A Mixed Discrete-Continuous Fragmentation Model

TL;DR

This paper introduces a novel hybrid discrete-continuous fragmentation model to better understand shattering phenomena, providing rigorous mathematical analysis and proving existence, uniqueness, and mass conservation of solutions.

Contribution

It develops a new integro-differential and ODE coupled model for fragmentation, with a comprehensive mathematical framework ensuring well-posedness and key properties.

Findings

Proved existence and uniqueness of solutions.

Demonstrated mass conservation and nonnegativity.

Established the model's mathematical robustness.

Abstract

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.