WIAS Preprint No. 1701, (2012)

Gradient structures and geodesic convexity for reaction-diffusion systems



Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35K57 53C21 53C23 60J60 82B35

Keywords

  • Geodesic convexity, gradient structures, gradient flow, Onsager operator, reaction-diffusion system, Wasserstein metric, relative entropy

Abstract

We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

Appeared in

  • Phil. Trans. R. Soc. A, 371 (2013) pp. 20120346/1--20120346/28.

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