On the Honesty in Nonlocal and Discrete Fragmentation Dynamics in Size and Random Position

A discrete initial-value problem describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and where new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. The existence of semigroups is established for each parameter and “glued” together so as to obtain a semigroup to the full space. Under certain conditions on each fragmentation rate, we used Kato’s Theorem in to show the existence of the generator and we provide sufficient conditions for honesty. 1. Introduction The process of fragmentation of clusters occurs in numerous domains of pure and applied sciences, such as the depolymerization, the rock fractures, and break of droplets. The fragmentation rate can be size and position dependent, and new particles resulting from the fragmentation are spatially randomly distributed according to some probability density. When it is supposed that every group of size (one -group) in a system of particles clusters consists of identical fundamental units (monomers), then the mass of every group is simply a multiple positive integer of the mass of the monomer. We focus here on clusters that are discrete; that is, they consist of a finite number of elementary (unbreakable) particles which are assumed to be of unit mass. The state at a given time is the repartition at that time of all aggregates according to their size and their position . The evolution of such particle-mass-position distribution is given by an integrodifferential [1] equation as we will see in this paper. Before going farther let us review what have already been done. Various types of fragmentation equations have been comprehensively analyzed in numerous works (see, e.g., [2–9]). Conservative and nonconservative regimes for fragmentation equations have been thoroughly investigated, and, in particular, the breach of the mass conservation law (called shattering) has been attributed to a phase transition creating a dust of zero-size particles with nonzero mass, which are beyond the model resolution. Shattering can be interpreted from the probabilistic point of view as the explosion in the Markov process describing fragmentation [8, 10] and from an analytic point of view as dishonesty of the semigroup associated with the model [2, 7]. Kinetic-Type Models with Diffusion were investigated in [11] where the author showed that the diffusive part does not affect the breach of the conservation laws. But