Convergence of a discrete Oort-Hulst-Safronov equation

Abstract

Communicated by M. Lachowicz SUMMARY A discrete version of the Oort-Hulst-Safronov (OHS) coagulation equation is studied. Besides the existence of a solution to the Cauchy problem, it is shown that solutions to a suitable sequence of those discrete equations converge towards a solution to the OHS equation.

A coagulation–disintegration model of Oort–Hulst cluster-formation

“…Note that the last two equations decouple from the rest of the system. In section 3.1.1 we describe the steady-state solution and, in the remainder of section 3.1, we analyse the special case where k a, taking a = O (1), and k 1, and assume initial conditions of the form…”

“…The steady solution of (3.1) is c j = c 1 e −λ( j− 1) , where e λ = 1 + kM 0 /ac 1 . The consistency condition that kM 2 0 = ac 2 1 implies e λ = 1 + √ k/a; and density conservation implies c 1 = k/( √ a + √ k) 2 .…”

“…since we have c 1 = and M 0 = 0 at t = 0 and, as t increases, c 1 decreases and M 0 increases. Towards the end of this timescale, but before we start the next, that is, in the range 1 t O(− log k), we have M 0 ∼ /e + o (1), and so c 1 decays according to c 1 ∼ exp(−a t/e). As t → +∞ this timescale will cease to be valid due to the exhaustion of monomers, hence in the next timescale, c 1 1.…”

“…Towards the end of this timescale, but before we start the next, that is, in the range 1 t O(− log k), we have M 0 ∼ /e + o (1), and so c 1 decays according to c 1 ∼ exp(−a t/e). As t → +∞ this timescale will cease to be valid due to the exhaustion of monomers, hence in the next timescale, c 1 1. We define the time at which this exhaustion of monomers occurs by −t 1c log k, this value can be found using the calculation given in section 3.1.6.…”

“…A model Oort-Hulst's proposed mechanisms of coagulation and fragmentation has been suggested earlier by Safranov [24], and this partial differential equation known as the OHS equation has been studied by other authors, most notably, by Lachowicz et al [13] and Bagland and Laurencot [2], who have shown the presence of self-similar solutions. A discrete version of the OHS equation has been proposed by Bagland [1], who showed strong connections between his discrete model and the continuum counterpart. This model allows a cluster to grow by modelling the outcome of a collision to be a disintegration of one cluster, immediately followed by several coagulation processes.…”