Finite volume approach for fragmentation equation and its mathematical analysis

This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

A note on the self-similar solutions to the spontaneous fragmentation equation

We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.

Estimating the division kernel of a size-structured population

We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rate α. At division, the cells divide at a constant rate R and share their content between the two daughter cells into fractions Γ and 1 − Γ where Γ has a symmetric density h on [ 0,1 ], since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval [ 0,T ] with fixed T. We address here the problem of estimating the division kernel h (or fragmentation kernel) when the division tree is completely observed. An adaptive estimator of h is constructed based on a kernel function K with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.

A theoretical explanation of grain size distributions in explosive rock fragmentation

We have measured grain size distributions of the results of laboratory decompression explosions of volcanic rock. The resulting distributions can be approximately represented by gamma distributions of weight per cent as a function of ϕ = − log 2 ⁡ d , where d is the grain size in millimetres measured by sieving, with a superimposed long tail associated with the production of fines. We provide a description of the observations based on sequential fragmentation theory, which we develop for the particular case of ‘self-similar’ fragmentation kernels, and we show that the corresponding evolution equation for the distribution can be explicitly solved, yielding the long-time lognormal distribution associated with Kolmogorov's fragmentation theory. Particular features of the experimental data, notably time evolution, advection, truncation and fines production, are described and predicted within the constraints of a generalized, ‘reductive’ fragmentation model, and it is shown that the gamma distribution of coarse particles is a natural consequence of an assumed uniform fragmentation kernel. We further show that an explicit model for fines production during fracturing can lead to a second gamma distribution, and that the sum of the two provides a good fit to the observed data.

EXISTENCE THEORY FOR THE KINETIC-FLUID COUPLING WHEN SMALL DROPLETS ARE TREATED AS PART OF THE FLUID

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incompressible Navier–Stokes equation and of the Vlasov–Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna–Lions theory for singular transport equations, and a compactness lemma specifically tailored for our study.

On a coagulation and fragmentation equation with mass loss

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is constant, the fragmentation-rate function is linearly bounded, and the continuous mass-loss-rate function is locally Lipschitz, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss, an explicit formula is given for the associated substochastic semigroup.