Large Time Asymptotics for a Continuous Coagulation-Fragmentation Model with Degenerate Size-Dependent Diffusion

Laurent Desvillettes, Klemens Fellner

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Abstract

We study a continuous coagulation-fragmentation model with constant kernels for reacting polymers (see [M. Aizenman and T. Bak, Comm. Math. Phys., 65 (1979), pp. 203-230]). The polymers are set to diffuse within a smooth bounded one-dimensional domain with no-flux boundary conditions. In particular, we consider size-dependent diffusion coefficients, which may degenerate for small and large cluster-sizes. We prove that the entropy-entropy dissipation method applies directly in this inhomogeneous setting. We first show the necessary basic a priori estimates in dimension one, and second we show faster-than-polynomial convergence toward global equilibria for diffusion coefficients which vanish not faster than linearly for large sizes. This extends the previous results of [J.A. Carrillo, L. Desvillettes, and K. Fellner, Comm. Math. Phys., 278 (2008), pp. 433-451], which assumes that the diffusion coefficients are bounded below. © 2009 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)2315-2334
Number of pages20
JournalSIAM Journal on Mathematical Analysis
Volume41
Issue number6
DOIs
StatePublished - Jan 2010
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Received by the editors March 13, 2009; accepted for publication (in revised form) October 20, 2009; published electronically January 8, 2010. The authors acknowledge partial support of the bilateral Austria-France project (Austria: FR 05/2007 France: Amadeus 13785 UA).Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien, Austria (klemens.fellner@univie.ac.at). Current address: DAMTP, Centre of Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK (K.Fellner@damtp.cam.ac.uk). This author has partially been supported by award KUK-I1-007-43 of Peter A. Markowich, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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