Eindhoven University of Technology Lecture (Dr. Rafael Bailo 1)

• Eindhoven University of Technology

This special lecture was held at Kanazawa University in March 2026.
Lecturer:
Rafael Bailo (Eindhoven University of Technology, the Netherlands)
Title:
The Aggregation-Diffusion Equation
Abstract:
　Aggregation–diffusion equations are a prototypical class of non-linear PDEs describing the competition between attractive interactions and diffusive effects. They arise in models of collective behaviour, granular media, biological swarms, and self-organising systems. In this talk, I will introduce the aggregation–diffusion equation, explain the role of the interaction kernel and the diffusion term, and discuss the main qualitative behaviours of solutions, including concentration, pattern formation, and steady states.
　Starting from interacting particle systems, I will show how these equations can be derived through mean-field limits, highlighting the link between microscopic dynamics and macroscopic PDEs. I will then present the Wasserstein gradient-flow structure of the equations, which provides a unifying variational framework for well-posedness, long-time behaviour, and stability. Finally, I will discuss structure-preserving numerical methods inspired by this gradient-flow formulation, focusing on finite-volume and particle schemes that respect key properties such as positivity, mass conservation, and energy dissipation.

This lecture was conducted as part of the exchange program under the Inter-University Exchange Project "Program for Developing Human Resources to Lead Mathematical and Physical Sciences for the Advancement of Cutting-Edge Science in Japan and EU Countries".
https://eutenkai.w3.kanazawa-u.ac.jp/