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Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel
About this Title
Sebastian Throm
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1328
ISBNs: 978-1-4704-4786-1 (print); 978-1-4704-6634-3 (online)
DOI: https://doi.org/10.1090/memo/1328
Published electronically: July 23, 2021
Keywords: Smoluchowski’s coagulation equation,
self-similarity,
fat tails,
uniqueness of profiles,
boundary layer
Table of Contents
Chapters
- 1. Introduction
- 2. Functional setup and preliminaries
- 3. Uniqueness of profiles – Proof of Theorem
- 4. Continuity estimates
- 5. Linearised coagulation operator – Proof of Proposition
- 6. Uniform bounds on self-similar profiles
- 7. The boundary layer estimate
- 8. The representation formula for $H_{0}(\cdot ,q)$
- 9. Integral estimate on $\mathfrak {H}_{0}(\cdot ,q)$
- 10. Asymptotic behaviour of several auxiliary functions
- A. Useful elementary results
- B. The representation formula for $W$
- C. Existence of profiles
Abstract
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 $K=2+\varepsilon W$. 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 $\varepsilon$ 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.
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