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Evolutionary Gamma convergence for gradient systems

Alex Mielke

Donnerstag, 2019-01-24 17:00Uhr

Many ordinary and partial differntial equations can be written as a gradient flow, which means that there is an energy functional that drives the evolution of the the solutions by flowing down in the energy landscape. The gradient is given in terms of a dissipation structure, which in the simplest case is a Riemannian metric. We discuss classical and nontrivial new examples in reaction-diffusion systems or friction mechanics. We will emphasize that having a gradient structure for a given differential equation means that we add additional physical information.

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 new notion of "EDP convergence" and show by examples that this theory is flexible enough to allow for situations where starting from quadratic dissipation potentials we arrive at physically relevant, effective dissipation potentials that are no longer quadratic, namely exponential laws for transmission at membranes or slip-stick motion on rough surfaces.