CONSERVATIVE AND SHATTERING SOLUTIONS FOR SOME CLASSES OF FRAGMENTATION MODELS

Abstract

Equations describing processes of cluster fragmentation have received considerable attention in recent years due to their importance in modelling e.g. the polymer degradation, droplet break-up, or rock crushing. In this paper we shall consider two mathematically most interesting features of them. One of these is the so-called shattering fragmentation, that is, the decrease of the total mass of the system that is formally conservative. The other is the existence of multiple solutions which indicates that in some cases the equations do not give the full picture of the dynamics of the model. Shattering fragmentation, that is considered to be an analogous but opposite process to a better understood gelation in coagulation models, has been investigated in several papers whose interest was, however, restricted to coefficients of a particular form that allowed either explicit solutions or yielded to a probabilistic approach. There have been few attempts to provide a deeper analysis of the existence of multiple solutions; most authors confined themselves to noticing them. In the present paper we present a systematic approach to both problems using the semigroup theory, and link them to the characterization of the generator of the solution semigroup of the fragmentation equation. As a result we provide criteria for both existence and absence of shattering fragmentation as well as of multiple solutions for large classes of coefficients that include those considered in all earlier works.