Discrete fragmentation with mass loss

Abstract

We examine an infinite system of ordinary differential equations that models a discrete fragmentation process in which mass loss can occur. The problem is treated as an abstract Cauchy problem, posed in an appropriate Banach space. Perturbation techniques from the theory of semigroups of operators are used to establish the existence and uniqueness of physically meaningful solutions under minimal restrictions on the fragmentation rates. In one particular case an explicit formula for the associated semigroup is obtained and this enables additional properties, such as compactness of the resolvent and analyticity of the semigroup, to be deduced. Another explicit solution of this particular fragmentation problem, in which mass is apparently created from a zero-mass initial state, is also investigated, and the theory of Sobolev towers is used to prove that the solution actually emanates from an initial infinite cluster of unit mass

Analytic Fragmentation Semigroups and Classical Solutions to Coagulation–fragmentation Equations — a Survey

“…Another approach, that is the subject of this paper, is based on semigroup theory, see e.g. [8], and consists in looking at (1) (or (6)) as a perturbation of the linear fragmentation semigroup by the nonlinear coagulation operator, denoted by C. This approach, initiated in [1], has been well developed for continuous coagulation-fragmentation models (though see [48,59,60] for the discrete version of the results), allowing for handling more singular fragmentation rates than the previously described method. However, since in a standard approach the nonlinear perturbation C is required to be Lipschitz continuous in the chosen state space, originally only bounded coagulation kernels were considered, see e.g.…”

“…Since, however, in general no simple description of the generator's domain was known, [8,Remark 8.16] or [4], the analyticity of the fragmentation semigroup was not studied. Fortunately, recently, [60], it has been proved that a simple fragmentation operator with uniform binary fragmentation generates an analytic semigroup in l 1 1 . This prompted interest in the topic and, subsequently, it was shown that a class of fragmentation operators, that includes physically relevant binary and homogeneous fragmentations, both in the discrete and continuous case, is sectorial albeit in a smaller space of densities that have finite higher moments (the space l 1 p := l 1 U with U = i p , p > 1, or its equivalent X 0,p = L 1 (R + , (1 + x p )dx) in the continuous case), [6,11].…”

“…We note that (32) cannot hold for p = 1 as ∆ (1) = 0. Nevertheless, there are analytic fragmentation semigroups in l 1 1 and X 0,1 , see [14,60], but the proofs in these spaces require direct estimates of the resolvent that is not explicitly available in general.…”

“…Thus, one can say that (16) is the solution corresponding to an infinitely large cluster of unit mass, originating (for p < 2) from the first Sobolev tower l 1 p,−1 . At the same time for p > 1 (actually, for p ≥ 1, see [60], but the proof for p = 1 is different) the semigroup (S Fp (t)) t≥0 is analytic, and hence is (S Fp,−1 (t)) t≥0 . Due to the immediate regularization property (25), the solution t → S Fp,−1 (t)f immediately appears in l 1 p so that it has nonzero components.…”

Analysis and simulations of the discrete fragmentation equation with decay

“…In this way, we obtain a pure fragmentation system that, however, is not mass conservative. Such systems were considered in Cai et al and Smith et al While mathematically they are equivalent to , physically they describe different models as in : The death process is independent of fragmentation, and in Cai et al, the mass loss is caused by the so‐called explosive fragmentation. Also, the study of death‐sedimentation process is of independent interest.…”