Mathematical theory for multi-dimensional coagulation models

Abstract

In this thesis, we consider problems that appear in the study of coagulation models. Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. The focus of this thesis consists in the analysis of models which incorporate additional information about the system of particles. <br /> In particular, we are interested in multi-component coagulation equations which offer information about the shape of the particles or about their position in space. We also study one-dimensional models where a source term that injects particles into the system is present. The common element of these new models is the appearance of a transport term in addition to the non-linear collision operator. <br /> We are mainly concerned with the existence of mass-conserving solutions and with the long-time behavior of solutions, which is often associated with the existence of self-similar profiles. In most of the mentioned cases, the contribution brought by the transport term can be used to control the contribution of the collision operator in order to prove the desired existence. In some instances, the transport term aids in extending the range of existence to collision rates for which in the case of the standard coagulation equation the system would not admit solutions or instantaneous loss of mass would occur. <br /> As such, another objective of this manuscript is to analyze the competition between the transport term and the collision term and to observe the new physical phenomena which may arise when one of the terms dominates over the other. In situations where the transport term is much larger than the collision term, we are able to show that the solutions to our models can be computed via solutions of the one-dimensional coagulation model.