Existence of solutions to the continuous RednerBen-AvrahamKahng coagulation equation

TL;DR

This paper proves the existence of solutions for a specific coagulation equation modeling particle dynamics where particles decrease in size upon collision, focusing on product-type kernels with certain decay properties.

Contribution

It establishes the existence of solutions for a class of coagulation equations with decreasing particle size, extending the mathematical understanding of such models.

Findings

Solutions exist for product-type coagulation kernels.

The kernels satisfy specific decay conditions at infinity.

The model captures a unique particle collision dynamic.

Abstract

We take into account a coagulation model that simulates a distinct kind of dynamics. In this model, two particles collide to produce a single particle, but the resulting particle decreases in size, allowing each particle to be fully identified by its size. It is demonstrated that the corresponding evolving integral partial differential equation has solutions for product-type coagulation kernels i.e. $0 \le \mathfrak{K}(\varrho, \varsigma)= \mathfrak{K}(\varsigma, \varrho)=r(\varsigma) r(\varrho)+\alpha(\varsigma, \varrho), (\varsigma, \varrho)\in \mathbb{R}_+^2$ and $\sup_{\varsigma \in [0,R]} \frac{\mathfrak{K}(\varsigma, \varrho)}{\varrho} \to 0 \ \mbox{as} \ \varrho \to \infty$.