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Author: Liero, Matthias × Author: Mielke, Alexander × Subject: Onsager operator ×

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    Optimal transport in competition with reaction: The Hellinger-Kantorovich distance and geodesic curves
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Mielke, Alexander; Savaré, Giuseppe
    We discuss a new notion of distance on the space of finite and nonnegative measures on Omega C Rd, which we call Hellinger-Kantorovich distance. It can be seen as an infconvolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Omega. We give a construction of geodesic curves and discuss examples and their general properties.
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    Gradient structures and geodesic convexity for reaction-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Liero, Matthias; Mielke, Alexander
    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.