Discrete coagulation--fragmentation systems in weighted $ll^1$ spaces

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove existence, uniqueness and positivity of solutions of a corresponding semi-linear Cauchy problem in a weighted $\ell^1$ space. This requires the application of novel results, which we prove for abstract semi-linear Cauchy problems in Banach lattices where the non-linear term is defined only on a dense subspace.

💡 Research Summary

The paper investigates an infinite system of ordinary differential equations that models the dynamics of clusters undergoing coagulation and fragmentation, where each cluster consists of identical monomer units. The state variable uₙ(t) denotes the number density of n‑mers at time t, and the evolution is governed by the classical coagulation‑fragmentation equations (1.1). Fragmentation rates aₙ and bₙ,ⱼ are assumed time‑independent, while coagulation rates kₙ,ⱼ(t) may depend on time. Unlike many previous works, the authors do not impose mass‑conservation conditions; they allow the total mass to increase or decrease during fragmentation events.

The core methodological contribution is the reformulation of (1.1) as a semilinear abstract Cauchy problem (ACP) in a weighted ℓ¹ space ℓ¹_w. The weight sequence w = (wₙ)ₙ consists of positive numbers; the physically most relevant choice is wₙ = n, for which the norm ‖u‖_{ℓ¹_w} coincides with the total mass M₁(t) = Σₙ n uₙ(t). The linear part A of the ACP corresponds to the pure fragmentation operator and is known to generate a sub‑stochastic C₀‑semigroup (as shown in earlier work