Fragmentation-coagulation equation with growth

Abstract

The theory of fragmentation-coagulation equations began around 1916 with a series of papers by Smoluchowski on pure coagulation and since then continued to incorporate other processes into the model. The intention was to study the evolution of objects undergoing breakdown and/or merging. The scientific goals are to determine the conditions under which solutions exist, are unique and identify them accordingly.

In this study, we considered the continuous fragmentation-coagulation equation with transport (decay or growth), subject to homogenous/McKendrick-von Foerster boundary condition in the latter case. The theory of semigroups of linear operators and, in particular, the Miyadera-Desch perturbation theorem are used to show the existence of semigroup solutions for the linear transport-fragmentation equation. We proved that the established semigroups have the moment improving property. The latter result plays a crucial role in the analysis of the complete transport-fragmentation-coagulation equation which is treated as a Lipschitz perturbation of the former linear problem. Under mild restrictions on the model coefficients, the existence of positive local classical solutions is established. Further, under additional conditions, their global in time existence is proved. Finally, a systematic technique is developed for obtaining closed-form solutions to continuous transport-fragmentation equations with homogenous boundary conditions and power-law coefficients. New solutions for the constant and linear decay/growth coefficients are presented. Furthermore, it is shown that the technique extends to some cases of the growth-fragmentation equation with the McKendrick-von Foerster boundary condition.