Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach

Wilson Lamb; Adam McBride; Louise Smith

doi:10.3934/dcds.2013.33.5177

Article Contents

2013, Volume 33, Issue 11&12: 5177-5187. Doi: 10.3934/dcds.2013.33.5177

This issue Previous Article On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators Next Article Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system

Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach

1.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH

Received: October 2011

Revised: August 2012

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.

Keywords:

Semigroups of operators,
semilinear Cauchy problems,
multicomponent coagulation-fragmentation processes.

Mathematics Subject Classification: Primary: 47J35, 47D06; Secondary: 45K05, 80A30.

Citation:

Related Papers

Cited by

References

[1]	J. Banasiak and L. Arlotti, "Positive Perturbations of Semigroups with Applications," Springer, London, 2006.
[2]	J. Banasiak, Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates, Nonlinear Analysis : Real World Applications, 13 (2012), 91-105.doi: 10.1016/j.nonrwa.2011.07.016.
[3]	J. Blum, Dust agglomeration, Advances in Physics, 55 (2006), 881-947.
[4]	J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations I. The strong fragmentation case, Proc. Roy. Soc. Edinburgh, Sect. A, 121 (1992), 231-244.doi: 10.1017/S0308210500027888.
[5]	F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, J. Math. Anal. Appl., 192 (1995), 892-914.doi: 10.1006/jmaa.1995.1210.
[6]	J. M. Fernández-Díaz and G. J. Gómez-García, Exact solution of a coagulation equation with a product kernel in the multicomponent case, Physica D, 239 (2010), 279-290.doi: 10.1016/j.physd.2009.11.010.
[7]	F. Gruy, Population balance for aggregation coupled with morphology changes, Colloids and Surfaces A : Physicochemical and Engineering Aspects, 374 (2011), 69-76.doi: 10.1016/j.colsurfa.2010.11.010.
[8]	M. Kostoglou, A. G. Konstandopoulos and S. K. Friedlander, Bivariate population dynamics simulation of fractal aerosol aggregate coagulation and restructuring, Aerosol Science, 37 (2006), 1102-1115.doi: 10.1016/j.jaerosci.2005.11.009.
[9]	A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation, Physica D, 239 (2010), 1436-1445.doi: 10.1016/j.physd.2009.03.013.
[10]	A. L. Smith, W. Lamb, M. Langer and A. C. McBride, Discrete fragmentation with mass loss, J. Evol. Equ., 12 (2012), 181-201.doi: 10.1007/s00028-011-0129-8.
[11]	A. L. Smith, "Mathematical Analysis of Discrete Coagulation-Fragmentation Equations," Ph.D. Thesis, University of Strathclyde, Glasgow, 2011.
[12]	R. D. Vigil and R. M. Ziff, On the scaling theory of two-component aggregation, Chemical Eng. Sci., 53 (1998), 1725-1729.doi: 10.1016/S0009-2509(98)00016-5.
[13]	J. A. D. Wattis, Exact solutions for cluster-growth kinetics with evolving size and shape profiles, J. Phys. A : Math. Gen., 39 (2006), 7283-7298.doi: 10.1088/0305-4470/39/23/007.