Well-posedness to the continuous coagulation processes with collision-induced multiple fragmentation

TL;DR

This paper proves the existence and uniqueness of mass-conserving weak solutions for a nonlinear continuous coagulation and fragmentation model with singular kernels, using weak L^1 compactness methods.

Contribution

It establishes the first existence and uniqueness results for weak solutions to a complex coagulation-fragmentation model with collision-induced multiple fragmentation and singular kernels.

Findings

Existence of weak solutions for the model.

Uniqueness of solutions under certain growth conditions.

Mass conservation of the solutions.

Abstract

An existence result on weak solutions to the continuous coagulation equation with collision-induced multiple fragmentation is established for certain classes of unbounded coagulation, collision and breakup kernels. In this model, a pair of particles can coagulate into a larger one if their confrontation is a completely inelastic collision; otherwise, one of them will split into many smaller particles due to a destructive collision. In the present work, both coagulation and fragmentation processes are considered to be intrinsically nonlinear. The breakup kernel may have a possibility to attain a singularity at the origin. The proof is based on the classical weak L^1 compactness method applied to suitably chosen approximating equations. In addition, we study the uniqueness of weak solutions under additional growth conditions on collision and breakup kernels which mainly relies on the integrability of higher moments. Finally, it is obtained that the unique weak solution is mass-conserving.