Optimal bounds for self-similar solutions to coagulation equations with product kernel

TL;DR

This paper rigorously proves the singular behavior of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with a product kernel, confirming a long-standing conjecture about their asymptotic form near zero.

Contribution

It establishes the precise asymptotic behavior of solutions with a multiplicative kernel, advancing understanding of their singularity structure.

Findings

Solutions behave like x^{-(1+2λ)} as x approaches zero

Confirmed the conjectured singular behavior for the first time

Provides rigorous bounds for self-similar solutions

Abstract

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with multiplicative kernel of homogeneity $2l\lambda \in (0,1)$. We establish rigorously that such solutions exhibit a singular behavior of the form $x^{-(1+2\lambda)}$ as $x \to 0$. This property had been conjectured, but only weaker results had been available up to now.