Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel

TL;DR

This paper investigates a specific coagulation-fragmentation model using a novel approach involving viscosity solutions to a singular Hamilton-Jacobi equation, providing insights into the well-posedness and long-term behavior of solutions.

Contribution

It introduces a new method based on viscosity solutions to analyze a critical coagulation-fragmentation equation with multiplicative and constant kernels, establishing well-posedness and long-time behavior results.

Findings

Proved well-posedness of the model

Analyzed regularity of solutions

Described long-time asymptotic behavior

Abstract

We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of \emph{mass-conserving} solutions to the Coagulation-Fragmentation equation.