On uniform decay of the entropy for reaction-diffusion systems : dedicated to the memory of Klaus Kirchgässner

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Date
2013
Volume
1768
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Journal
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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

In this work we derive entropy decay estimates for a class of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the log-Sobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions

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Keywords
Reaction-diffusion, mass-action law, log-Sobolev inequality, exponential decay of relative entropy
Citation
Mielke, A., Haskovec, J., Markowich, P. A., & Kirchgässner, K. (2013). On uniform decay of the entropy for reaction-diffusion systems : dedicated to the memory of Klaus Kirchgässner (Vol. 1768). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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