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Author: Mielke, Alexander × Subject: entropy functional ×

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    A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems : dedicated to Herbert Gajewski on the occasion of his 70th birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Mielke, Alexander; Gajewski, Herbert
    In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.
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    Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities : dedicated to Peter Markowich on the occasion of his sixtieth birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Mittnenzweig, Markus
    We discuss how the recently developed energy-dissipation methods for reaction-diffusion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, with the important feature, that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.