Well-posedness of the coagulation-fragmentation equation with size diffusion

Abstract

describes the dynamics of the size distribution function φ = φ(t, x) ≥ 0 of particles of size x ∈ (0,∞) at time t > 0. Particles modify their sizes according to three different mechanisms: random fluctuations, here accounted for by size diffusion at a constant diffusion rate D > 0 (hereafter normalized toD = 1), spontaneous fragmentation with overall fragmentation rate a ≥ 0 and daughter distribution function b ≥ 0, and binary coalescence with coagulation kernel k ≥ 0. Nucleation is not taken into account in this model, an assumption which leads to the homogeneous Dirichlet boundary condition (1.1b) at x = 0. Let us recall that the coagulation-fragmentation equation without size diffusion, corresponding to setting D = 0 in (1.1a), arises in several fields of physics (grain growth, aerosol and raindrops formation, polymer and colloidal chemistry) and biology (hematology, animal grouping) and has been studied extensively in the mathematical literature since the pioneering works