Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation

In this paper we give an elementary proof of the unique, global-in-time solvability of the coagulation-(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument

A distributional approach to fragmentation equations

We consider a linear integro-di®erential equation that models multiple fragmentation with inherent mass-loss. A systematic procedure is presented for constructing a space of generalised functions Z0 in which initial-value problems involving singular initial conditions such as the Dirac delta distribution can be analysed. The procedure makes use of results on sun dual semigroups and quasi-equicontinuous semigroups on locally convex spaces. The existence and uniqueness of a distributional solution to an abstract version of the initial-value problem are established for any given initial data u0 in Z0

Coagulation, fragmentation and growth processes in a size structured population

An equation describing the dynamical behaviour of hytoplankton cells is considered in which the effects of cell division and aggregration are incorporated by coupling the coagulationfragmentation equation with the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial boundary value problem is shown to be well posed in a physically relevant Banach space

Stability for a class of equilibrium solutions to the coagulation-fragmentation equation

We consider the behaviour of solutions to the continuous constant-rate coagulation-fragmentation equation in the vicinity of an equilibrium solution. Semigroup methods are used to show that the governing linear equation for a perturbation epsilon(x,t) has a unique globally defined solution for suitable initial conditions. In addition, Laplace transforms and the method of characteristics lead to an explicit formula for epsilon. The long-term behavior of epsilon is also discussed

Fractional calculus of periodic distributions

Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Gr¨unwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotient. The equivalence of the two approaches is established and an application to a fractional diffusion equation, posed in a space of periodic distributions, is also discuss

Fractional transformations of generalised functions

A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by A. H. Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalised functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak continuous group on the space of generalised functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory

Growth--fragmentation--coagulation equations with unbounded coagulation kernels

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

Vinegar fermentation and home production of cider vinegar

Vinegar is an important by-product on many Iowa farms, utilizing a considerable part of the apple crop which would otherwise be wasted by spoiling, often on account of difficulty in marketing. The apple crop of Iowa amounts to between three and four million dollars per year and a more efficient home production of vinegar from culls and windfalls would result in a large saving. Many inquiries are received by the Iowa Agricultural Experiment Station regarding difficulties or failure in vinegar making. In many cases the vinegar never reaches the legal standard of strength for marketable vinegar, which is 4 percent of acetic acid. This is gene rally due to contamination of the fermenting vinegar with bacteria or other micro-organisms which cause what are called foreign” or unfavorable fermentations, thus changing the sugars to something else than the acetic acid which is desired. These contaminating bacteria also sometimes impart an unfavorable or disagreeable flavor to the vinegar, as well as preventing the development of the proper amount of acidity, which is necessary for proper preservation as well as to meet the legal standard if the vinegar is to be offered for sale. In a few cases the low acidity produced is due to the use of apples too low in sugar content, or to dilution of the cider with water. The cause and prevention of these difficulties is to be discussed in this bulletin

Discrete fragmentation systems in weighted ℓ1 spaces

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 ℓ1 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 ℓ1 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