Existence of Gelling Solutions for Coagulation-Fragmentation Equations

Weak Compactness Techniques and Coagulation Equations

“…Of particular importance is the contribution by Stewart [81] where weak L 1 -compactness techniques were used for the first time and turned out to be a very efficient tool which was extensively used in subsequent works. As for the conjecture for gelling kernels (1.9), it was solved rather recently in [27,40]. An intermediate step is the existence of solutions to (1.1)-(1.2) and (1.5) with non-increasing finite mass, that is, satisfying M 1 (f (t)) ≤ M 1 (f (0)) for t ≥ 0, see [24,45,47,52,69,80] and the references therein.…”

“…We conclude the introduction with a few words on related interesting issues: we focus in these notes on the deterministic approach to the modeling of coagulation and leave aside the stochastic approach which has been initiated in [54,55,78,79] and further developed in [2,8,9,14,23,24,30,40,41,69] and the references therein.…”

Self-similar solutions to a coagulation equation with multiplicative kernel

“…In addition, ω(τ ) > 3 since r(τ ) > (3 − τ )/2. Finally, since Y 0 = ∞, standard computations allow us to show that (6) implies that f τ defined by (15) Owing to the previous identity, it is straightforward to check that f τ defined by (15) solves (3) in the weak sense defined in Theorem 1, which completes the proof of Theorem 1. …”

“…For such kernels, there is actually a runaway growth which produces particles with infinite mass in a finite time, a phenomenon called the occurrence of gelation (see, e.g., the review articles [1,3,8,10] for more information). Let us mention at this point that, though formal arguments predicting the occurrence of gelation have been known for some time, a rigorous proof has only been supplied recently in [6] by probabilistic arguments and in [5] by deterministic arguments. Now, introducing the gelation time T gel := inf {t ≥ 0 such that M 1 (t) < M 1 (0)} and assuming that T gel is finite, a detailed analysis of the behaviour of f (t) just before the gelation time is required to elucidate the gelation mechanism.…”

“…In the next section, we show how to reduce (at least formally) the nonlinear integro-differential equation (6) to a singular ordinary differential equation. We solve this equation in Section 3 and establish some properties of its solutions, from which Theorem 1 follows.…”