Discrete fragmentation systems in weighted ℓ1 spaces

Lyndsay Kerr, Wilson Lamb, Matthias Langer

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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.
Original languageEnglish
Pages (from-to)1419-1451
Number of pages15
JournalJournal of Evolution Equations
Volume20
Issue number4
Early online date7 Feb 2020
DOIs
Publication statusPublished - Dec 2020

Keywords

  • discrete fragmentation
  • positive semigroup
  • analytic semigroup
  • long-time behaviour
  • Sobolev towers

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