Singularities in the kinetics of coagulation processes

Weak Compactness Techniques and Coagulation Equations

“…As already mentioned this property fails to be true in general for coagulation kernels which grows sufficiently fast for large x and y, a fact which has been known/conjectured since the early eighties [25,39,53,90] but only proved recently in [27,40]. In fact, the occurrence of gelation was first shown for the multiplicative kernel K 2 (x, y) = xy by an elementary argument [52] and conjectured to take place for coagulation kernels K satisfying K(x, y) ≥ κ m (xy) λ/2 for some λ ∈ (1, 2] and κ m > 0 [25,39,53,90]. This conjecture was supported by a few explicit solutions constructed in [13,17,50].…”

“…However, one has to keep in mind that the previous computation is only formal as it uses Fubini's theorem without justification. That some care is indeed needed stems from [52] where it is shown that (1.6) breaks down in finite time for the multiplicative kernel K 2 (x, y) = xy for all non-trivial solutions. An immediate consequence of this result is that the mass-conserving solution constructed on a finite time interval in [57,58] cannot be extended forever.…”

“…An immediate consequence of this result is that the mass-conserving solution constructed on a finite time interval in [57,58] cannot be extended forever. Soon after the publication of [52] a particular solution to (1.1)-(1.2) was constructed for K(i, j) = (ij) α , α ∈ (1/2, 1) which fails to satisfy (1.6) for all times [50]. At the same time, it was established in [53,88] that the condition K(i, j) ≤ κ(i + j) was sufficient for (1.1)-(1.2) to have global mass-conserving solutions, that is, solutions satisfying (1.6).…”

“…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.…”

Coagulation equations with mass loss

“…Leyvraz & Tschudi [5] have noted that the coagulation equations without mass loss could be solved exactly if the coagulation rates were given by one of two special kernels, that is a function which specifies how the rate of coagulation depends on the size of the aggregating clusters. The solvable cases which they describe are the size-independent kernel a i,j = a and the size-dependent kernel a i,j = a(i + j).…”