Discrete fragmentation systems in weighted ℓ1 spaces

Kerr, Lyndsay and Lamb, Wilson and Langer, Matthias (2020) Discrete fragmentation systems in weighted ℓ1 spaces. Journal of Evolution Equations, 20 (4). pp. 1419-1451. ISSN 1424-3199 (https://doi.org/10.1007/s00028-020-00561-6)

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Abstract

We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted ℓ1 space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted ℓ1 space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well-posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.

ORCID iDs

Kerr, Lyndsay ORCID logoORCID: https://orcid.org/0000-0002-6667-7175, Lamb, Wilson ORCID logoORCID: https://orcid.org/0000-0001-8084-6054 and Langer, Matthias ORCID logoORCID: https://orcid.org/0000-0001-8813-7914;
  • Item type:Article
    ID code:71402
    Dates:
    Date
    Event
    December 2020
    Published
    7 February 2020
    Published Online
    8 January 2020
    Accepted
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:10 Feb 2020 09:19
    Last modified:11 Nov 2024 12:33
    URI:https://strathprints.strath.ac.uk/id/eprint/71402
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