We investigate an integro-differential equation that models the evolution of fragmenting clusters. We assume cluster size to be a continuous variable and allow for situations in which mass is not necessarily conserved during each fragmentation event. We formulate the initial-value problem as an abstract Cauchy problem (ACP) in an appropriate weighted L^1 space, and apply perturbation results to prove that a unique, physically relevant classical solution of the ACP is given by a strongly continuous semigroup for a wide class of initial conditions. Moreover, we show that it is often possible to identify a weighted L^1 space in which this semigroup is analytic, leading to the existence of a unique, physically relevant classical solution for all initial conditions belonging to that space. For some specific fragmentation coefficients, we provide examples of weighted L^1 spaces where our results can be applied.

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $\ell^1$ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients.

In this paper, we prove the global in time solvability of the continuous growth–fragmentation–coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth–fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue ‘Semigroup applications everywhere’.

We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers), and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted ℓ ^1 space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted ℓ ^1 space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.

An integro-differential equation is constructed which describes the evolution of a system of masses evolving by completely inelastic collisions and spontaneous fragmentations. The development of a characteristic size spectrum is demonstrated for a simple example.

An integro–differential equation modelling coagulation and multiple–fragmentation is investigated. Under the assumptions that the coagulation kernel is constant and the fragmentation kernel takes the form (ν + 2)(y/x) ν x −1, where ν > − 1, global existence and uniqueness of mass–conserving solutions are established. This extends similar results obtained elsewhere, such as [1,4,8,10,12,13] since the fragmentation kernel considered here is singular at the origin. In the case of pure fragmentation, when no coagulation occurs, semigroup methods are used to derive a closed–form solution of the resulting linear equation. This solution agrees with that obtained in [9] by means of a non–rigorous limiting argument.

Analytic Methods for Coagulation-Fragmentation Models is a two-volume set that provides a comprehensive exposition of the mathematical analysis of coagulation-fragmentation models. Initially, an in-depth survey of coagulation-fragmentation processes is presented, together with an account of relevant early results obtained on the associated model equations. These provide motivation for the subsequent detailed treatment of more up-to-date investigations which have led to significant theoretical developments on topics such as solvability and the long-term behaviour of solutions. To make the account as self-contained as possible, the mathematical tools that feature prominently in these modern treatments are introduced at appropriate places. The main theme of Volume I is the analysis of linear fragmentation models, with Volume II devoted to processes that involve the nonlinear contribution of coagulation. Features of Volume I: The main models of the theory together with their derivations and early methods of solution A detailed presentation of the operator theoretical methods and semigroup theory that play an essential role in the theory of fragmentation processes A comprehensive theory of fragmentation processes, including fragmentation with growth and decay in both the discrete and continuous particle size cases An analytical explanation of the `pathologies’ of the fragmentation equation, such as the shattering phase transition and non-uniqueness of solutions An analysis of the long-term dynamics of the discrete size fragmentation equation with growth

Mathematical models arising in the natural sciences often involve equations which describe how the phenomena under investigation evolve in time. In these notes, some mathematical techniques will be presented for analysing a range of evolution equations that can arise in a number of applied disciplines such as biomathematics and population dynamics. Different types of equations will be examined, but a unifying theme will be provided by developing methods from a dynamical systems point of view and using some elegant results from functional analysis. To fix ideas, we will begin with some simple finite-dimensional models from population dynamics which are expressed in terms of ordinary differential equations. We then go on to consider dynamical systems in an infinite-dimensional setting, and provide a gentle introduction to the theory of strongly continuous semigroups of operators. This theory is applied to an infinite system of nonlinear ordinary differential equations that models the time-evolution of the size distribution of a collection of particles that can coagulate to form larger particles or fragment into smaller particles.

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher-order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when fragmentation is described by power-law rates, which are commonly used in engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L 1 spaces.

We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.

We consider the multiple fragmentation equations with polynomially bounded fragmentation rates, both in the discrete and continuous cases. The theory of semigroups of operators on Fréchet spaces is used to produce a simple proof that if moments of all non-negative orders of solutions are initially finite then they remain finite for all future times. Moreover, a class of fragmentation processes is identified in which the existence of the first moment of the initial distribution suffices for the existence of all other moments for positive times.

A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets , where the index set may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory.

The distributions of interisland gaps and captures zones for islands nucleated on a one-dimensional substrate during submonolayer deposition are considered using a novel retrospective view. This provides an alternative perspective on why scaling occurs in this continuously evolving system. Distributional fixed-point equations for the gaps are derived both with and without a mean-field approximation for nearest neighbor gap-size correlation. Solutions to the equations show that correct consideration of fragmentation bias justifies the mean-field approach, which can be extended to provide closed-from equations for the capture zones. Our results compare favorably to Monte Carlo data for both point and extended islands using a range of critical island size i=0,1,2,3. We also find satisfactory agreement with theoretical models based on more traditional fragmentation theory approaches.

The nucleation and growth of point islands during submonolayer deposition on a one-dimensional substrate is simulated for critical island size i=0,1,2,3. The small- and large-size asymptotics for the gap-size and capture-zone distributions (CZDs) are studied. Comparisons to theoretical predictions from fragmentation equation analyses are made, along with those from the recently proposed generalized Wigner surmise (GWS). We find that the simulation data can be fully understood in the framework provided by the fragmentation equations, while highlighting the theoretical areas that require further development. The GWS works well for the small-size CZD behavior, but completely fails to describe the large-size CZD asymptotics of the one-dimensional system.