Moment bounds for the solutions of the Smoluchowski equation with coagulation and fragmentation

Abstract

We prove various bounds on moments

where f_m is a solution of the discrete (respectively, continuous) Smoluchowski coagulation–fragmentation equations with diffusion. In a previous paper we proved similar results for all weak solutions to the discrete Smoluchowski equation provided that there is no fragmentation and certain moments are bounded in suitable L^q-spaces initially. We prove the corresponding results in the case of the continuous Smoluchowski equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to the Smoluchowski equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β we assume that sup_n sup_m≤l β(m, n)/n < ∞ for every positive l, and that there exist constants a₀ ≥ 0 and c₀ such that β(n,m) ≤ c⁰(n + m)a₀^a0.