# A nonlocal coagulation-fragmentation model

Mirosław Lachowicz; Dariusz Wrzosek

Applicationes Mathematicae (2000)

- Volume: 27, Issue: 1, page 45-66
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topLachowicz, Mirosław, and Wrzosek, Dariusz. "A nonlocal coagulation-fragmentation model." Applicationes Mathematicae 27.1 (2000): 45-66. <http://eudml.org/doc/219259>.

@article{Lachowicz2000,

abstract = {A new nonlocal discrete model of cluster coagulation and fragmentation is proposed. In the model the spatial structure of the processes is taken into account: the clusters may coalesce at a distance between their centers and may diffuse in the physical space Ω. The model is expressed in terms of an infinite system of integro-differential bilinear equations. We prove that some results known in the spatially homogeneous case can be extended to the nonlocal model. In contrast to the corresponding local models the analysis can be carried out in the $L_1(Ω)$ setting. Our purpose is to study global (in time) existence, mass conservation and well-posedness of the model.},

author = {Lachowicz, Mirosław, Wrzosek, Dariusz},

journal = {Applicationes Mathematicae},

keywords = {integro-differential equations; diffusion; coagulation; nonlocal interaction; fragmentation; kinetic models; integro-differential bilinear equations},

language = {eng},

number = {1},

pages = {45-66},

title = {A nonlocal coagulation-fragmentation model},

url = {http://eudml.org/doc/219259},

volume = {27},

year = {2000},

}

TY - JOUR

AU - Lachowicz, Mirosław

AU - Wrzosek, Dariusz

TI - A nonlocal coagulation-fragmentation model

JO - Applicationes Mathematicae

PY - 2000

VL - 27

IS - 1

SP - 45

EP - 66

AB - A new nonlocal discrete model of cluster coagulation and fragmentation is proposed. In the model the spatial structure of the processes is taken into account: the clusters may coalesce at a distance between their centers and may diffuse in the physical space Ω. The model is expressed in terms of an infinite system of integro-differential bilinear equations. We prove that some results known in the spatially homogeneous case can be extended to the nonlocal model. In contrast to the corresponding local models the analysis can be carried out in the $L_1(Ω)$ setting. Our purpose is to study global (in time) existence, mass conservation and well-posedness of the model.

LA - eng

KW - integro-differential equations; diffusion; coagulation; nonlocal interaction; fragmentation; kinetic models; integro-differential bilinear equations

UR - http://eudml.org/doc/219259

ER -

## References

top- [A1] H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), 225-254. Zbl0535.35017
- [A2] H. Amann, Coagulation-fragmentation processes, to appear. Zbl0977.35060
- [ABL] L. Arlotti, N. Bellomo and M. Lachowicz, Kinetic equations modelling population dynamics, Transport Theory Statist. Phys., to appear. Zbl0946.92019
- [AL] L. Arlotti and M. Lachowicz, Qualitative analysis of an equation modelling tumor-host dynamics, Math. Comput. Modelling 23 (1996), 11-29. Zbl0859.92011
- [BC] J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation, J. Statist. Phys. 61 (1990), 203-234. Zbl1217.82050
- [BHV] P. Baras, J. C. Hassan et L. Véron, Compacité de l'opérateur définissant la solution d'une équation d'évolution non homogène, C. R. Acad. Sci. Paris Sér. I 284 (1977), 799-802. Zbl0348.47026
- [BL] N. Bellomo and M. Lachowicz, Mathematical biology and kinetic theory: Evolution of the dominance in a population of interacting organisms, in: Nonlinear Kinetic Theory and Hyperbolic Systems, V. Boffi et al. (eds.), World Sci., Singapore, 1992, 11-20.
- [BP] N. Bellomo and J. Polewczak, The generalized Boltzmann equation, existence and exponential trend to equilibrium, C. R. Acad. Sci. Paris Sér. I 319 (1994), 893-898. Zbl0813.76078
- [BW] P. Bénilan and D. Wrzosek, On an infinite system of reaction-diffusion equations, Adv. Math. Sci. Appl. 7 (1997), 349-364.
- [Ca] 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. Zbl0760.34044
- [CD] J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Statist. Phys. 77 (1994), 89-123. Zbl0838.60089
- [CP] J. F. Collet and F. Poupaud, Existence of solution to coagulation-fragmentation system with diffusion, Transport Theory Statist. Phys. 25 (1996), 503-513. Zbl0870.35117
- [DKB] E. B. Dolgosheina, A. Yu. Karulin and A. V. Bobylev, A kinetic model of the agglutination process, Math. Biosciences 109 (1992), 1-10. Zbl0825.92155
- [vD] P. G. J. van Dongen, Spatial fluctuations in reaction-limited aggregation, J. Statist. Phys. 54 (1989), 221-271.
- [Dr] R. Drake, A general mathematical survey of the coagulation equation, in: Topics in Current Aerosol Research, G. M. Hidy and J. R. Brock (eds.), Pergamon Press, Oxford, 1972, 202-376.
- [HEZ] E. M. Hendriks, M. H. Ernst and R. M. Ziff, Coagulation equations with gelation, J. Statist. Phys. 31 (1983), 519-563.
- [JS] E. Jäger and L. Segel, On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl. Math. 52 (1992), 1442-1468. Zbl0759.92011
- [LP] M. Lachowicz and M. Pulvirenti, A stochastic particle system modeling the Euler equation, Arch. Rational Mech. Anal. 109 (1990), 81-93. Zbl0682.76002
- [LW] P. Laurençot and D. Wrzosek, Fragmentation-diffusion model. Existence of solutions and asymptotic behaviour, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 759-774. Zbl0912.35031
- [Ma] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.
- [Mo] D. Morgenstern, Analytical studies related to the Maxwell-Boltzmann equation, Arch. Rational Mech. Anal. 4 (1955), 533-555. Zbl0068.31505
- [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1982.
- [PS] M. Pierre and D. Schmitt, Blowup in reaction-diffusion system with dissipation of mass, SIAM J. Math. Anal. 28 (1997), 259-269. Zbl0877.35061
- [Po] A. Y. Povzner, The Boltzmann equation in the kinetic theory of gases, Amer. Math. Soc. Transl. 47 (1962), 193-216. Zbl0188.21204
- [Sl] M. Slemrod, Coagulation-diffusion systems: derivation and existence of solution for the diffuse interface structure equations, Phys. D 46 (1990), 351-366. Zbl0732.35103
- [Sm] M. Smoluchowski, Versuch einer mathematischen Theorie der kolloiden Lösungen, Z. Phys. Chem. 92 (1917), 129-168.
- [Wr] D. Wrzosek, Existence of solution for the discrete coagulation-fragmentation model with diffusion, Topol. Methods Nonlinear Anal. 9 (1997), 279-296. Zbl0892.35077
- [Z1] A. Ziabicki, Generalized theory of nucleation kinetics. IV. Nucleation as diffusion in the space of cluster dimensions, positions, orientations, and internal structure, J. Chem. Phys. 85 (1986), 3042-3057.
- [Z2] A. Ziabicki, Configurational space for clusters in the theory of nucleation, Arch. Mech. 42 (1990), 703-715.
- [ZJ] A. Ziabicki and L. Jarecki, Cross sections for molecular aggregation with positional and orientational restrictions, J. Chem. Phys. 101 (1993), 2267-2272.
- [Zf] R. M. Ziff, Kinetics of polymerization, J. Statist. Phys. 23 (1980), 241-263.
- [ZM] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerisation, J. Phys. A 18 (1985), 3027-3037.

## Citations in EuDML Documents

top- Emile F. Doungmo Goufo, Morgan Kamga Pene, Jeanine N. Mwambakana, Duplication in a model of rock fracture with fractional derivative without singular kernel
- Stanisław Brzychczy, Existence of solutions and monotone iterative method for infinite systems of parabolic differential-functional equations
- Emile Franc Doungmo Goufo, Stella Mugisha, Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.