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Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel

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Abstract

We consider mass-conserving self-similar solutions for Smoluchowski’s coagulation equation with kernel K(ξ,η)=(ξη)λ with λ∈(0,1/2). It is known that such self-similar solutions g(x) satisfy that x −1+2λ g(x) is bounded above and below as x→0. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function h(x)=h λ x −1+2λ g(x) in the limit λ→0. It turns out that \(h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)\) as x→0. As x becomes larger h develops peaks of height 1/λ that are separated by large regions where h is small. Finally, h converges to zero exponentially fast as x→∞. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    J. B. McLeod & B. Niethammer

  2. Institute of Applied Mathematics, Endenicher Allee 60, 53115, Bonn, Germany

    J. J. L. Velázquez

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  1. J. B. McLeod
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  2. B. Niethammer
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  3. J. J. L. Velázquez
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Correspondence to B. Niethammer.

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McLeod, J.B., Niethammer, B. & Velázquez, J.J.L. Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel. J Stat Phys 144, 76–100 (2011). https://doi.org/10.1007/s10955-011-0239-2

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  • DOI: https://doi.org/10.1007/s10955-011-0239-2

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