Topics on Aggregation-Diffusion PDEs
MATH-748 / 2 crédits
Enseignant: Invited lecturers (see below)
Langue: Anglais
Remark: Spring semester: 15.4-30.06.2027
Frequency
Only this year
Summary
The main goal is to analyse aggregation-diffusion PDEs. These ubiquitous models are descriptions of many body interacting particle systems with applications in physics, biology and data science. Calculus of variations, optimal transport and PDE tools will be used for understanding their properties.
Content
Part I Introduction and preliminaries
1. Motivation and examples: bacterial chemotaxis, granular flows, damping dominated systems, opinion dynamics, data science.
2. Aggregation-Diffusion Equations: a brief introduction
3. A primer on optimal transport theory
Part II The Aggregation Equation
4. Nonlocal interactions via smooth potentials
5. Nonlocal interaction via mildly singular potentials and one dimensional aggregation equations
6. The case of singular interaction potentials
7. Minima of nonlocal interaction energies
Part III The case with diffusion
8. Linear diffusion: Periodic case and bifurcations.
9. A primer on nonlinear diffusions: Classical and Variational Approach
10. A priori estimates and functional inequalities
11. Asymptotic behavior: the case of homogeneous potentials
Keywords
Nonlinear Fokker-Planck equations, optimal transport, gradient flows, free energies, minimizers, HLS inequalities, asymptotic behavior, bifurcations.
Learning Prerequisites
Required courses
Basic PDE course
Recommended courses
Optimal transport course but not necessary
Learning Outcomes
By the end of the course, the student must be able to:
- Analyze Qualitative properties of solutions of nonlocal aggregation-diffusion equations, Long time aysmptotics. Mininimizers of interaction energies.
Resources
Bibliography
Own material, course given at the University of Oxford and book in preparation.
Moodle Link
Dans les plans d'études
- Forme de l'examen: Exposé (session libre)
- Matière examinée: Topics on Aggregation-Diffusion PDEs
- Cours: 22 Heure(s)
- Projet: 12 Heure(s)
- Type: optionnel