Asymptotics of self-similar solutions to coagulation equations with product kernel

TL;DR

This paper analyzes the asymptotic behavior of self-similar solutions to Smoluchowski's coagulation equation with a product kernel as the parameter lambda approaches zero, revealing detailed oscillatory and peak structures.

Contribution

It provides a detailed formal asymptotic description of the qualitative behavior of rescaled solutions in the limit lambda to zero, including oscillations and peak formations.

Findings

h(x) ~ 1 + C x^{lambda/2} cos(sqrt(lambda) log x) as x -> 0

h develops peaks of height 1/lambda separated by large regions

h converges to zero exponentially fast as x -> infinity

Abstract

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.