Existence, Uniqueness and Mass Conservation for the Coagulation-Fragmentation Equation

Abstract

We prove global existence and uniqueness to the initial value problem for the coagulation–fragmentation equation for an unbounded coagulation kernel with possible linear growth at infinity and a fragmentation kernel from a very large class of unbounded functions. We show that the solutions satisfy the mass conservation law.

Weak Compactness Techniques and Coagulation Equations

“…For this class of coagulation kernels, we establish the existence of mass-conserving solutions to (3.1)-(3.2) [4,21,47,81,83]. Theorem 3.6.…”

“…Actually two approaches have been developed to establish the uniqueness of solutions to (3.1)-(3.2): a direct one which consists in taking two solutions and estimating a weighted L 1 -norm of their difference and another one based on a kind of Wasserstein distance. To be more specific, since the pioneering works [59,61], uniqueness has been proved in [4,21,35,41,46,69,82] by the former approach and is summarized in the next result. Let us point out two immediate consequences of Proposition 3.16.…”

“…Following the contributions [53,88] the conjecture for non-gelling coagulation kernels (1.8) is completely solved in [4] for the discrete coagulation equations (1.1)-(1.2), an alternative proof being given in [46]. It took longer for this conjecture to be solved for the continuous coagulation equation (1.5), starting from the pioneering works [1,61] for bounded kernels and continuing with [21,47,81,83]. 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.…”

From the discrete to the continuous coagulation–fragmentation equations

“…In fact, it can be slightly relaxed, see Section 6. [12,33]. Here we relax the assumptions made in [33] and make the following growth assumption: for each R ∈ R + there holds …”

Scaling limit of a discrete prion dynamics model

“…xU (0,x)dx, see [8]. Hence, (2.9) has to be incorporated in the model as a constraint to select the physically relevant solution, as suggested in [8] and [13].…”