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November  2013, 33(11&12): 5177-5187. doi: 10.3934/dcds.2013.33.5177

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, United Kingdom, United Kingdom, United Kingdom

Received  October 2011 Revised  August 2012 Published  May 2013

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.
Citation: Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177
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.

show all references

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.

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