WIAS Preprint No. 1768, (2013)

On uniform decay of the entropy for reaction-diffusion systems



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Haskovec, Jan
  • Markowich, Peter A.

2010 Mathematics Subject Classification

  • 35K57 35B40 92E20

Keywords

  • Reaction-diffusion, mass-action law, log-Sobolev inequality, exponential decay of relative entropy

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.

Appeared in

  • J. Dynam. Differential Equations, 27 (2015) pp. 897--928.

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