Research Projects

The coagulation-fragmentation (C-F) equations play a fundamental role in many scientific and engineering disciplines, such as polymer formation and degradation, aerosol science, raindrop formation and neurodegenerative disease. These equations are studied for discrete or continuous particle sizes, depending on the physical model. In neurodegenerative diseases such as the Alzheimer’s disease, the transport of β-amyloid oligomers through brain tissue significantly influences disease progression. These oligomers undergo diffusion, coagulation, and fragmentation within the brain’s complex extracellular environment. To model this behavior, the classical C-F equation is extended by adding a spatial diffusion term, resulting in the diffusive C-F equation. This equation captures the interplay between the spatial spread of β-amyloid (diffusion), the formation of larger aggregates (coagulation), and the breakage of fibrils into smaller, potentially toxic oligomers (fragmentation). Fragmentation can be classified into two types depending on the physical setting: linear and nonlinear. In the study of β-amyloid’s evolution in the cerebral tissue, there are works which consider these diffusive C-F equations on perforated domains (representing the brain tissue) where the perforations represent the neurons. The study of effective behavior of phenomena in perforated domains when the perforations reduce in size (but increase in number within the domain) can be handled by the theory of homogenization. The homogenization of the discrete diffusive coagulation equation with and without linear fragmentation in the perforated domains has been studied in the literature. This project aims to derive the homogenized equation for the discrete diffusive coagulation equation with nonlinear fragmentation. In the literature so far, homogenized equations have only been obtained for the discrete coagulation equations with linear fragmentation, and not for its continuous counterpart. While existence and uniqueness results are well established for the continuous diffusive coagulation equation with linear fragmentation, the corresponding homogenized equation has not been derived yet. Thus, the next goal would be to obtain this homogenized equation for the continuous version of the equation in perforated domains. The connection between discrete and continuous C-F equations has been studied and it shown that the continuous model can be approximated by a sequence of discrete models. We emphasise here that such results are available only in the context of binary fragmentation. Our objective will be to establish such a “discrete-to-continuum” limit for general (non-binary) processes. After achieving our aforementioned objectives regarding the derivation of homogenized models for discrete and continuous diffusive C-F equations, we aim to explore the commutativity property of the two limits, i.e. homogenization limit and discrete-to-continuum limit.