The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel

TL;DR

This paper proves that solutions to a perturbed Smoluchowski's coagulation equation with a bounded, zero-homogeneity kernel approach a universal self-similar profile at an exponential rate, extending the scaling hypothesis to non-explicit kernels.

Contribution

It provides the first full proof of the scaling hypothesis for a family of kernels that are not explicitly solvable, using a constructive perturbation approach.

Findings

Solutions converge to a universal self-similar profile

Convergence occurs at an exponential rate in self-similar variables

Explicit estimates of constants are provided

Abstract

We consider Smoluchowski's coagulation equation with a kernel of the form $K = 2 + \epsilon W$, where $W$ is a bounded kernel of homogeneity zero. For small $\epsilon$, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.