  Motivated by the occurrence of "shattering" mass-loss observed in purely
continuous fragmentation models, this work concerns the development and the
mathematical analysis of a new class of hybrid discrete--continuous
fragmentation models. Once established, the model, which takes the form of an
integro-differential equation coupled with a system of ordinary differential
equations, is subjected to a rigorous mathematical analysis, using the theory
and methods of operator semigroups and their generators. Most notably, by
applying the theory relating to the Kato--Voigt perturbation theorem, honest
substochastic semigroups and operator matrices, the existence of a unique,
differentiable solution to the model is established. This solution is also
shown to preserve nonnegativity and conserve mass.
