On the Fundamental Solution of a Homogeneous Linearized Coagulation Equation

TL;DR

This paper derives an explicit fundamental solution for a linearized coagulation equation with a specific kernel, providing detailed asymptotic estimates and analyzing particle fluxes between sizes.

Contribution

It presents the first explicit representation and asymptotic analysis of the fundamental solution for the linearized Smoluchowski coagulation equation with a multiplicative kernel.

Findings

Explicit formula for the fundamental solution

Asymptotic behavior of the solution analyzed

Particle fluxes between sizes characterized

Abstract

In this paper we study the fundamental solution of the equation obtained by the linearisation of the Smoluchowski coagulation equation with the multiplicative kernel $(x y)^{\lambda/2}$ with $\lambda\in (1, 2)$ around the steady state $f(x)=x^{-(3+\lambda)/2}$. An explicit representation formula as well as detailed estimates on its asymptotics are obtained. We also describe in a detailed form particle fluxes between different sizes for this linearised equation.