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Mathematical Aspects of Coagulation-Fragmentation Equations

  • Conference paper
Mathematics of Energy and Climate Change

Part of the book series: CIM Series in Mathematical Sciences ((CIMSMS,volume 2))

Abstract

We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Sect. 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Sect. 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the function spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sects. 3 and 4 we are concerned with several aspects of the solutions behaviour. We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof.

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Notes

  1. 1.

    The notation is not very good since it suggests there can be at most a countable number of daughter particles (y k ): in fact, there is no a priori reason preventing the distribution to be continuous.

  2. 2.

    As it should, in a binary fragmentation…

  3. 3.

    In this work we shall use the notation \(x \wedge y =\min \{ x,y\}\) and x ∨ y = max{x, y} and analogously for the comparison of more than two numbers.

  4. 4.

    In other versions of this coagulation process of stochastic particles it is assumed the resulting particle is located at the centre of mass \(\frac{x_{i}m_{i}+x_{j}m_{j}} {m_{i}+m_{j}}\) [176], in still others coagulation can happen within a whole interval of distances between the particles and not only at the distance \(\varepsilon\) [104]

  5. 5.

    The precise technical condition used in [94], possibly not necessary, is that ρ satisfies the inequality

    $$\displaystyle{128\frac{K_{c}\rho } {L} + 2\left (\frac{32K_{c}\rho } {L} \right )^{2+ \frac{1+2\alpha } {\gamma +2(1-\alpha )} } <1.}$$
  6. 6.

    The value of the constant is irrelevant for the result since it can always be transformed into another value by a time rescaling. The choice we make simplifies the computations a bit.

  7. 7.

    There is an analogous behaviour of loss of mass in the pure fragmentation system, called “shattering” and interpreted as a loss of mass to infinitesimal clusters. We shall not consider this behaviour here, directing the interested reader to the literature, e.g., [7, 89, 152].

  8. 8.

    The Banach space Y 1 was defined in page 108.

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Acknowledgements

I would like to thank Prof. Conceição Carvalho for the invitation to give an overview talk about the mathematics of coagulation equations at the thematic session on Non-Equilibrium Statistical Mechanics: Kinetics, Chemistry and Coagulation she organized as part of the International Conference on Mathematics of Energy and Climate Change, held in Lisbon, Portugal, in March 2013 and organized by CIM-Centro Internacional de Matemática.

I am also grateful to CIM president, Prof. Alberto Pinto, who enthusiastically supported my suggestion of writing this chapter by updating and translating into English an unpublished document I wrote in Portuguese a few years ago, and was very understanding in accepting the somewhat oversized result.

This work was partially supported by FCT under Strategic Project - LA 9 - 2013–2014.

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  1. Departamento de Ciências e Tecnologia, Universidade Aberta, Rua da Escola Politécnica 141-147, 1269-001, Lisboa, Portugal

    F. P. da Costa

  2. Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    F. P. da Costa

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  1. IHES Le Bois-Marie, Bures-sur-Yvette, France

    Jean-Pierre Bourguignon

  2. Department of Mathematics, ETH Zürich; Seminar für Angewandte Mathematik, Zürich, Switzerland

    Rolf Jeltsch

  3. Department of Mathematics, University of Porto; Faculty of Sciences, Porto, Portugal

    Alberto Adrego Pinto

  4. Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil

    Marcelo Viana

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da Costa, F.P. (2015). Mathematical Aspects of Coagulation-Fragmentation Equations. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Mathematics of Energy and Climate Change. CIM Series in Mathematical Sciences, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-16121-1_5

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