Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T02:01:49.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Fragmentation–diffusion model. Existence of solutions and their asymptotic behaviour*

Published online by Cambridge University Press:  14 November 2011

Philippe Laurençot  and
Dariusz Wrzosek
Philippe Laurençot
Affiliation:
Institut Elie Cartan-Nancy, Université de Nancy I, BP 239, F-54506 Vandœuvre les Nancy cedex, France e-mail: [email protected]
Dariusz Wrzosek
Affiliation:
Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2,02-097 Warszawa, Poland e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An infinite system of reaction–diffusion equations that represents a particular case of the discrete coagulation–fragmentation model with diffusion is studied. The reaction part of the model describes the rate of clusters break-up into smaller particles. Diffusion constants are assumed to be different in each equation and concentration-dependent fragmentation coefficients are considered. Existence of solutions is studied under fairly general assumptions on fragmentation coefficients and initial data. Uniqueness in the class of mass-preserving solutions is proved. Convergence of solutions to spatially homogeneous equilibrium state is obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 759 - 774
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H.. Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45 (1983), 225–54.CrossRefGoogle Scholar
2Ball, J. M. and Carr, J.. The discrete coagulation–fragmentation equations: existence, uniqueness, and density conservation. J. Statist. Phys. 61 (1990), 203–34.CrossRefGoogle Scholar
3Baras, P.. Compacite de l'opérateur f ↦ u solution d'une équation non linéaire (du/dt) + Au∍f. C. R. Acad. Sci. Paris, Sér. A 286 (1978), 1113–16.Google Scholar
4Bénilan, Ph. and Wrzosek, D.. On an infinite system of reaction–diffusion equations. Adv. Math. Sci. Appl. 7 (1997), 349–64.Google Scholar
5Carr, J.. Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case. Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 231–44.CrossRefGoogle Scholar
6Carr, J. and Costa, F. P. da. Asymptotic behaviour of solutions to the coagulation–fragmentation equations. II. Weak fragmentation. J. Statist. Phys. 77 (1994), 89–123.CrossRefGoogle Scholar
7Collet, J. F. and Poupaud, F.. Existence of solutions to coagulation–fragmentation systems with diffusion. Transport Theory Statist. Phys. 25 (1996), 503–13.CrossRefGoogle Scholar
8Collet, J. F. and Poupaud, F.. Asymptotic behaviour of solutions to the diffusive coagulation fragmentation system. Phys. D. 114 (1998), 123–46.CrossRefGoogle Scholar
9Deuflhard, P. and Wulkow, M.. Towards an efficient computational treatment of heterogeneous polymer reactions. In Computational Ordinary Differential Equations, ed. Fatuula, S. O., 287–306 (Paris: University of Paris, 1992).Google Scholar
10Dongen, P. G. J. van. Spatial fluctuations in reaction-limited aggregation. J. Statist. Phys. 54 (1989), 221–71.CrossRefGoogle Scholar
11Drake, R.. In Topics in Current Aerosol Research, eds. Hidy, G. M. and Brock, J. R., A general mathematical survey of the coagulation equation, 202–376. (Oxford, Pergamon Press, 1972).Google Scholar
12Hendriks, E. M., Ernst, M. H. and Ziff, R. M.. Coagulation equations with gelation. J. Statist. Phys. 31 (1983), 519–63.CrossRefGoogle Scholar
13Martin, R. H. and Pierre, M.. Nonlinear reaction–diffusion systems. In Nonlinear Equations in Applied Science, eds. Ames, W. F. and Rogers, C., (Boston: Academic Press, 1992).Google Scholar
14Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations (New York: Springer, 1983).CrossRefGoogle Scholar
15Slemrod, M.. Coagulation–diffusion systems: derivation and existence of solutions for the diffuse interface structure equations. Phys. D 46 (1990), 351–66.CrossRefGoogle Scholar
16Smoluchowski, M.. Versuch einer mathematischen Theorie der kolloiden Lösungen. Z. Phys. Chem. 92 (1917), 129–68.Google Scholar
17Wrzosek, D.. Existence of solutions for the discrete coagulation–fragmentation model with diffusion. Topol. Methods Nonlinear Anal, (to appear).Google Scholar
18Ziff, R. M.. Kinetics of polymerization. J. Statist. Phys. 23 (1980), 241–63.CrossRefGoogle Scholar
19Ziff, R. M. and McGrady, E. D.. The kinetics of cluster fragmentation and depolymerisation. J. Phys. A 18 (1985), 3027–37.CrossRefGoogle Scholar