On Uniform Decay of the Entropy for Reaction–Diffusion Systems

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Journal of Dynamics and Differential Equations, ISSN: 1572-9222, Vol: 27, Issue: 3-4, Page: 897-928

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Mielke, Alexander; Haskovec, Jan; Markowich, Peter A.
Springer Nature; Springer Science + Business Media
Mathematics; Exponential decay of relative entropy; Log-Sobolev inequality; Mass-action law; Reaction–diffusion
article description
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.