers. The remaining white region in Zone II corresponds to the set of parameters in which the separation of integrable and partial mass concentrated global minimisers is not known yet.
• Figure 1: Metastability in the aggregation-diffusion equation \ref{['aggeqn0']} under a force with exponential decay from BCH2020. Long periods of slow motion are interspersed by rapid merging phases until a steady state is reached.

Theorems & Definitions (3)

• Theorem 4.1: Homogeneous Interaction Potential CarrilloDelgadinoPatacchini2019carrillo2019ReverseHardyLittlewoodCCH17
• Proposition 5.1: Conservation of mass and non-negativity
• Theorem 5.2: Energy dissipation