A kinetic model for coagulation–fragmentation

The aim of this paper is to show an existence theorem for a kinetic model of coagulation–fragmentation with initial data satisfying the natural physical bounds, and assumptions of finite number of particles and finite Lp$L^p$-norm. We use the notion of renormalized solutions introduced by DiPerna and Lions (1989) [3], because of the lack of a priori   estimates. The proof is based on weak-compactness methods in L¹$L^1$, allowed by Lp$L^p$-norms propagation.     

Mass-conserving self-similar solutions to coagulation–fragmentation equations

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.     

A short remark on a growth–fragmentation equation

An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in [5] about the existence of global solutions in the so-called self-similar case is not satisfied. The solution is local and blows up in finite time.     

A one-dimensional coagulation–fragmentation process with a dynamical phase transition

We introduce a reversible Markovian coagulation–fragmentation process on the set of partitions of {1,…,L}$\{1,\dots ,L\}$ into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model (Giacomin (2007) [10]). Depending on a parameter λ$\lambda$, the typical configuration can be either dominated by a single big interval (delocalized phase), or composed of many intervals of order 1$1$ (localized phase), or the interval length can have a power law distribution (critical regime). In the three cases, the time required to approach equilibrium (in total variation) scales very differently with L$L$. In the localized phase, when the initial condition is a single interval of size L$L$, the equilibration mechanism is due to the propagation of two “fragmentation fronts” which start from the two boundaries and proceed by power-law jumps.     

Approximative solution of the coagulation–fragmentation equation by stochastic particle systems

This paper studies stochastic particle systems related to the coagulation fragmentation equation. For a certain class of unbounded coagulation kernels and fragmentation rates, relative compactness of the stochastic systems is established and weak accumulation points are characterized as solutions. These results imply a new existence theorem. Finally a simulation algorithm based on the particle systems is proposed     

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L²$L^2$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.     

Analytic Fragmentation Semigroups and Classical Solutions to Coagulation–fragmentation Equations——a Survey

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation–coagulation equations.