Uniqueness of Fat-Tailed Self-Similar Profiles to Smoluchowski?s Coagulation Equation for a Perturbation of the Constant Kernel

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski's coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel K which can be written as K = 2 + epsilon W. The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on W, we will show that for sufficiently small epsilon there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile. Establishing uniqueness of self-similar profiles for Smoluchowski's coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

Automated summary

This article addresses the uniqueness of self-similar profiles for Smoluchowski's coagulation equation with algebraic decay (fat tails) at infinity. By considering a rate kernel and perturbation, it shows that under certain regularity assumptions, there exists at most one self-similar profile for sufficiently small perturbations. This is the first statement of uniqueness for a non-solvable kernel in the context of fat-tailed self-similar profiles.