Aggregation-diffusion equations in biology with a gradient flow structure

Thesis

This thesis is concerned with the analysis of non-linear partial differential equations arising naturally from biological models. These models include non-linearities, long-range interactions, or local effects which pose a challenge that require new PDE techniques. In this thesis, the main tools that we need are Otto calculus, Wasserstein gradient flows, viscosity solutions, C₀-semigroup theory, and finite volumes.

The models we are interested in show a dichotomy between diffusion and aggregation. Therefore, one of the main question is to understand the long-time dynamics and to check wether diffusion, aggregation, or a mix of both dominates the behaviour of the equation. Hence, this work contains various parabolic PDEs of aggregation-diffusion type for which we analyse different properties. For example existence, uniqueness, longtime behaviour, steady states, or minimisers of the associated free energy functional, among others.

Chapter 1 is an introduction, presenting the mathematical context, motivations and necessary tools for the chapters to follow. Chapter 2 to 4 each correspond to a manuscript. Chapter 5 presents new outcomes and perspectives.

Authors: Fernández Jiménez, A

• DOI: 10.5287/ora-gw4o9kdpp
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: nonlinear parabolic equations; variational problems; Cahn-Hilliard; saturation; aggregation-diffusion; Wasserstein gradient flows; long-time behaviour
• Subjects: Differential equations, Partial