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Uniform Exponential Decay for Reaction-Diffusion Systems with Complex-Balanced Mass-Action Kinetics

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Patterns of Dynamics (PaDy 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 205))

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Abstract

We consider reaction-diffusion systems on a bounded domain with no-flux boundary conditions. The reaction system is given by mass-action law kinetics and is assumed to satisfy the complex-balance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing. We discuss three methods to obtain energy-dissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the log-Sobolev estimate and suitable handling of the reaction terms as well as the mass-conservation relations. The three methods are (i) a convexification argument based on the author’s joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument developed by Glitzky, Gröger, and Hünlich.

Dedicated to Bernold Fiedler on the occasion of his sixtieth birthday.

The research was partially supported by DFG via SFB 910 (project A5).

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Acknowledgements

With great pleasure, the author thanks Bernold Fiedler for many years of friendship and a multitude of rich and entertaining interactions. The research was partially supported by the DFG Collaborative Research Center 910 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application via Subproject A5 “Pattern Formation in Systems with Multiple Scales” and by the Erwin-Schrödinger-Institut für Mathematische Physik (ESI) in Vienna, where part of this work was prepared. The author is grateful for stimulating and helpful discussion with Laurent Desvillettes, Klemens Fellner, and Annegret Glitzky.

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  2. Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Alexander Mielke

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  1. Alexander Mielke
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Correspondence to Alexander Mielke .

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Editors and Affiliations

  1. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Pavel Gurevich

  2. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Juliette Hell

  3. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Björn Sandstede

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Arnd Scheel

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Mielke, A. (2017). Uniform Exponential Decay for Reaction-Diffusion Systems with Complex-Balanced Mass-Action Kinetics. In: Gurevich, P., Hell, J., Sandstede, B., Scheel, A. (eds) Patterns of Dynamics. PaDy 2016. Springer Proceedings in Mathematics & Statistics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-64173-7_10

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