The discrete diffusive coagulation-fragmentation equations with scattering
type: Item
instance of: scholarly article
title: The discrete diffusive coagulation-fragmentation equations with scattering
zbMATH Open document ID: 1052.35082
author: Christoph Walker
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-08-19T00:00:00Z
review text: The author considers a model of coagulation-fragmentation with diffusion in which the coagulation clusters are assumed to be multiples of an elementary identical unit. Moreover, the model is endowed with a suitable scattering mechanism that prevents from exceeding a maximal cluster size, see \textit{A. Fasano} and \textit{F. Rosso} [Lectures on applied mathematics, Proceedings of the symposium organized by the Sonderforschungsbereich 438 on the occasion of Karl-Heinz Hoffmann's 60th birthday, 123--141 (2000; Zbl 1049.82525)]. The state of the system at time \(t\) is thus described by \(u_i = u_i(x,t)\), \(i = 1,2,\dots,M\), where \(x \in \Omega \subset {\mathbb R}^n\) is the position variable and \(i\) is the cluster size. The set of functions \(u = (u_1,u_2,\dots,u_n)\) is assumed to satisfy a set of reaction-diffusion equations of the following form: \[ \dot u_i - d_i \Delta u_i = f_i(x,u), \quad i = 1,2,\dots,M, \] with assigned initial and Neumann boundary conditions. The functions \(f_i\) account for (mass-preserving) coagulation-fragmentation and scattering. The problem is recast into a semilinear Cauchy problem in a suitable \(L^p\)-like space, with \(2 \leq p < \infty\) and \(2p > n\). Local-in-time existence and uniqueness of mild and strong solutions is proved under fairly general assumptions. Global-in-time existence and uniqueness is proved under the assumption \(d_1 = d_2 = \cdots =d_M\). Under the same assumption, in the last part of the paper a careful analysis of the asymptotic behaviour for \(t \to +\infty\) is carried on, focusing on the three special (but physically relevant) cases of dominating coagulation, dominating fragmentation, and detailed-balance condition.
reviewed by: L. Barletti
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2092996
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: collisional breakage
zbMATH Keywords: coagulation clusters
zbMATH Keywords: local-in-time existence
zbMATH Keywords: mild and strong solutions
zbMATH Keywords: global-in-time existence
MaRDI profile type: Publication
full work available at URL: j.na.2004.05.003
OpenAlex ID: W2001595932
cites work: Convergence to equilibrium in a system of reacting polymers
cites work: Dual semigroups and second order linear elliptic boundary value problems
cites work: Q4838180
cites work: Q4836474
cites work: Q4327845
cites work: Quasiconvexity at the boundary, positivity of the second variation and elastic stability
cites work: Q4118367
cites work: Q4344330
cites work: Q4546012
cites work: Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case
cites work: Asymptotic behavior of solutions to the coagulation-fragmentation equations. II: Weak fragmentation
cites work: Existence of solutions to coagulation-fragmentation systems with diffusion
cites work: Asymptotic behaviour of solutions to the diffusive fragmentation-coagulation system
cites work: Q5583334
cites work: Q4513827
cites work: The continuous coagulation-fragmentation equations with diffusion
cites work: Global existence for the discrete diffusive coagulation-fragmentation equations in \(L^1\).
cites work: Convergence to equilibrium for the continuous coagulation-fragmentation equation
cites work: Fragmentation–diffusion model. Existence of solutions and their asymptotic behaviour
cites work: The Becker-Döring model with diffusion: II. Long time behavior
cites work: Coagulation model with partial diffusion
cites work: The discrete coagulation equations with collisional breakage.
cites work: FROM THE NONLOCAL TO THE LOCAL DISCRETE DIFFUSIVE COAGULATION EQUATIONS
cites work: A new model for the dynamics of dispersions in a batch reactor: Numerical approach
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Global solutions of reaction-diffusion systems
cites work: Approach to Equilibrium for the Coagulation-Fragmentation Equation via a Lyapunov Functional
cites work: Q4836063
cites work: Coalescence and breakage processes
cites work: Asymptotic behaviour of liquid–liquid dispersions
cites work: Existence of solutions for the discrete coagulation-fragmentation model with diffusion
DOI: 10.1016/J.NA.2004.05.003
description: scientific article; zbMATH DE number 2092996