Local mass-conserving solution for a critical Coagulation-Fragmentation equation

TL;DR

This paper demonstrates the existence of local mass-conserving solutions for a critical coagulation-fragmentation equation with specific initial conditions by employing a singular perturbation approach and analyzing the limiting behavior through the Bernstein transform.

Contribution

It introduces a novel singular perturbation method to establish local mass conservation for a critical coagulation-fragmentation equation with finite second moment initial data.

Findings

Existence of a unique mass-conserving solution up to a finite time T*

Use of Bernstein transform to analyze solution limits

Perturbation of fragmentation kernel enables mass conservation analysis

Abstract

The critical coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels is known to not have global mass-conserving solutions when the initial mass is greater than $1$. We show that for any given positive initial mass with finite second moment, there is a time $T^*>0$ such that the equation possesses a unique mass-conserving solution up to $T^*$. The novel idea is to singularly perturb the constant fragmentation kernel by small additive terms and study the limiting behavior of the solutions of the perturbed system via the Bernstein transform.