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    On Uniform Decay of the Entropy for Reaction–Diffusion Systems

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    Type
    Article
    Authors
    Mielke, Alexander
    Haskovec, Jan cc
    Markowich, Peter A. cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Date
    2014-09-11
    Online Publication Date
    2014-09-11
    Print Publication Date
    2015-12
    Permanent link to this record
    http://hdl.handle.net/10754/621407
    
    Metadata
    Show full item record
    Abstract
    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.
    Citation
    Mielke A, Haskovec J, Markowich PA (2014) On Uniform Decay of the Entropy for Reaction–Diffusion Systems. Journal of Dynamics and Differential Equations 27: 897–928. Available: http://dx.doi.org/10.1007/s10884-014-9394-x.
    Sponsors
    The authors are grateful for helpful comments and stimulating discussions with Klemens Fellner, Annegret Glitzky and Konrad Groger. The research was partially supported by DFG under SFB910 Subproject A5 and by the European Research Council under ERC-2010-AdG 267802. Partially supported by DFG under SFB910 Subproject A5 and by the European Research Council under ERC-2010-AdG 267802.
    Publisher
    Springer Nature
    Journal
    Journal of Dynamics and Differential Equations
    DOI
    10.1007/s10884-014-9394-x
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10884-014-9394-x
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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