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Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

Part of: Nonlinear integral equations General theory in ordinary differential equations Miscellaneous special kernels Integro-ordinary differential equations

Published online by Cambridge University Press:  19 February 2019

Prasanta Kumar Barik ,
Ankik Kumar Giri  and
Philippe Laurençot
Prasanta Kumar Barik
Affiliation:
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India (prasant.daonly01@gmail.com); (ankikgiri.fma@iitr.ac.in)
Ankik Kumar Giri
Affiliation:
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India (prasant.daonly01@gmail.com); (ankikgiri.fma@iitr.ac.in)
Philippe Laurençot
Affiliation:
Institut de Mathématiques de Toulouse UMR 5219 Université de Toulouse, CNRS F-31062, Toulouse Cedex 9, France (philippe.laurencot@math.univ-toulouse.fr)
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Abstract

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Keywords

Coagulationsingular coagulation kernelsexistencemass-conserving solutions

MSC classification

Primary: 45J05: Integro-ordinary differential equations 45K05: Integro-partial differential equations
Secondary: 34A34: Nonlinear equations and systems, general 45G10: Other nonlinear integral equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1805 - 1825
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Aldous, D.J.. Deterministic and stochastic model for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), 3–48.CrossRefGoogle Scholar
2Ball, J. and Carr, J.. The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation. J. Statist. Phys. 61 (1990), 203–234.CrossRefGoogle Scholar
3Barik, P.K. and Giri, A.K.. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinet. Relat. Models 11 (2018), 1125–1138.CrossRefGoogle Scholar
4Bourgade, J.P. and Filbet, F.. Convergence of a finite volume scheme for coagulation-fragmentation equations. Math. Comp. 77 (2008), 851–882.CrossRefGoogle Scholar
5Camejo, C.C. and Warnecke, G.. The singular kernel coagulation equation with multifragmentation. Math. Methods Appl. Sci. 38 (2015), 2953–2973.CrossRefGoogle Scholar
6Camejo, C.C., Gröpler, R. and Warnecke, G.. Regular solutions to the coagulation equations with singular kernels. Math. Methods Appl. Sci. 38 (2015), 2171–2184.CrossRefGoogle Scholar
7Châu-Hoàn, L.. Etude de la classe des opérateurs m-accrétifs de L 1(Ω) et accrétifs dans L ∞ (Ω) (Thèse de 3ème cycle, Université de Paris VI, 1977).Google Scholar
8Clark, J.M.C. and Katsouros, V.. Stably coalescent stochastic froths. Adv. Appl. Probab. 31 (1999), 199–219.CrossRefGoogle Scholar
9Dubovskii, P.B. and Stewart, I.W.. Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 (1996), 571–591.3.0.CO;2-Q>CrossRefGoogle Scholar
10Escobedo, M. and Mischler, S.. Dust and self-similarity for the Smoluchowski coagulation equation. Ann. Inst. H. Poincaré Anal. non Linéaire 23 (2006), 331–362.CrossRefGoogle Scholar
11Escobedo, M., Mischler, S. and Perthame, B.. Gelation in coagulation and fragmentation models. Comm. Math. Phys. 231 (2002), 157–188.CrossRefGoogle Scholar
12Escobedo, M., Laurençot, Ph., Mischler, S. and Perthame, B.. Gelation and mass conservation in coagulation-fragmentation models. J. Diff. Eqs. 195 (2003), 143–174.CrossRefGoogle Scholar
13Escobedo, M., Mischler, S. and Rodriguez Ricard, M.. On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. non Linéaire 22 (2005), 99–125.CrossRefGoogle Scholar
14Filbet, F. and Laurençot, Ph.. Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation. Arch. Math. 83 (2004a), 558–567.CrossRefGoogle Scholar
15Filbet, F. and Laurençot, Ph.. Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25 (2004b), 2004–2028.CrossRefGoogle Scholar
16Giri, A.K., Laurençot, Ph. and Warnecke, G.. Weak solutions to the continuous coagulation with multiple fragmentation. Nonlinear Anal. 75 (2012), 2199–2208.CrossRefGoogle Scholar
17Kapur, P.C.. Kinetics of granulation by non-random coalescence mechanism. Chem. Eng. Sci. 27 (1972), 1863–1869.CrossRefGoogle Scholar
18Laurençot, Ph.. Weak compactness techniques and coagulation equations. In Evolutionary equations with applications in natural sciences (ed. Banasiak, J. and Mokhtar-Kharroubi, M.). Lecture Notes Math., vol. 2126, pp. 199–253 (Springer, 2015).CrossRefGoogle Scholar
19Laurençot, Ph. and Mischler, S.. From the discrete to the continuous coagulation-fragmentation equations. Proc. Roy. Soc. Edinburgh 132A (2002), 1219–1248.CrossRefGoogle Scholar
20Laurençot, Ph. and Mischler, S.. On coalescence equations and related models. In Modeling and computational methods for kinetic equations. Model. Simul. Sci. Eng. Technol., pp. 321–356 (Boston: Birkhaüser, 2004).CrossRefGoogle Scholar
21Leyvraz, F. and Tschudi, H.R.. Singularities in the kinetics of coagulation processes. J. Phys. A 14 (1981), 3389–3405.CrossRefGoogle Scholar
22Norris, J.R.. Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1999), 78–109.Google Scholar
23Smoluchowski, M.. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift f. physik. Chemie 92 (1917), 129–168.Google Scholar
24Stewart, I.W.. A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 (1989), 627–648.CrossRefGoogle Scholar
25Stewart, I.W.. A uniqueness theorem for the coagulation-fragmentation equation. Math. Proc. Camb. Phil. Soc. 107 (1990), 573–578.CrossRefGoogle Scholar
26Vrabie, I.I.. Compactness methods for nonlinear evolutions, 2nd edn Pitman Monogr. Surveys Pure Appl. Math., vol. 75 (Longman, 1995).Google Scholar