A uniqueness result for self-similar profiles to Smoluchowski's coagulation equation revisited
type: Item
instance of: scholarly article
title: A uniqueness result for self-similar profiles to Smoluchowski's coagulation equation revisited
zbMATH Open document ID: 1356.82034
published in: Journal of Statistical Physics
publication date: 2016-09-22T00:00:00Z
review text: In this note, the authors correct the proof of a previous result [the first and the third author, J. Stat. Phys. 157, No. 1, 158--181 (2014; Zbl 1302.82136)] on the uniqueness question of self-similar profiles of the Smoluchowski's coagulation equation. This is an integro-differential evolution equation, derived from a mean-field model, that describes a system of coagulating particles via their density function, where the variables of this function are the time and the size of the particles. The gap in the previous uniqueness proof was a wrongly used contraction argument for a very weak norm. They explain why this argument is wrong and why new ideas are needed. Here, the authors point out the need of some additional assumptions on the kernel in the integral operator, in order to obtain the uniqueness result for self-similar solutions with given mass (the result which was stated in the above-mentioned reference). Actually, this note only explains the new strategy how to fill the gap in the previous paper. The complete details of the proof are contained in a third paper [the authors, ``A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant'', Preprint, \url{arXiv:1510.03361}].
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35C06
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35Q82
zbMATH DE Number: 6629231
zbMATH Keywords: Smoluchowski's coagulation equation
zbMATH Keywords: uniqueness of solutions
zbMATH Keywords: self-similar profiles
author: Barbara Niethammer
author: S. Throm
author: Juan J. Velázquez
reviewed by: Alpár R. Mészáros
MaRDI profile type: Publication
full work available at URL: s10955-016-1553-5
OpenAlex ID: W2408822474
cites work: On self-similarity and stationary problem for fragmentation and coagulation models.
cites work: Existence of self-similar solutions to Smoluchowski's coagulation equation
cites work: Approach to self‐similarity in Smoluchowski's coagulation equations
cites work: Exponential Tail Behavior of Self-Similar Solutions to Smoluchowski's Coagulation Equation
cites work: Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant
DOI: 10.1007/S10955-016-1553-5
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description: scientific article; zbMATH DE number 6629231