This thesis is dedicated to the variational and numerical study of a particular class of continuity equations called aggregation-diffusion equations. They model the evolution of a continuum body whose total mass is conserved in time, undergoing up to three distinct phenomena: diffusion, confinement and aggregation. Diffusion describes the motion of the body’s particles from crowded regions of space to sparser ones; confinement results from an external potential field independent of the mass distribution of the body; and aggregation describes the nonlocal particle interaction within the body. Due to this wide range of effects, aggregation-diffusion equations are encountered in a large variety of applications coming from, among many others, porous medium flows, granular flows, crystallisation, biological swarming, bacterial chemotaxis, stellar collapse, and economics. An aggregation-diffusion equation has the very interesting and rich mathematical property of being the gradient flow for some energy functional on the space of probability measures, which formally means that any solution evolves so as to decrease this energy every time as much as possible. In this thesis we exploit this gradient-flow structure of aggregation-diffusion equations in order to derive properties of solutions and approximate them by discrete particles. We focus on two main aspects of aggregation-diffusion gradient flows: the variational analysis of the pure aggregation equation, i.e., the study of minimisers of the energy when only nonlocal aggregation effects are present; and the particle approximation of solutions, especially when only diffusive effects are taken into account. Regarding the former aspect, we prove that minimisers exist, enjoy some regularity, are supported on sets of specific dimensionality, and can be approximated by finitely supported discrete minimisers. Regarding the latter aspect, we illustrate theoretically and numerically that diffusion can be interpreted at the discrete level by a deterministic motion of particles preserving a gradient-flow structure.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:724186 |
| Date | January 2017 |
| Creators | Patacchini, Francesco Saverio |
| Contributors | Carrillo de la Plata, José Antonio ; Huang, Yanghong |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/50180 |
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