Long-time behaviour for kinetic models of clustering and non-local diffusion equations

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Agglomeration and clustering are quite universal phenomena which appear in a broad range of different natural and industrial processes. Some typical examples are the formation of rain drops, the creation of polymers, formation of planets but also biological processes such as the creation of so-called marine snow. The common feature of all these phenomena, which occur on a very broad range of length scales, is the formation of larger clusters due to the coalescence with smaller particles. One of the fundamental problems in this context, which is also the main goal of this project, consists in understanding the long-time behaviour of this coarsening process as precisely as possible. One of the most important mathematical models for the description of the phenomena above is Smoluchowski's coagulation equation. For the latter, numerical simulations suggest that the long-time behaviour is self-similar. With the exception of very few special cases, for which explicit solution formulas exist, this so-called scaling hypothesis has not yet been proven. The main concern of this project is thus the proof of this conjecture for certain selected models for which no explicit solutions can be given. Moreover, the long-time behaviour of a simplified model for agglomeration is to be studied, where a target cluster moves through a random background of particles and coalesces with them upon touching. The expected evolution of the size of the target cluster for large times is again self-similar. Supplementary, non-local diffusion equations are to be considered as they arise on the one hand naturally in the consideration of coagulation models while on the other hand their study exhibits similar difficulties.