Mass-conserving solutions to coagulation-fragmentation equations with non-integrable fragment distribution function

TL;DR

This paper proves the existence of mass-conserving solutions to coagulation-fragmentation equations with a non-integrable fragment distribution, allowing for infinite fragments and minimal assumptions on growth rates.

Contribution

It introduces a framework for solutions where the fragmentation produces infinitely many fragments and handles non-integrable distributions without growth restrictions.

Findings

Established existence of mass-conserving solutions

Handled non-integrable fragment distributions

Allowed for infinite fragmentation scenarios

Abstract

Existence of mass-conserving weak solutions to the coagulation-fragmentation equation is established when the fragmentation mechanism produces an infinite number of fragments after splitting. The coagulation kernel is assumed to increase at most linearly for large sizes and no assumption is made on the growth of the overall fragmentation rate for large sizes. However, they are both required to vanish for small sizes at a rate which is prescribed by the (non-integrable) singularity of the fragment distribution.