Scaling theory and exactly solved models in the kinetics of irreversible aggregation

Self-similar solutions to a coagulation equation with multiplicative kernel

“…However, as τ c > 2, ψ τ has infinite mass for any τ ∈ [τ c , 3). A similar property was already observed for the Smoluchowski coagulation equation (1) [2,3,12] and is related to the fact that the self-similar behaviour (2) is rather expected to be relevant for large values of y, see [10] for a more detailed discussion. In that connection, let us point out that, while the second moment of f τ blows up at time T , it follows from (12) that f τ (t, y) has a finite limit γ 0 y −τ as t → T for each y ∈ (0, ∞).…”

“…It is however well-known by now that this property fails to be true for coagulation kernels growing sufficiently rapidly for large y, y * , such as a(y, y * ) = (y y * ) λ/2 for λ ∈ (1, 2]. 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.…”

“…a(ξy, ξy * ) = ξ λ a(y, y * ) for (ξ, y, y * ) ∈ (0, ∞) 3 ), the value τ = (λ + 3)/2 has been proposed in [2] but numerical simulations performed in [9,11] seem to indicate a different value of τ . We refer to the review article [10] for a thorough discussion of this issue. In fact, for the only case which is solved, namely a(y, y * ) = y y * , there is an interval of values of τ for which there exists a self-similar solution f τ (t, y) := s(t) −τ ϕ τ (y/s(t)) to (1) [12].…”

Fibrin gel formation in a shear flow

“…In Section 2.1 we discuss gelation resulting from an initial distribution of polymers when there are no sources or sinks of polymers (Ziff and Stell 1980). There have been many reviews on models of coagulation and gelation, see for example Leyvraz (2003). The effect of removal terms on the gel time was studied by Singh and Rodgers (1996), and similarly, source terms were considered by Davies, King, and Wattis (1999).…”

Weak Compactness Techniques and Coagulation Equations

“…For the sake of completeness, we supplement the existence results with the occurrence of gelation in finite time for gelling kernels [27] (Section 3.3) and with uniqueness results (Section 3.4). For further information on coagulation equations and related problems we refer to the books [9,20] and the survey articles [2,48,51,87].…”