Coagulation-fragmentation Equations Research Articles

A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels

AbstractIn this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X+ = {c∈L1: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non‐negative initial data belong to X+. The proof is based on weak L1 compactness methods applied to suitably chosen approximating equations.

Analytic expression for the distribution of gel species in spatially homogeneous systems

The equilibrium solution of the stochastic coagulation-fragmentation equations, for a spatially homogeneous system containing M monomers, is studied. If the coagulation-fragmentation rates Kij and Fij satisfy the condition Fij/Kij= lambda aiaj/ai+j (needed to guarantee detailed balance) with ak varies as Ak- delta zc-k and 2< delta <3, then the ensemble average size distribution (xk) has a clearly bimodal character in the gel phase. The distribution of gel species is sharply peaked near k=Mg, where g is the gel fraction for the infinite system, with a width varies as M1( delta -1/). The scaling function describing its precise shape is determined analytically. This function is highly non-symmetric; its second moment diverges.

An existence theorem for the discrete coagulation–fragmentation equations

This paper proves the existence of solutions for the discrete coagulation–fragmentation equation over all times, even when source and efflux terms are present. The hypotheses required cover most physical applications. Roughly speaking, the hypotheses ensure a finite flux of mass through the system. The techniques used, which extend those in I of this series, may apply to other infinite systems of differential equations.

An existence theorem for the discrete coagulation-fragmentation equations

This paper gives an existence result for the discrete coagulation-fragmentation equations:(If k = 1, the first and last sums are 0.)

A geometrical interpretation of the coagulation equation

The coagulation-fragmentation equation describes geodesic motion in an infinite-dimensional space. This space has a symmetric affine connection but no metric in general. Some advantages of the new approach are indicated.