On Uniform Decay of the Entropy for Reaction–Diffusion Systems

  • Alexander MielkeEmail author
  • Jan Haskovec
  • Peter A. Markowich


This work provides entropy decay estimates for classes of nonlinear reaction–diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate 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: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.


Reaction–diffusion Mass-action law Log-Sobolev inequality  Exponential decay of relative entropy 

Mathematics Subject Classification

M35K57 35B40 92E20 



The authors are grateful for helpful comments and stimulating discussions with Klemens Fellner, Annegret Glitzky and Konrad Gröger. The research was partially supported by DFG under SFB 910 Subproject A5 and by the European Research Council under ERC-2010-AdG 267802. Partially supported by DFG under SFB 910 Subproject A5 and by the European Research Council under ERC-2010-AdG 267802.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alexander Mielke
    • 1
    Email author
  • Jan Haskovec
    • 2
  • Peter A. Markowich
    • 2
  1. 1.WIAS BerlinBerlinGermany
  2. 2.King Abdullah University of Science and TechnologyThuwalKSA

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