A formula for the post-gelation mass of a coagulation equation with a separable bilinear kernel
type: Item
instance of: scholarly article
title: A formula for the post-gelation mass of a coagulation equation with a separable bilinear kernel
zbMATH Open document ID: 1129.82029
published in: Physica D
publication date: 2006-11-15T00:00:00Z
review text: The authors consider the coagulation equation \[ \dfrac{\partial c}{\partial t}(\lambda, t)=\dfrac{1}{2}\int^{\lambda}_0 K(\lambda-\mu, \mu)c(\lambda-\mu, t)c(\mu,t)\,d\mu-c(\lambda,t)\int^\infty_0 K(\lambda, \mu)c(\mu,t)\,d\mu\tag{1} \] \( c(\lambda, 0)=c_0(\lambda)\) with the specific choice of the kernel \[ K(\lambda, \mu)=\theta(\lambda)\theta(\mu),\quad\theta(\lambda)=\alpha +\beta\lambda,\quad \alpha, \beta\geq 0. \] Their attention is focussed on the so-called gelation phenomenon, corresponding to breakdown of mass conservation, due to the formation of a ``superparticle'', which necessarily occurs if \(\beta>0\). In particular they want to describe the behaviour of the system after gelation. Using Laplace transform equation (1) is reduced to a nonlinear first order PDE, which however is of non-standard type, since it includes the unknown value of the solution along one of the characteristics of the equation. The authors manage to provide the solution of this problem, obtaining the explicit expression of gelation time, depending on the initial data. Then they proceed further, deriving the time evolution of mass \(M(t)\) after gelation by means of an elegant method. Examples for various choices of initial mass distributions are illustrated. the asymptotic behaviour of \(M(t)\) for large \(t\) is also discussed.
Mathematics Subject Classification ID: 82C22
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 5073055
zbMATH Keywords: integro-differential equations
zbMATH Keywords: gelation
zbMATH Keywords: coagulation
author: H. van Roessel
author: M. Shirvani
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: j.physd.2006.08.003
OpenAlex ID: W2057421924
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DOI: 10.1016/J.PHYSD.2006.08.003
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description: scientific article; zbMATH DE number 5073055