"Gradient structures and EDP convergence for reaction and diffusion"Mielke, AlexanderWe on recent developments concerning gradient structures and their uses for diffusion processes and chemical reaction systems. We highlight that different gradient structures can be used for the same differential equation, which may reflect different physical applications of the same equation. For chemical reactions systems we discuss the unique properties of the cosh-type gradient structure involving a non-quadratic dissipation potential. Considering a family of gradient systems depending on a small parameter, it is natural to ask for the limiting (also called effective) gradient system if the parameter tends to 0. This can be achieved on the basis of De Giorgi's Energy-Dissipation Principle (EDP). We discuss the notion of "EDP convergence" and show by examples that this theory is flexible enough to allow for situations where starting from a linear kinetic relation (or quadratic dissipation potentials) we arrive at physically relevant, nonlinear effective kinetic relations. This is joint work many co-authors including Thomas Frenzel, Matthias Liero, Mark Peletier, Michiel Renger, and Artur Stephan. |
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