On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour

TL;DR

This paper proves the existence, uniqueness, and long-term behavior of mass-conserving solutions to the Safronov-Dubovskii coagulation equation with at most linear growth kernels, using a novel approach based on the de la Vallee-Poussin theorem.

Contribution

It introduces a new proof technique for well-posedness of the coagulation equation relying on the finiteness of the first moment, expanding understanding of solution regularity and asymptotics.

Findings

Existence of mass-conserving weak solutions for kernels with linear growth.

Weak solutions are shown to be classical under regularity conditions.

Solutions depend continuously on initial data and exhibit specific asymptotic behavior.

Abstract

The global existence of mass-conserving weak solutions to the Safronov-Dubovskii coagulation equation is shown for the coagulation kernels satisfying the at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallee-Poussin theorem [8, Theorem 7.1.6], which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions con-structed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large-time behaviour of solutions are also addressed.