A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant

TL;DR

This paper revises and corrects the proof of the uniqueness of self-similar solutions to Smoluchowski's coagulation equation for kernels close to constant, under specific regularity and asymptotic conditions, for small perturbations.

Contribution

It provides a corrected proof establishing the uniqueness of self-similar profiles for near-constant kernels with detailed analytic and asymptotic assumptions.

Findings

Uniqueness holds for kernels close to constant with small perturbation parameter.

The proof requires the kernel to have an analytic extension and specific asymptotic behavior.

Results apply to kernels with homogeneity degree zero and perturbations within a controlled range.

Abstract

In this article we correct the proof of a uniqueness result for self-similar solutions to Smoluchowski's coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that \begin{equation*} -\varepsilon \leq K(x,y)-2 \leq \varepsilon \left( \Big(\frac{x}{y}\Big)^{\alpha} + \Big(\frac{y}{x}\Big)^{\alpha}\right) \end{equation*} for $\alpha \in [0,\frac 1 2)$. Assuming in addition that $K$ has an analytic extension to $\mathbb{C}\setminus(-\infty,0]$ and prescribing the precise asymptotic behaviour of $K$ at the origin, we prove that self-similar solutions with given mass are unique if $\varepsilon$ is sufficiently small.