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Proceedings of the Royal Society of Edinburgh Section A: Mathematics Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Article contents

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

Part of: Miscellaneous special kernels General theory in ordinary differential equations Integro-ordinary differential equations Nonlinear integral 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)
Corresponding
E-mail address:
prasant.daonly01@gmail.com
ankikgiri.fma@iitr.ac.in
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

Coagulation singular coagulation kernels existence mass-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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