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Large time behaviour of exchange-driven growth
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Esenturk, Emre and Velazquez, Juan (2021) Large time behaviour of exchange-driven growth. Discrete and Continuous Dynamical Systems. Series A, 41 (2). pp. 747-775. doi:10.3934/dcds.2020299 ISSN 1078-0947.
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WRAP-large-time-behaviour-exchange-driven-growth-Esenturk-2020.pdf - Accepted Version - Requires a PDF viewer. Download (1020Kb) | Preview |
Official URL: https://www.aimsciences.org/article/doi/10.3934/dc...
Abstract
Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel K(j,k). Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel (i) K(j,k)=bjak and (ii) K(j,k)=jak+bj+εβjαk. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass ρs above which there is no equilibrium. We prove that if the system has an initial mass below ρs then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above ρs then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Engineering > WMG (Formerly the Warwick Manufacturing Group) |
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Library of Congress Subject Headings (LCSH): | Probabilities, Convergence , Symmetric operators | ||||||||||||
Journal or Publication Title: | Discrete and Continuous Dynamical Systems. Series A | ||||||||||||
Publisher: | American Institute of Mathematical Sciences | ||||||||||||
ISSN: | 1078-0947 | ||||||||||||
Official Date: | February 2021 | ||||||||||||
Dates: |
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Volume: | 41 | ||||||||||||
Number: | 2 | ||||||||||||
Page Range: | pp. 747-775 | ||||||||||||
DOI: | 10.3934/dcds.2020299 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Reuse Statement (publisher, data, author rights): | This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Large time behaviour of exchange-driven growth following peer review. The definitive publisher-authenticated version Esenturk, Emre and Velazquez, Juan (2020) Large time behaviour of exchange-driven growth. Discrete and Continuous Dynamical Systems. Series A . doi:10.3934/dcds.2020299 is available online at: http://dx.doi.org/10.3934/dcds.2020299 | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 14 August 2020 | ||||||||||||
Date of first compliant Open Access: | 18 August 2021 | ||||||||||||
RIOXX Funder/Project Grant: |
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